sparse grid method in the libor market model. option valuation and the

SPARSE GRID METHOD IN THE LIBOR MARKET
MODEL. OPTION VALUATION AND THE ’GREEKS’
MSc Thesis (Afstudeerscriptie)
written by
Vladislav Sergeev
(born January 25th, 1980 in Omsk city, Russian Federation)
under the supervision of Dr. Drona Kandhai, and submitted to the Board of
Examiners in partial fulfillment of the requirements for the degree of
MSc in Grid Computing
at the Universiteit van Amsterdam.
Date of the public defense:
November 28th, 2006
Members of the Thesis Committee:
Dr. Peter van Emde Boas
Prof. Dr. Chris Jesshope
Dr. Drona Kandhai
Mr. Breanndán Ó Nualláin, MSc.

Abstract
In this project we study numerical approximations of the Libor Market Model by using
Sparse Grid techniques. This approach is a promising method to solve high dimensional
partial differential equations. The LIBOR Market Model is a model for
interest rate movements and typically suffers from the curse of dimensionality when
finite-difference techniques are used. As an extension of the earlier work by J. Blackham
[Blackham, 2004], we target a more detailed study of the accuracy of the sparse
’Greeks’ (∆ and V) for 2D Chooser Option, 3D Bermudan Swaption and a 2D Digital
Option. Given the results of experiments, we conclude that, despite satisfactory convergence
of sparse grid values, further improvements are required to enhance the quality of
the ’Greeks’ in the discussed setting. Applicable recommendations of possible further
developments are included accordingly.

Acknowledgements
This project would not have been possible without the support of many people. First of
all, I would like to express repect and gratitude to my supervisor, dr. Drona Kandhai,
whose professionalism, bonded with remarkable personality, has guided and tutored me
throughout the whole year. Thanks for making sense out of confusion and parenting
my interest in Derivative Finance. Words of acknowledgment are due to Dick van
Albada, Joke Blom, Walter Hoffmann, Dmitri Vasunin and Vladimir Korkhov, who at
different times provided their insightful comments. A large ’thank you’ goes to Kees
Oosterlee and Coen Leentvaar from TU Delft, whose expertise and feedback proved to
be invaluable. I am blessed to be a friend to many who had their hands on the pulse
of this project. Nelly’s patience, love and care secured my sanity. Finally and most
importantly, I am deeply indebted to my parents.
Äîðîãèå ðîäèòåëè, ñïàñèáî çà âñå ÷òî âû äëÿ ìåíÿ ñäåëàëè. ß âàñ î÷åíü
ëþáëþ.

Contents
1 Introduction 1
2 Asset Pricing Overview 3
2.1 Arbitrage and Risk-Neutral World . . . . . . . . . . . . . . . . . . . 3
2.2 From Discrete Time to Continuous Time . . . . . . . . . . . . . . . . 6
2.3 Pricing with Martingales and Measures . . . . . . . . . . . . . . . . 13
3 LIBOR Market Model 17
3.1 Forward LIBOR Rates . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 LIBOR Market Model . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Interest Rate Derivatives. Caplets, Caps and Swaptions. . . . . . . . . 22
3.4 Calibration of LIBOR Market Model . . . . . . . . . . . . . . . . . . 24
3.4.1 Calibration to caplets . . . . . . . . . . . . . . . . . . . . . . 25
3.4.2 Correlation structure of forward rates. . . . . . . . . . . . . . 25
3.5 The LIBOR Market Model PDE . . . . . . . . . . . . . . . . . . . . 28
4 Computational Methods 30
4.1 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . 33
5 Sparse Grids 37
5.1 Full grid approximations. . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Sparse grid approximations. . . . . . . . . . . . . . . . . . . . . . . 41
5.3 A Combination technique . . . . . . . . . . . . . . . . . . . . . . . . 42
6 Numerical Results. Valuation and the Greeks 43
6.1 Two Dimensional Case. 2D Chooser Option . . . . . . . . . . . . . . 44
6.1.1 Convergence of MC, Full and Sparse Grid solutions. . . . . . 45
6.1.2 Discussion: Convergence of Pricing. . . . . . . . . . . . . . . 49
6.1.3 Greeks: Delta . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.1.4 Greeks: Vega . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.2 Three Dimensional Case. 3D Bermudan Swaption . . . . . . . . . . . 54
6.2.1 Convergence Study . . . . . . . . . . . . . . . . . . . . . . . 56
6.2.2 Greeks. Delta and Vega. . . . . . . . . . . . . . . . . . . . . 58
6.3 Discontinuous Payoff. 2D Digital option. . . . . . . . . . . . . . . . 61
ii

6.3.1 Valuation Results. . . . . . . . . . . . . . . . . . . . . . . . 61
6.3.2 Greeks. Delta and Vega. . . . . . . . . . . . . . . . . . . . . 64
6.4 Discussion and Recommendations. . . . . . . . . . . . . . . . . . . . 66
7 Conclusion and Future Work 67
A Implementation Details 69
iii

List of Figures
2.1 From Single Period to Multi-Period . . . . . . . . . . . . . . . . . . 7
3.1 Tenor Structure of Interest Rates . . . . . . . . . . . . . . . . . . . . 18
4.1 Monte Carlo. Delta of the option with a digital feature. . . . . . . . . 32
4.2 2D Solution mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Band coefficient matrix. . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.1 Standard Basis functions S 3 . . . . . . . . . . . . . . . . . . . . . . 38
5.2 Hierarchical representation. . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 Mapping of hierarchical representation S ∗ 3 onto a full grid. . . . . . . 40
5.4 Mapping of hierarchical representation Ŝ∗ 3 onto a sparse grid. . . . . . 41
6.1 2D Chooser Option. MC vs. FD Convergence. . . . . . . . . . . . . . 45
6.2 2D Chooser. Convergence of Full and Sparse Grid. . . . . . . . . . . 46
6.3 2D Chooser. Solution Surface for level 6. . . . . . . . . . . . . . . . 47
6.4 2D Chooser. Sparse vs. full solution. Slices. . . . . . . . . . . . . . . 48
6.5 2D Chooser. Absolute error surface on level 8. . . . . . . . . . . . . . 49
6.6 2D Chooser. Sparse delta surface on level 7. . . . . . . . . . . . . . . 50
6.7 2D Chooser. Full vs. sparse delta profile on level 7 . . . . . . . . . . 51
6.8 2D Chooser. Sparse vega surface on level 8. . . . . . . . . . . . . . . 52
6.9 2D Chooser. Full vs. sparse vega slices on level 8. . . . . . . . . . . . 53
6.10 3D Bermudan Swaption. Full and Sparse solution surfaces. . . . . . . 55
6.11 3D Bermudan Swaption. Benchmarking against [Blackham, 2004] and
Monte Carlo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.12 3D Bermudan Swaption. Solution slices corresponding to fixings in L 1
and L 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.13 3D Bermudan Swaption. Delta and Vega Surfaces for L 1 =5.62%. Full
grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.14 3D Bermudan Swaption. L 3 -Delta. Slices corresponding to a fixing in
L 1 = 5% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.15 3D Bermudan Swaption. L 3 -Vega. Slices corresponding to a fixing in
L 1 = 5.62% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.16 2D Digital Option. Terminal payoff discretized on a L6 full grid. . . . 61
6.17 2D Digital Option. Convergence of Monte Carlo and full grid FD. . . 61
iv

6.18 2D Digital Option. Full and sparse grid solution surfaces. Level 7. . . 62
6.19 2D Digital Option. Full grid Delta on L7. . . . . . . . . . . . . . . . 64
6.20 2D Digital Option. Sparse Delta L7 and L9 slices. . . . . . . . . . . . 64
6.21 2D Digital Option. Full grid vega on L7. . . . . . . . . . . . . . . . . 65
6.22 2D Digital Option. Sparse Vega L7 and L9 slices. . . . . . . . . . . . 65
v

List of Tables
6.1 Setup of LMM. Input data for LMM . . . . . . . . . . . . . . . . . . 44
6.2 Setup of LMM. LIBOR rates and caplet volatilies . . . . . . . . . . . 44
6.3 2D Chooser. Convergence values for full and sparse grid solutions. . . 45
6.4 2D Chooser. At-the-money and in-the-money option values. . . . . . 46
6.5 2D Chooser. Deep out-of-the-money and at-the-money with low strike. 46
6.6 3D Bermudan Swaption. Spot Rates as Table (6.2) . . . . . . . . . . 56
6.7 3D Bermudan Swaption. Full and sparse on-grid solution values for
L 1 =3.75%. 500000 MC simulation trials. . . . . . . . . . . . . . . . 56
6.8 3D Bermudan Swaption. Full and sparse on-grid solution values for
L 1 =5.62%. 500000 MC simulation trials. . . . . . . . . . . . . . . . 57
6.9 2D Digital. Full and sparse grid convergence of on-grid points. 500000
MC simulation trials. . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.10 2D Digital Continued. Full and sparse grid convergence of on-grid
points. 500000 MC simulation trials. . . . . . . . . . . . . . . . . . . 63
vi

Chapter 1
Introduction
The modern field of Derivative Finance is a very broad and highly scientific subject
that spans several areas of knowledge. First of all, it naturally requires profound understanding
of the dynamics of financial markets and financial products traded in the
real world. Secondly, over the past several decades it has grown into an accomplished
branch of applied mathematics that heavily employs probability theory and the theory
of continuous-time stochastic processes. Thirdly, the problems of derivative pricing,
growing in complexity along with the demands of the market players, increasingly rely
on efficient numerical approximation and simulation methods. Last but not least, high
performance and parallel computing environments have become an indispensable tool
for the process of valuation and risk-analysis of complex multi-asset financial instruments.
The main focus of this thesis is on the family of interest-rate derivatives, a very
important class of products used to hedge exposure to interest-rate risk. Since the
demand for such insurance is generated by large financial institutions (e.g. pension
funds, insurance companies), banks and market makers selling interest-rate derivatives
have to deal with the risks of serious losses. The necessity of controlling these risks,
and thus, finding the correct value of a derivative instrument, formulates the motivation
for the theory behind Derivative Finance. The second chapter of the thesis is meant to
provide a brief introduction to the main concepts of option pricing in general, as well
as mathematical theory needed while operating in the economy of interest rates.
Both in academia and industry, a great deal of research is devoted to the development
of realistic models for interest rate dynamics 1 and frameworks for pricing and
hedging options. Among those discussed in the literature, one can distinguish two fundamental
approaches: spot rate models and market rate models. The former approach
takes a fictional instantaneous spot rate (alive for an infinitesimal time-period dt) as a
basis for modelling of the term structure 2 . The rest of the rates are obtained by integrating
with respect to a spot interest rate. From a mathematical standpoint, formulation
and pricing products in terms of this abstraction is quite convenient. However, calibra-
1 of course, these models are being designed to simulate and not to forecast the future developments in
the interest rate behaviour.
2 the set of consequent interest rates with increasing maturity
1

tion of spot-rate model to the observed market data is far from straight-forward, since
such rates do not exist in practice [Pelsser, 2000].
In the third chapter we turn our attention to market rate models, namely the LIBOR
Market Model (LMM), where LIBOR stands for London Interbank Offered Rate. Published
as [Brace et al., 1997], it has been widely accepted and became one of the most
popular models used in practice. In contrast to spot-rate models, it takes the process
for a real market rate as the basic modelling unit. This fact simplifies fitting the model
to market parameters to a great extent. On the other hand, the number of interest rates
underlying the derivative has a direct impact on the dimensionality of the problem.
In fact, without sophisticated extensions, e.g. [Pietersz et al., 2005], the number of
dimensions is equal to the number of underlying forward rates.
Since the analytical valuation formulae are rarely available in a multi-dimensional
setting, one needs to approximate the solution with a computational method of choice.
The fourth chapter contains the discussion of Monte Carlo simulation (MC) and Finite
Difference Method (FDM) from the perspective of LMM valuation. As of this moment,
valuation of interest-rate derivatives lies in the domain of Monte Carlo simulation,
since it is weakly dependent on the dimensionality of the problem and allows to price
claims contigent on a large (up to 30) number of rates. Yet, as discussed in Chapter 4,
FDM has an number of important advantages over MC and the main reason for it to be
underfavoured is its ’curse of dimensionality’. The exponential growth in the number
of mesh points with the increase in the dimensionality makes it unusable in more than
three dimensions. The challenge of breaking the curse motivated researchers to seek
for alternative discretisation methods. One such method, known as the Sparse Grid
method was introduced in [Zenger, 1991] and has been able to show promising results
in application to high-dimensional problems in many areas of science. It is discussed
in the fifth chapter of this text.
The idea of application of sparse grid technique to the field of option pricing is not
new [Reisinger, 2004]; nor is it such with respect to the derivatives in the LIBOR Market
Model [Blackham, 2004]. The author of the latter reports price convergence of full
and sparse grid solutions for products contingent on 2,3,4 and possibly 5 forward rates.
The current master thesis was conceived as an extension of the work of [Blackham,
2004] in lower (2D and 3D) dimensions, with the threefold objective for the practical
part:
1. Independent verification of the findings regarding the option valuation for 2D
and 3D cases of [Blackham, 2004] and exploration of a larger parameter space.
2. Investigation of the behaviour for a discontinuous payoff in case of a sparse grid
method.
3. A more rigorous study of the quality of the sparse ’Greeks’. Essential to risk
management, ’Greeks’ are the quantities that show sensitivity of the computed
price to the changes in the input parameters.
The results and due observations are presented in the final chapter, followed by the
summary of recommendations for future work.
2

Chapter 2
Asset Pricing Overview
The most important goal of Derivative Finance is finding the fair price of a contingent
claim. In this chapter, we will provide a brief introduction to discrete and continuous
models used in this area of applied mathematics, define the notion of no-arbitrage markets,
risk-neutral world and replicating portfolios. The last section is going to prepare
the ground for further discussion of LIBOR market model by putting option valuation
into a more general framework of martingales and measures.
2.1 Arbitrage and Risk-Neutral World
Simple Market In order to introduce several important concepts, we will start with
a very simple setting. The market should consist of only two assets: zero-coupon 1
cash bond B, and a stock S. The evolution of these two assets can be observed at
two discrete time points: the current moment T 0 and the time of option maturity T 1 .
As continuously-compounded interest rates are assumed to be constant, there is no
uncertainty about the value of B at T 1 . The price of the stock, on the other hand, is
subject to uncertainty and evolves to one of the two possible states: ”up” - the stock
price increases and ”down” - the stock price goes down. The market, described above,
is known as a one-step binary model.
Inspired by [Etherage, 2002], this view of the economy can be presented in the
matrix-vector form. Prices of N assets at T 0 are denoted by vector I; values of N
assets being in n possible states at T 1 are incorporated into matrix D. Specifically for
our case, I and D take the following form:
I =
(
S0
B 0
)
(
up down
)
S0 u S
D =
0 d
B 0 e rT B 0 e rT
where r - is the risk-free rate, u and d coefficients correspond to the upward and downward
price movements of S 0 with u > e rT > d.
1 providing no interim coupon payments.
3

Before starting to operate in this economy, we need to finish its setup by imposing
several assumptions:
• short-selling 2 of securities is allowed;
• there are negligible or no transaction costs;
• arbitrage opportunities are absent;
While the first two assumptions are easy to perceive, the assumption of no-arbitrage
requires our special attention and is going to have a significant impact on the way we
see derivative valuation. Let’s define arbitrage.
Arbitrage Let’s set 3 θ = (θ 0 , . . . , θ N ) t to be a portfolio of N assets, where θ i is
the quantity of the i th asset, held in the portfolio. The products θ t·I and D t θ, are,
therefore, the price of establishing such a portfolio at time T 0 and its value at time T 1 ,
respectively. Arbitrage can be loosely defined as a portfolio θ, whose cost θ t·I≤ 0
at T 0 , and whose vector D t θ contains positive values at T 1 . In other words, one has a
chance of collecting positive payoffs without initial investment, and no chance of losing
any money. The assumption of no-arbitrage is characteristic of efficient, liquid markets,
where, in theory, the time of arbitrage existence strives to zero. This happens because
there are many market participants in search of the underpriced and overpriced assets.
Arbitrage is exploited the instant it has been found, driving the prices of securities to
their fair values, eliminating the possibilities of riskless profits and bringing the market
to the state of equilibrium.
Pricing by Arbitrage Consider a binary economy where such arbitrage opportunities
do not last. What would be the correct price for an instrument, whose payoff is
contingent on the assets comprising the economy? Obviously, this value is strongly
correlated with its final payoff. Let’s take, for example, plain-vanilla call option c and
write c(S T1 ) for a payoff of c as a function of stock price at time T 1 . If the option
writer could establish a portfolio θ = (ζ, γ) t , such that D t θ = (c(S 0 u), c(S 0 d)) t ,
she would completely replicate the payoff of c at T 1 , and, as a result, get its value
θ t·I= c 0 . A claim, whose payoff can be replicated by such a portfolio, is said to be
attainable. Clearly, having a no-arbitrage economy in mind, our market model should
not allow for existence of several replicating strategies that generate the same payoff
pattern and have different initial costs. The technique of determining the value of contingent
claims by finding the value of their replicating portfolio is known as pricing
by arbitrage. Writing χ = (c(S 0 u), c(S 0 d)) t for the c’s payoff vector, the cost of a
strategy θ that replicates χ and consequently the option’s value at T 0 can be calculated
as following:
θ = (D t ) −1 χ therefore c 0 = θ t · I
2 To sell an asset short means borrowing it from other investors having guaranteed to return it on a specific
date in the future plus some interest payment
3 a superscript t denotes a transpose operation of a matrix or a vector
4

Risk-Neutral Pricing Another important implication of no-arbitrage assumption is
reflected in the Arbitrage Theorem:
Theorem 2.1.1. In a market with N assets and n states at time T 1 , there is no arbitrage
if and only if there is a vector ψ, strictly positive in all coordinates, such that I = Dψ
For the proof, an interested reader is referred to [Etherage, 2002].
The vector ψ is known as a state price vector. Let us define |ψ| 1 = ∑ n
i=1 ψ i. If each
side of the relation I = Dψ is multiplied with a scalar:
1
I = Dψ 1
(2.1)
|ψ| 1
|ψ| 1
then the product ψ 1
|ψ| 1
can be interpreted as a probability distribution vector. Given
such a vector, the right-hand side of (2.1) can be interpreted as the expectation operator
with respect to the probability distribution ψ 1
|ψ| 1
. As a result, the following relation
holds for each asset comprising our economy:
I i T 0
= |ψ| 1 E(I i T ), i = 1 . . . N.
Conveniently enough, it appears 4 that the coefficient |ψ| 1 is nothing but a risk-free
discount factor e −rT .
For example, we can look at the expected rate of return of the stock setting the
probabilities of up and down movements to be the p and 1 − p, and given the previous
result:
E(S T1 ) = pS T0 u + (1 − p)S T0 d = S T0 e rT (2.2)
which implies that:
p = erT − d
u − d
As we can see, the expectation does not rely on real-world probabilities of up or
down movements of the stock, just on the magnitude of change in the price, u and d.
The same price will be observed whether P(u) = 0.5 or, for instance, P(u) = 0.1.
Such perspective on price formation correlates with the key notion of Efficient Markets
Theory - Efficient Markets Hypothesis (EMH). The so-called ’weak form’ of EMH
states that the current price of the stock already reflects the investors’ expectations of
possible price movements [Bodie et al., 2004]. Even though probabilistic distribution
of stock price movements is not present in this valuation framework, we could treat p
and 1 − p as pseudo-probabilities of future payoff value. With this result, we can conclude
that given a probability distribution vector, time zero price of any asset defining
the market can be found as the expectation with respect to this probability measure,
discounted at a risk-free rate. The world where the stock-prices grow on average at a
risk-free rate we shall call “risk-neutral” and the normalized state price vector will be
addressed as risk-neutral probability measure Q.
4 This can be justified by pricing a portfolio that replicates a zero-coupon discount bond, paying one unit
of currency at time T 1 . E.g.:
B 0 = B 0 e rT (ψ 0 + · · · + ψ n ) =⇒ |ψ| 1 = (ψ 0 + · · · + ψ n ) = e −rT .
5

Definition 2.1.2. An economy is said to be complete if every derivative security is
attainable.
By a very simple argument, a market will be complete if N ≥ n. If a number of
possible states exceeds a number of marketed assets then solving a system of equations
will not give unique values for portfolio weights and arbitrage opportunities will
arise. We shall end this section with the first, light-weight edition of the Fundamental
Theorem of Asset Pricing.
Theorem 2.1.3. An economy is complete and free of arbitrage if and only if there exists
a unique equivalent probability measure.
We will rely on the reader’s intuition until this theorem is restated and explicated
later in the chapter to fit into the martingale valuation framework. Given a risk-neutral
probability measure, the T 0 price of option c 0 in a no-arbitrage complete market equals:
c 0 = e −r E Q [c 1 ] (2.3)
2.2 From Discrete Time to Continuous Time
A single period model is, of course, very far from practical use. Nevertheless, it serves
well as a building block for a more realistic discrete multi-period model of recombinant
binomial trees, as suggested in [Cox et al., 1979].
Recombinant Trees
Consider a better refined timeline with N timesteps
t = t 0 < t 1 < . . . < t i < . . . < t N = T
This allows us to monitor the stock price time series at a finite number of discrete
time points prior to maturity of an option. The stock price at time t i+1 is now S i u
or S i d, where the same growth coefficients u and d apply to every time period in the
partition above. Therefore, the same price can be obtained by simple regrouping of the
coefficients, e.g. S 4 = S 0 uudu = S 0 uduu. As one can see from Figure (2.1), the
result of all recombinations is the binomial recombinant tree of possible stock price
movements.
Since the single period model plays a role of a construction primitive in a binomial
recombinant world, we already know how to find a fair value of an option. It is nothing
but a recursive application of no-arbitrage techniques of the previous section by means
of stepping backwards in the tree. Again, in the option pricing context the probabilistic
views of the market players are disregarded and expectations of the future option
payoffs are found under the unique equivalent risk-neutral measure.
Replication in Discrete Time This chapter has started from pricing a derivative instrument
by construction of a replicating portfolio in a single period model. In a binomial
tree, where the evolution of a stock price and a bond is a multi-step process, the
replication procedure has to be performed at each time step. This means the weights of
6

S 0 u
d
u S 0
S u 2
0
Su
3
0
S du 2
0
4
S0u
S
du 3
0
S0
S0
ud S 0
2
S0ud
S u d
2
0
2
T 0
T
1
S 0
d S 0
2
S0d
S ud 3
0
T
0
T
1
T
2
3
S0d
T3
T4
S d 4
0
Figure 2.1: From Single Period to Multi-Period
assets in a replicating portfolio have to be adjusted according to what is happening with
the prices of the assets. Intuitively, a replication strategy would be meaningful for the
writer of the claim only in case this strategy does not require any maintaining costs, i.e.
is self-financing. In other words, after spending a certain amount of cash to establish
positions in S 0 and B 0 , the replication procedure should involve only changing their
weights and exclude any additional cash inputs. Denote a vector of portfolio weights
as θ = (ζ, γ) t . A one-step example of the self-financing property for a replicating
portfolio V would be:
V i = ζ i S i + γ i B i = ζ i+1 S i + γ i+1 B i (2.4)
where ζ i+1 and γ i+1 are the quantities of S and B respectively, held over the period
t ∈ [ i∆t, (i + 1)∆t ) . A change in portfolio value over one-step period gives:
For an N-step recombinant tree:
V i+1 − V i = ζ i+1 (S i+1 − S i ) + γ i+1 (B i+1 − B i ) (2.5)
N−1
∑
N−1
∑
V N = V 0 + ζ i+1 (S i+1 − S i ) + γ i+1 (B i+1 − B i ) (2.6)
i=0
If we assume interest rates to be zero, then it can be easily shown that for V N B −1 =
Ṽ N :
N−1
∑
Ṽ N = Ṽ0 + ζ i+1 ( ˜S i+1 − ˜S i ) or ∆Ṽi = ζ i+1 ∆ ˜S i (2.7)
i=0
7
i=0

where ˜S i is discounted stock price 5 . Detailed account for pricing and hedging of options
in a recombinant tree can be found in such references as [Hull, 1999] and [Etherage,
2002].
Log-normal distribution of stock prices Up until this moment, we haven’t been
concerned with the degree of realism for the stock price development model. This
simplification has been intentional for the sake of showing the no-arbitrage principle in
the most ingenuous setting. At this point we shall try to remedy this and introduce the
more or less realistic process for the stock price evolution. There are three important
facts about the process of stock prices that can help us address this issue:
S
• Stock price returns, T
S0
= ∑ N S i
i=1 S i−1
are assumed to be markovian random
variables (independent from history, or, equivalently, EMH)
• The expectation of a future stock price for a single step can be computed with
respect to a certain probability measure (Q or P). Note, that the same measure
applies to the whole sequence of price changes.
• In the real world, the trades on stock markets occur almost continuously.
Unlike the first two, continuity of trades has not been accounted for in our binomial
view of the market so far. This can be changed if we allow N → ∞, or likewise,
let dt → 0, where dt is T N
. These results taken together empower the Central Limit
Theorem to show us that stock price returns converge in distribution to the Gaussian
d
−→ N (·, ·) with some mean and variance. When returns are normally
process, or S T
S0
distributed random variables, the stock prices can theoretically take negative values,
which is clearly unacceptable for the asset price. How can we overcome this obstacle
imposed by the nature of the Gaussian distribution?
It must be obvious, that expected future values of the underlying are determined by
the choice of the coefficients u and d, where u > e rdt > d. Let’s set u and d such that:
u = e µdt+σ√ dt
d = e µdt−σ√ dt
The volatility parameter σ quantifies the amount of risk characterictic of a specific
stock and is calculated to be the stock’s standard deviation from the expected
return. The drift term µ, same for both up and down movement, quantifies the constant
tendency of growth in stock, or expected return. Having u and d as exponential
functions of dt, we conveniently work with natural logarithms of u and d and denote
R i = log(S i ) − log(S i−1 ) to be stock’s single period log-return. The choice of parameters
as exponential functions of dt should affect the distribution of the overall process.
Taking the up and down movements to be equally possible, i.e. P(u) = P(d) = 0.5,
we arrive at the very important result for option pricing theory:
Under a probability measure P stock returns have a lognormal distribution:
( ST
)
log ∼ N
S 0
(
µT, σ 2 T
)
(2.8)
5 The discounted process ∆Ṽi can, in fact, be characterized as a discrete martingale process. We shall
skip discussion of discrete martingales to introduce them in continuous time later in the text.
8

or
log(S T )
d
−→ log(S 0 ) + µT + σ √ T Z. (2.9)
where Z ∼ N (0, 1) and our expected return on the stock µ being a mean of a single
period log-return 6 .
Among numerous properties of lognormally distributed random variables, there is
one we have been looking for: they cannot take negative values. This type of distribution
is believed to come close in attempt to mimic the real-world processes and is
widely used in modelling of different underlying assets.
Brownian Motion Due to the fact that log-returns, as an algebraic transformation
of stock returns, are independent, identically distributed markovian random variables,
the price change from time T 0 to time T N can be thought of as a sequence of N such
variables or a stochastic process with notation {R t } N 1 . If we let dt → 0, {R t } t≥1 is a
continuous-time stochastic process. As in [Etherage, 2002], consider a specific type of
a continuous stochastic process:
Definition 2.2.1. A real-valued stochastic process {W t } t≥1 is a standard P-Brownian
motion (or a P-Wiener process) if given a probability measure P,
1. for each s ≥ 0 and t > 0 the random variable W t+s − W s ∼ N (0, t − s)
2. for each n ≥ 1 and any times 0 ≤ t 0 ≤ t 1 · · · ≤ t n , the random variables
{W tr − W tr−1 } are independent.
3. W 0 = 0. This property may be relaxed, as the process x + W t is a standard
Brownian motion with W 0 = x, where x is a real constant.
4. W t is continuous in t ≥ 0
Given this definition, we can construct various stochastic processes Y t , that are
functions of standard Brownian motion W and share its characteristic properties. For
example:
• Y t = Y 0 + σW t is a Brownian motion process with variance σ 2 t.
• Y t = Y 0 + νt + σW t is a Brownian motion with drift. In other words, Y t ∼
N (Y 0 + νt, σ 2 t).
• Y t = Y 0 exp(νt + σW t ) is a geometric Brownian motion. A useful result that
will be needed later is :
E[e νt+σW ] = e (ν+ 1 2 σ2 )t
(2.10)
One can tell that our process for log-returns (2.9) is a stochastic process that is identical
to Brownian motion with a drift 7 . On the other hand, the process for stock returns is a
geometric Brownian motion, taking strictly positive values with constant variance.
)
S t = S 0 exp
(νt + σW t (2.11)
6 Here we make use of the fact that the sum of identically distributed, independent random variables is a
random variable for which the mean is the sum of the means and the variance is the sum of the variances.
7 dW = (W t+dt − W t ) ∼ N (0, dt), according to properties 1 and 2 of Brownian Motion
9

A glimpse of stochastic integrals As we have moved to continuous world, the discrete
sums and differences of equations like (2.6) turn into integrals and differentials:
∫ T ∫ T
V T = V 0 + ζ t dS t + γ t dB t (2.12)
0
0
Computation ∫ T
0 γ tdB t means taking a Riemann-Stieltjes integral with respect to a
smooth, differentiable function B t = e rt . On the contrary, finding ∫ T
0 ζ tdS t requires
integration with respect to geometric Brownian motion and is beyond the reach of ordinary
“Newtonian” calculus. The paths of Brownian motion, despite their continuity,
are not differentiable. Computation of integrals w.r.t. dW , also known as Itô integrals,
lies in the domain of Stochastic Calculus and requires some theoretical material this
thesis was not meant to provide. Nevertheless, some most important properties and
results will be stated without proofs:
Itô’s Processes and Itô’s Lemma We shall postpone interpretation of (2.12) till next
section. For now, let’s consider a different process:
∫ T
∫ T
X T = X 0 + µ(t, X)dt + σ(t, X)dW t (2.13)
0
0
or, equivaletely, in differential form:
dX t = µ(t, X)dt + σ(t, X)dW t (2.14)
Processes that are described by equations like (2.13) and (2.14), involving Itô and Riemann
integrals are refered to as Itô processes. Here µ(t, X) is a drift of process and
σ(t, X) is the volatility. The characteristic feature of an Itô process is the fact that
the drift and volatility terms of (2.13) appear to be functions of time and the process
itself. [Hull, 1999] provides clear intuition why such dependancy must be the case for
the process of the stock price. He argues that the expected return per unit of time, µ,
should be the same regardless of the current stock value. The same argument is used
for the volatility parameter - the swings in the asset’s development are proportional to
its price. This vision of the market transforms (2.14) into a more specific form :
dX t = µX t dt + σX t dW t (2.15)
Here µ(t, X t ) = µX t and σ(t, X t ) = σX t . The Itô Process (2.15) is widely used as
a model for stock price evolution and represents a Stochastic Differential Equation, or
SDE. A very useful tool from Stochastic Calculus, that can sometimes be employed to
obtain solutions to SDE’s such as (2.15) is Itô’s Lemma. Itô’s Lemma can be thought
of as an extension of Taylor approximation formula into stochastic world. More specifically,
it states that if the sequence of random variables follows a Itô’s process X t with
drift ν x and volatility σ t , then the function of both time and this process f(t, X t ) is an
Itô’s process as well. One dimensional version follows:
df t,Xt = ∂f(t, X t)
dt + ∂f(t, X t)
dX t + 1 ∂t ∂X t 2
10
∂ 2 f(t, X t )
∂Xt
2 σt 2 dt (2.16)

There is an important difference between Itô’s formula and its deterministic counterpart.
The laws of stochastic calculus 8 stipulate that dWt
2 = σt 2 dt, where W t is a
Brownian motion. As a result, the second derivative term with respect to a stochastic
factor is of the same order of accuracy as ∂f(t,X t)
∂t
and cannot be neglected.
As a demostration of Itô’s lemma, let’s try to acquire the process for geometric
Brownian motion as given by X t = W t and f(t, W t ) = S t = S 0 exp ( )
νt + σW t ,
which is also (2.11). The partial derivatives of S t with respect to time and Brownian
motion are:
and therefore:
∂f(t, X t )
∂t
∂f(t, X t )
∂ 2 f(t, X t )
= νS t = σS t
∂X t ∂Xt
2 = σ 2 S t dt
dS t = (ν + 1 2 σ2 )S t dt + σS t dW
}{{} t
} {{ }
σ(t,S t )
µ(t,S t)
Thus, the process for prices given by geometric Brownian motion is an Itô process. We
shall call ν + 1 2 σ2 - an instantaneous expected return and ν - an overall expected return.
Black-Scholes-Merton Model Having accepted a log-normal distribution for the
continuous process of an underlying asset, we have eventually prepared ourselves for
discussion of Black-Scholes-Merton approach to option valuation. In their seminal paper
[Black and Scholes, 1973], Black and Scholes have suggested a model that dwells
on several assumptions:
• Trading proceeds continuously;
• Stock prices follow a log-normal distribution, the process described by (2.15);
• Absense of arbitrage opportunities;
• No rebalancing costs;
• Limitless lending and borrowing is allowed;
Having imposed this setting, Black and Scholes argued that a short position in a derivative
c t can be made riskless in case it is a part of a portfolio Π t such that:
Π t = −c t + ∆ t S t (2.17)
Both c t and S t depend on the same source of uncertainty, which is a Brownian Motion
term W t of the stock price process. Therefore if Π t is constructed by taking reverse positions
in a derivative and a certain amount ∆ t of its underlying, the Brownian motion
terms cancel, eliminating the risk. Π t is called a Hedge portfolio. According to our
no-arbitrage assumption, a position in a porfolio that has no risk must be of the same
nature as a position in a risk-free cash bond B t , i.e. earn a risk-free interest rate during
its lifetime:
dΠ t = rΠ t dt
8 Namely, the quadric variation result. See [Etherage, 2002].
11

Suppose this is not true and Π t earns more or less than prescribed by no-arbitrage. Then
we would have two different prices for a risk-free investment, driving market players
to buy underpriced and sell overpriced positions, which naturally enforces the laws of
supply and demand bring the market back to equilibrium. The second step towards the
price of an option in a continuous world would be finding the weight ∆ t such, that the
value of the hedge porfolio would be exactly zero. A player with a short position in
an option would then be able to compensate the claim against him by selling his stock
holdings. This sounds like exactly what we did in discrete time - a replicating strategy.
The only thing missing is a self-financing property of the portfolio. In discrete time, the
changes in ∆ t were financed by a ψ t amount of cash. The question is: if we add a cash
bond holding to our hedge portfolio, will it safeguard our continuous-time replicating
strategy from the need of extra money input? Our initial assumptions give a positive
answer.
Π 0 = −ψ t B 0
Notation put into words, establishing a hedge portfolio, which replicates an option sold,
requires additional investment in cash.
Rearranging:
dΠ 0 = d(−c 0 + ∆ t S t ) = −rψ t B 0 dt
dc t = ∆ t dS t + r(c t − ∆ t S t )dt (2.18)
Subscripts t should remind us that the holdings in assets ∆ t and ψ t change with time
just like the prices of assets themselves.
On the other hand, we know that a derivative c has a payoff that depends on time
and a stock price S t . We can write this as c t = F (t, S t ). What we also know is that
a process for the stock price evolution is being modelled as an Itô process. Therefore
we can apply Itô’s lemma to show how development in c t is expressed in terms of S t .
Taking first derivative with respect to time and derivatives with respect to the stock
price, we find
dc t = dF (t, S t ) = ∂F (t, S t)
∂t
+ ∂F (t, S t)
dS t + 1 ∂S t 2 σ2 t St
2
∂ 2 F (t, S t )
∂St
2 dt (2.19)
Equating (2.18) and (2.19) and rearranging the terms we arrive at Black-Scholes Partial
Differential Equation (PDE):
∂F (t, S t )
∂t
+ ∂F (t, S t)
∂S t
rS t + 1 2
∂ 2 F (t, S t )
∂St
2 σt 2 St 2 = rF (t, S t ) (2.20)
This PDE shows us, that a derivative c t = F (t, S t ), while evolving in time and its
underlying asset, earns a risk-free rate. As already observed, risk preferences of investors
do not serve as a parameter required for finding the fair value of a claim. In
other words, the assumption of a risk-neutral world, where investors do not require
compensations for taking risks, is valid in continuous-time as well.
12

Delta. In order to set up a riskfree hedge, as in (2.17), one needs to have the knowledge
of the process for ∆ t . Recall, that ∆ t shows how much of the underlying has to
be held at a timepoint t as a part of the hedge portfolio. On the other hand, ∆ t can
be interpreted as the sensitivity of the option’s price to the movement in its underlying
and, therefore, is equivalent to the first derivative of the option value with respect to
the price shifts in the asset. Note, that Black-Scholes PDE, given by (2.20), implicitly
provides us with such information.
Considering its utmost importance for replication of a claim, ∆ t will reappear in
the focus of our attention later in this text.
2.3 Pricing with Martingales and Measures
Let’s try to fit the main ideas of derivative pricing developed so far into a couple of
sentences. The risk of selling a derivative security can be hedged away by establishing
a replicating, self-financing portfolio of market assets. The cost of setting up a strategy
that eliminates the risk for a short side is, by no-arbitrage, the only correct price for
a product being replicated. According to no-arbitrage, any riskless asset or a porfolio
must earn a risk-free rate.
We have just seen that an option value can satisfy a partial differential equation,
therefore it can be approximated numerically with Finite Difference or Finite Volume
Methods. Another approach that has been discussed is risk-neutral valuation. It involves
taking expectations of a future payoff with respect to a unique risk-neutral probability
measure, possibly resulting in a closed form expression.
Complementing each other, both approaches are used intensively and ultimately
arrive at the same result. In this section we will introduce some mathematical tools
that are employed by risk-neutral valuation in continuous time, namely, martingale
processes and the change of measure result.
Some definitions
• Numeraire. A Numeraire is any asset B, whose price is strictly positive at any
moment in time. For example, S t
B t
is a price of a discounted stock price S t . The
price of the cash bond B t serves as a numeraire for the stock price.
• Martingales. A martingale is a F t -adapted stochastic process X t such that under
a certain measure, e.g. Q:
E Q = [ X t |F t
]
= Xs , ∀s < t
In words, Q-expectation of the future value of the process is always a current
value of the process. As time evolves, the process does not drift away from
its starting point, but fluctuates in its neighbourhood. Again we have already
encountered such processes previously in the text. The process for discounted
stock price e rt S t is a martingale under the risk-neutral measure, see (2.2).
13

Change of Measure In order to make use of the risk-neutral valuation principle, we
need to take expectations under Q. In case of one-step discrete setting we were able
to find the probability weights by solving for a price-state vector. Needless to say, in
continuous time the same approach is hardly applicable. Yet, after we’ve taken the time
step to the limit, the probability vectors become probability density functions. Without
going much into detail, we’ll state that changing the measure amounts to changing the
probability density function. Let’s sketch how expectation under one measure can be
taken with respect to another measure:
where
E P[ ∫
] ∞
X t |F t = X t f P (z)dz =
−∞
∫ ∞
−∞
dQ
dP = f Q (z)
f P (z)
X t
dQ
dP f P (z)dz = E Q[ X t |F t
]
(2.21)
is simply a ratio of density functions or, in financial terminology, a Radon-Nikodym
derivative. It can be easily shown that:
( ∫
dQ
t
dP = exp κ s dWs P − 1 ∫ t
)
κ 2
2
sds
(2.22)
0
is a geometric brownian motion with drift − 1 2 κ2 s and volatility κ s . Before elaborating
on the meaning of κ s term, we provide continuous time-version of Arbitrage theorem,
as promised:
Theorem 2.3.1. A continuous, complete market is free of arbitrage if and only if for
every choice of numeraire there exists a unique equivalent martingale measure.
Restricting our choice of numeraire to a cash bond B t , this theorem states that
there exists only one measure under which our discounted stock process Bt
−1 S t will
be a martingale process. This measure is our risk-neutral measure Q. Another very
important theorem that is going to be crucial to discussion of the next chapter is Girsanov’s
Theorem.
Girsanov’s Theorem We’ll start with a simple example. Let’s denote ˜S t = Bt −1 S t ,
i.e. a stock price with B t as a numeraire. By application of Itô’s lemma, the Itô process
for ˜S t is:
d ˜S t = (µ − r) ˜S t dt + σ ˜S t dWt
P
The equivalent martingale theorem claims that arbitrage is not feasible if there exists a
unique martingale measure for ˜S t is a martingale process. Rearranging the terms:
( )
d ˜S t = σ ˜S µ − r
t
σ
dt + dW t
P = σ ˜S t (κdt + dWt P ) = σ ˜S t dXt P (2.23)
Here, the resulting process appears to be a martingale with respect to a new process
dX P t . Of course, this appearance is only superficial as required martingale behaviour
is ruined by the drift κ in the Brownian motion dX P t . So, we wouldn’t have made any
progress without help from a so-called Girsanov’s Theorem:
0
14

Theorem 2.3.2. Let W P (t) = (W1 P (t), W2 P (t), . . . , Wd P (t)), t ≥ 0 be a d-dimensional
vector of correlated Brownian motions under some measure P, with correlation values
given by matrix ρ. If κ(t) = (κ 1 (t), κ 2 (t), . . . , κ d (t)), is a vector of pre-visible
stochastic processes adapted to the same filtration F t , with
[ ( ∫ 1 T
)]
E P exp κ(s) 2 ds < ∞ (2.24)
2 0
and Y t defined as
( ∫ t
Y t = exp κ s dWs P − 1
0
2
∫ t
0
)
κ 2 sds
then Y t is the Radon-Nikodym derivative dQ
dP
, and under the measure Q the process
is also a d-dimensional Brownian motion.
(2.25)
dW Q (t) = dW P (t) + κ(t)ρ dt, t ≥ 0 (2.26)
Shaking off notation 9 , it simply states that by changing the measure to some measure
Q, we can make the drift in dX P disappear. In other words, dX Q becomes a
standard Brownian motion and, consequently, ˜S t is a true martingale. It is worth noting,
that the quantity µ−r
σ
has an independent interpretation in CAPM. Our drift κ is
commonly known as market price of risk. It shows the extent to which the volatility in
the asset is compensated by the excess return. From this prospective one can conclude
that eliminating the drift, or making the market price of risk κ zero, we find ourselves
in the risk-neutral world where risk requires no compensation.
Before we wind up our discussion of preliminaries, there is one more result that we
need to address for this chapter to be more or less complete.
Brownian Martingale Representation Theorem Several pages ago, we have encountered
a transformation of a discrete integral (2.6) to a continuous form (2.12)
without providing any further explanation. Now, after becoming familiar with a notion
of a martingale process, we can loosely state the Brownian Martingale Representation
theorem, which proves to be an indispensable tool in construction of self-financing
replicating portfolios in continuous time.
Theorem 2.3.3. Let W t be a Q-martingale process w.r.t. a filtration F t and with
bounded variation E[Wt 2 ] < ∞. Then any Q-martingale M t w.r.t. F t and E[Mt 2 ] <
∞, can be written as:
where ζ s is an F t -adapted process.
M t = M 0 +
∫ t
0
ζ s dW t
In other words, any two martingales with respect to the same filtration are scaled
versions of each other. As an illustration of this result, let’s think that a demand exists
for a new exotic option D with some complex payoff. If there is a way that a payoff
9 Here we state the more general, multi-dimensional version, of Girsanov’s theorem.
15

of D can be decomposed into a number of less complex payoffs A,B and C that are
already traded in the market, and a cash position, then this product is attainable. Since,
the derivatives A,B and C are all known to have a martingale property 10 , then there
is a self-financing way of replicating the exotic claim by continuously rebalancing a
portfolio of these less-exotic assets, speaks Theorem (2.3.3). The price of such a new
product will, by no-arbitrage, be equal to the cost of establishing such a portfolio. As
for SDE, given by (2.12), by using a martingale property of d ˜S t from (2.23) and the
fact that at any given time t: V t = ζ t S t + γ t B t , we can show that the discounted
portfolio Ṽt is a martingale:
dṼt = d(Bt
−1 V t ) = V t dBt
−1
+ B −1
t
dV t = −rṼtdt + ζ t B −1
= −rṼtdt + ζ t (d ˜S t − S t dBt
−1 ) + γ t rB t dtBt
−1
= ζ t d ˜S t − rṼtdt + r(ζ t S t + γ t B t )Bt
−1 dt
t
dS t + γ t Bt
−1 dB t
= ζ t d ˜S t . (2.27)
In the light of Theorem (2.3.1), it is important to note that dṼt and d ˜S t are, in fact,
numeraire-rebased processes with a cash bond B t as a numeraire asset. The result
of (2.27) should remind us of (2.7), where a discrete process ∆Ṽi appeared to be a discrete
martingale. Theorems (2.3.3) and (2.3.1) do ensure that self-financing replication
works in continuous time as well.
After discussion of martingale pricing approach, our short introduction to the general
subject of asset pricing is more or less complete. All the tools developed with the
stock economy in mind will be applied to the interest rate markets in next chapter.
10 according to the risk-neutral valuation principle
16

Chapter 3
LIBOR Market Model
Having left the equity world, evolution of a stock is not relevant to the valuation process
any more. From now on, we will be operating on a tenor structure of forward LIBOR
rates, described in the first section of this chapter. While the importance of a numeraire
concept has been disguised by ubiquitous discounting so far, asset pricing in the economy
of stochastic interest rates appeals to numeraires explicitly. The mathematical
apparatus developed in the previous chapter will prove indispensable as we introduce
LIBOR Market Model and pricing under terminal measure. We shall spend some time
discussing the characteristic interest rate products and calibration of the pricing framework
to the current market data. In the finale, the LIBOR Market PDE will be loosely
derived.
3.1 Forward LIBOR Rates
Tenor Structure. In the previous chapter we have been discussing a setting that implicated
a risk-free rate r which is constant regardless of the bond maturity. When
pricing claims that are contingent on interest rate movements, this assumption turns
into an obstacle and has to be dismissed. If we want to acquire a more realistic model
of interest rate evolution, its setup will have to be able to account for the shape of the
curve, that is currently observable in the real-world markets.
The underlying asset in an interest rate economy is a zero-coupon, discount bond
that pays one unit of currency at maturity. Let’s denote D(t, T ) to be the price of such
a bond at time t and with maturity at T . The specific term structure, characterictic of
the market being modelled at time t = 0, is defined by a vector of N discount bonds
ordered with respect to their increasing maturity dates, i.e.:
(
)
D(0) = D(0, 1), . . . , D(0, 1 ≤ i < N), . . . , D(0, N) (3.1)
where D(0, i) = D(t, T i ) and t < T 1 < · · · < T N−1 < T N . Note that our discount
zero-coupon bond is a mere abstraction of a zero rate. Writing L 1 for a one-year zero
17

ate and Y for a notional amount, the following result must hold in a market free of
arbitrage:
Y ≡ Y (1 + L 1 )D(0, 1)
i.e. compounding Y with L 1 and then discounting with D(0, 1) produces the same
notional.
We shall write α i = T i − T i+1 for the time period between any two consecutive
bonds, namely tenor, and for our model assume it to be constant. Therefore, the ordering
(3.1) of N bonds can be referred to as interest rate tenor structure.
Forward Rates. The LIBOR market Model operates in terms of forward rates, rather
than zero rates. We shall illustrate the notion of a forward LIBOR rate with help of
a Forward Rate Agreement example. As in [Hull, 1999], a forward rate agreement
(FRA) is an over-the-counter agreement that a certain interest rate will apply to a certain
principal during a specified future period of time. Given a FRA:
Definition 3.1.1. Forward LIBOR rate L i (t) is an interest rate one can contract for
at time t to lend or borrow money for the future time period [T i , T i+1 ]. Given prices of
discount bonds maturing at T i and T i+1 , the forward rate is defined by:
1 + α i L i (t) = D(0, T i)
D(0, T i+1 )
where t ≤ T i (3.2)
a
N-1
0
1
2
N-1
N
T
D 0,1
1
L N-1
D 0,2
1
D 0,N-1
1
D 0,N
1
Figure 3.1: Tenor Structure of Interest Rates
Keeping our assumptions of no bid-ask spread and no-arbitrage, a FRA should cost
exactly nothing if the contracted interest rate equals a forward rate implied by a current
yield curve. Instead of taking one of the sides in the FRA, a trader can achieve the
same goal by manipulating bonds in the money-market. Let’s illustrate this statement
by rewriting (3.2):
D(0, T i ) − D(0, T i+1 )
= α i L i (0) (3.3)
D(0, T i+1 )
18

The right-hand side of the equality above can be perceived in the following way. A
market player sets up a portfolio comprised of a long position in a bond with shorter
maturity T i and a short position with a T i+1 bond, borrowing the difference between
the two at the observed zero interest rate until T i+1 . When the clock hits T i , the player
receives the principal due to the short position. As α i more time passes, she has to
deliver the principal on the shorted bond −D(0, T i+1 ) plus the money borrowed to
1
establish a portfolio, compounded at
D(0,T t+1 ). The eventual extra money outflow is
the right-hand side of (3.3), or the interest payment she could have contracted for by
entering a short side of the FRA. Taking reverse positions in the portfolio assets at time
zero would be equivalent to taking a long side of the FRA and receiving the forward
LIBOR rate for T i,i+1 .
The scene for LMM and the concepts discussed so far in the chapter can be shown
visually in Figure (3.1) From the figure above one could clearly see: the price of the
discount bond can be expressed as a product of forward LIBOR rates, implied by the
tenor structure:
D(0, T i ) =
N−1
∏
i=0
1
1 + α i L i (T i )
(3.4)
3.2 LIBOR Market Model
LIBOR Process The foundation of this section resides on the proposition that forward
LIBOR rates L i (T i ) should follow a martingale process. Indeed, looking back
at (3.3) we note that a discount bond D(t, T i+1 ) has a strictly positive value and therefore
can be treated as a numeraire. With arbitrage-free market in mind, the Fundamental
Theorem of Asset Pricing postulates there must exist a unique measure Q i+1 , under
which a numeraire-denominated portfolio is a martingale. This implication applies to
a LIBOR rate α i L i (T i ) with α i which is a constant and leads us to the definition of a
LIBOR process.
In LMM, the process of continuous evolution of market rates is described by the
following stochastic differential equation:
dL i (T i ) = σ i (t)L i (T i )dW Qi+1 (3.5)
where σ i is a continuous bounded volatility function and dW Qi+1 is a Wiener process
under a martingale measure Q i+1 . Itô’s formula can be applied to acquire the exact
solution to (3.5):
(
L i (t) = L i (0)exp − 1 ∫ t
∫ t
)
σ i (s) 2 ds + σ i (s)dW Qi+1 (s) (3.6)
2
0
With some theory from the second chapter and one of the properties of Itô integrals 1 ,
we can conclude that in LMM forward rates are modelled to be geometric Brownian
motion processes with variance κ 2 = ∫ t
0 σ i(s) 2 ds and mean − 1 2 κ2 .
1 If σ i (s) is a bounded, non-stochastic function, then the Itô integral of the formRt
0 σ i(s)dW (s) converges
in distribution to N(0,Rt
0 σ i(s) 2 ds)
19
0

Change of Numeraire Before we can discuss the key result of LMM - modelling
forward rates under terminal measure, it is necessary to address such issue as change
of numeraire. This result should remind us of the change of measure result from the
first chapter, as changing the numeraire implicates change of measure, according to
Theorem 2.3.1. Similar to narrative of [Pelsser, 2000], consider some derivative product
with payoff V (T ) and two numeraire assets D and B. Then, in the Risk-Neutral
world, value of such product at time zero is defined as:
[ ] V (T )
E D D(T, T ) ∣ F t = V (t)
[ ] V (T )
or E B D(t, T )
B(T, T ) ∣ F t = V (t) (3.7)
B(t, T )
where D and B are equivalent martingale measures imposed by the choice of numeraire
D and B. With no arbitrage assumption:
[ ]
[ ]
V (T )
V (t) = E D V (T )
D(T, T ) ∣ F t D(t, T ) = E B B(T, T ) ∣ F t B(t, T ) (3.8)
Let V D (T ) be D-denominated payoff of the product. Now, if we want to express e.g.
the D-denominated value of our product as the B-expectation, the equation (3.8) can
be rewritten as
E D[ [
] [
]
V D (T )|F t = E
B D(T, T )/D(t, T )
V D (T ) B(T, T )/B(t, T ) ∣ F t = E B V D (T ) dD
]
dB ∣ F t (3.9)
The last expectation operator should look familiar. It has the same meaning as the
D(T,T )/D(t,T )
change of measure result (2.21) for stochastic processes. Variable
B(T,T )/B(t,T ) is
nothing more than a “numeraire” notation for a Radon-Nikodym derivative dD/dB,
previously defined as (2.22). To summarize, we have derived a recipe of how the
expectation of a numeraire discounted value can be expressed in terms of expectation
under a measure imposed by a different choice of numeraire. This result will be needed
in the discussion of pricing under terminal measure.
Terminal Measure Still keeping the discussion quite abstract from derivatives with
any specific payoff pattern, we shall address the issue of pricing interest rate products
that are contingent on several forward rates. While simple, one-rate derivatives allow
for closed-form expressions 2 , pricing options on several rates requires some extra effort.
The complications in valuing such derivatives is the fact that every LIBOR rate
is a martingale process under its own measure. These measures are imposed by the
discount bonds with associated maturity dates, or numeraire assets.
The key idea of LMM is modelling a process for an arbitrary valid forward rate
in terms of the probability measure ruling the Brownian motion of the rate with most
distant payoff. In other words, all the uncertainty in the development of the price
for derivative should be defined solely by the martingale measure of the last forward
rate, or “terminal” measure. We shall assume Brownian Motions driving the individual
rates to be independent processes and, for now, no correlation exists between any two
Brownian motion terms.
2 as we shall see further on in this text
20

At this point of the text all necessary mathematical tools have been covered to
proceed with derivation. It is the recursive application of multi-dimensional Girsanov’s
theorem and Change of Measure results that will help collect all the LIBOR rates under
the same measure. First, recall that according to Theorem (2.3.2), we can go from
one measure to another by introducing the drift into the original LIBOR process, as
in (2.26). In order to do so, we need to obtain a drift-correction term κ, which is at the
same time the volatility of a Radon-Nikodym derivative.
1. Working out drift-correction term
It was agreed that discount bonds D(t, T i ) serve as numeraire assets that impose
martingale measures on separate forward rates. Let Q i denote such numeraireborn
measure, therefore making dL i (t) a martingale. So, from (3.9):
Y t =
dQi
dQ i+1 = D(t, T i)/D(0, T i )
D(t, T i+1 )/D(0, T i+1 ) = D(0, T i+1)
D(0, T i ) (1 + α iL i (t)) (3.10)
At the same time, from (2.25):
( ∫ t
Y t =
dQi
dQ i+1 = exp κ(s)dW Qi+1 (s) − 1 2
0
∫ t
Note that Y t is a martingale process described by the SDE:
0
)
κ(s) 2 ds
(3.11)
dY t = κ(t)Y t dW i+1 (3.12)
The final expression for κ(t) now requires application of Itô’s Lemma to (3.10):
dY t = D(0, T i+1)
D(0, T i ) α idL i (t) = D(0, T i+1)
D(0, T i ) α iσ i (t)L i (t)dW i+1 (3.13)
The last equality comes directly from definition of a LIBOR process. Equating
(3.12) and (3.13):
κ(t)Y t dW i+1 = D(0, T i+1)
D(0, T i ) α iσ i (t)L i (t)dW i+1
D(0,
κ(t)
✟ ✟✟✟ T i+1 )
✟ D(0, T i ) (1 + α iL i (t))✘✘ dW i+1 ✘ D(0,
=
✟ ✟✟✟ T i+1 )
✟ D(0, T i ) α iσ i (t)L i (t)✘✘ dW i+1 ✘
κ(t) = α iσ i (t)L i (t)
1 + α i L i (t)
(3.14)
2. Adjusting the drift.
Now that we have a drift correction term κ(t) we can apply Girsanov’s theorem
to get a change of measure relation 3 :
dW i = dW i+1 − κ(t)ρ i,i+1 dt = dW i+1 − α iρ i,i+1 σ i (t)L i (t)
dt
1 + α i L i (t)
3 In case of a single underlying Brownian Motion, correlation term ρ i,j = 1
21

For instance, let’s set κ T i = αiσi(t)Li(t)
1+α i L i (t)
to be the drift-correction term for the terminal
measure and find the L i−3 (t) forward rate under the terminal measure.
dL i−3 (t) = σ i−3 (t)L i−3 (t)dW i−2
[
]
= σ i−3 (t)L i−3 (t) dW i−1 − κ T i−2(t)ρ i−3,i−2 dt
( )
]
= σ i−3 (t)L i−3 (t)[
dW i − κ T i−1(t)ρ i−3,i−1 dt − κ T i−2(t)ρ i−3,i−2 dt
We proceed with opening the brackets and rearranging the terms:
(3.15)
dL i−3 (t)
[
]
L i−3 (t) = −σ i−3(t)κ T i−1(t)ρ i−3,i−1 dt + σ i−3 (t) dW i − κ T i−2(t)ρ i−3,i−2 dt
]
= −σ i−3 (t)dt
[κ T i−2(t)ρ i−3,i−2 + κ T i−1(t)ρ i−3,i−1 + σ i−3 (t)dW i (3.16)
The general form:
where
µ i (t) =
dL i (t)
L i (t) = µ i(t)dt + σ i (t)dW N (3.17)
{
−σi (t) ∑ N−1
j=i+1 κT j (t)ρ i,j i < N − 1
0 i = N − 1
(3.18)
3.3 Interest Rate Derivatives. Caplets, Caps and Swaptions.
So far in this chapter we have been mostly busy with the description of the model that
tries to immitate forward rates. However, our idea of a claim contingent on such rates
has been quite an abstract one. Now it’s good time to dismiss the assumption of “some”
payoff pattern and discuss several more specific interest rate products, that are traded
in the markets.
A Caplet. A simpliest of all interest-rate derivatives. It is analogous to a European
call option from the equity derivatives family. It’s payoff is as follows:
c i (T i+1 ) = α i (L i (T i ) − K i ) + (3.19)
One can’t help but notice an important difference between a caplet and it’s equity counterpart.
At maturity, the holder of a caplet has the option to take a long side of the
forward contract to borrow money at an interest rate K, also known as a strike rate for
α i units of time. This is in contrast to an in-the-money call option that immediately
22

pays off at expiry. Conveniently, the exact solution for such product becomes available
via analytical risk-neutral valuation along the following lines:
[
] [ ( ) ]
c i (t)
D(t, T i+1 ) = c i (T i+1 )
L i (T i ) − K i EQi+1 = α i E Qi+1 +
(3.20)
D(T i+1 , T i+1 )
1
In market practice, the process for dL i is commonly assumed to be geometric Brownian
Motion (3.6), and since it’s payoff is plain-vanilla, we compute this expectation by
integrating with respect to log-normal distribution of dL i , that results in:
c i (t) = D(t, T i+1 ) [ L i (t)Φ(d 1 ) − K i Φ(d 2 ) ] (3.21)
where Φ(·) is cumulative normal distribution function and
d 1 = log(L i(t)/K i ) + 0.5σ 2 i (T i − t)
σ i
√
Ti − t
d 2 = d 1 − σ i
√
Ti − t
The equation (3.21) is commonly refered to as Black’s formula and is widely used in
the industry. The popularity of this formula has enabled the traders to quote caplet
prices in terms of Black implied volatilities.
A Cap. It is worth mentioning that individual caplets are the instruments that are not
traded in the markets. Instead, caps, i.e. portfolios consisting of consecutive caplets
are bought and sold. The holder of a cap has an option to enter into a forward contract
at maturity of each caplet. It’s value at time t is therefore:
C(t) =
N−1
∑
i=1
c i (t) (3.22)
Individual caplet prices and volatilities can be extracted from quoted cap prices via
boot-strapping algorithm.
Swaps and Swaptions.
• A Swap is a contract between two market players to exchange a series of floatingrate
payments for a series of fixed-rate payments. This instrument can help hedge
against high interest rates via entering a payer swap, in which the party has to
pay a fixed rate in exchange for the floating. So, the cash outflow for the fixed
side and an arbitrary payment date is:
V fix
i (t) = D(t, T i+1 )α i K (3.23)
On the other hand, engagement in a reciever swap, where the fixed rate is received
and floating is payed, would mean having strong reasons to believe that
23

the interest rates are going to fall. The expected cash outflow for one payment at
T i in this case would be:
Vi flo (t) = D(t, T i+1 )α i E Qi+1 [L i (T )] = D(t, T i ) − D(t, T i+1 ) (3.24)
} {{ }
L i (t)
Given a set of payment dates {T i } N t=0, the value of, e.g. payer swap, is:
V pswap
t =
N∑
i=t
V flo
i
−
N∑
i=t
V fix
i (3.25)
The strike rate is set such that the current price of the swap is zero. Such rate K
is called par swap rate:
K par
t = D(t, T n) − D(t, T N )
∑ N
i=n+1 a i−1D(t, T i )
• European Swaption gives a holder the right to take a side in a swap contract at
maturity of the contract. It’s payoff in case of a payer swap option:
V ESw
T
= max(V pswap
T
, 0) (3.26)
• Bermudan Swaption gives a holder the right to enter a swap on a predefined
number of dates in the future. We shall assume these dates are the fixing dates
of the tenor structure. It’s payoff is as follows:
V BSw
t = max(H t (t), E t (t)) (3.27)
where H N−1 (T N−1 ) = 0 and E t (t) = V swap
t . H t (t) is a common notation
for the hold value (a contract has not been exercised up to and including time
t. E t (t) is a common notation for immediate exercise value, which in case of
Bermudan swaption equals value of the swap [Peterbarg, 2003].
Chooser Option. This product gives a holder the right to enter into a forward contract
at the rate equal to the difference between the maximum of the previous forward rates
and a strike rate K. The payoff follows:
V Cho
T
= α T −1 ( max
0

1. Tenor structure of N interest rates
2. Instantaneous volatility function σ i (t), i ∈ [1..N]
3. Correlation structure of LIBOR rates.
Tenor structure of interest rates is available to us immediately but volatility and correlation
require some extra work.
3.4.1 Calibration to caplets
LIBOR Market Model assumes that forward LIBOR rates follow a lognormal distribution,
i.e L i (T i ) = L i (t) exp(Z i+1 ), where Z i+1 ∼ N (− 1 2 κ2 , κ 2 ) under Q T i+1
,
and:
κ 2 =
∫ Ti
t
||σ(s)|| 2 ds
One can see that, in order to get a value for variance of a LIBOR process, we need to
integrate an instantaneous volatility function, that is unfortunately is not at our disposal
by default. What we do have is Black implied volatilites that are available to public.
Calibration to caplets implies that we should take quoted caplet volatility values as a
template/skeleton and fit the curve that matches the view of the market. Discussion on
approximating the instantaneous volatility function can be found in [Rebonato, 1998].
In this thesis, a simplified approach of a flat instantaneous volatility function will
be adopted. To be more specific, we shall assume LIBOR rate volatility to be constant
for a given tenor. Calibration to caplets in this case is particularly easy. By setting:
or
σ 2 BLACK(T i − t) = κ 2
√
1
σ BLACK =
T t − t
∫ Ti
t
||σ(s)|| 2 ds
we can accept Black volatilities as an average of an instantaneous volatility function
throughout the given tenor and use them directly.
3.4.2 Correlation structure of forward rates.
Having calibrated to caplet volatilities, the setup of the model is still not complete. In
our discussion of pricing under terminal measure we have neglected any correlation
that might exist between the LIBOR processes. As for this moment, any given process
for a LIBOR rate is driven solely by its Brownian Motion with drift, imposed by
change of measure. If we are modelling such tenor structure of N LIBOR rates, the
resulting processes can be shown in the following matrix-vector form with matrix A as
25

an identity matrix 4 :
⎡
⎢
⎣
dL 1 (t)
L 1 (t)
.
dL N (t)
L N (t)
⎤
⎡
⎥
⎦ = ⎢
⎣
µ 1 (t)
.
µ N−1 (t)
⎤
⎡
⎥ ⎢
⎦ dt+ ⎣
σ 1 (t)
.
σ N (t)
⎤ ⎛
⎥ ⎜
⎦ ⎝
a 11 . . . a 1N
. . . . . .
a N1 . . . a NN
⎞
⎟
⎠
A
⎡
⎢
⎣
dW 1 (t)
.
dW N (t)
One might find it hard to believe that all forward rates in the modelled term structure
are completely independent. For this specific reason, the model needs to be calibrated
to account for multiple sources of uncertainty in the behaviour of an interest rate.
It is worth trying to understand the idea by looking at only 2 forward rates dL 1 (t)
and dL 2 (t), where dW 1 (t) and dW 2 (t) are two independent Brownian motions. The
Itô process that is driven by N-factor Brownian Motion takes the following form:
dL i (t)
L i (t) = µ i(t)dt +
⎤
⎥
⎦
Q N+1
N∑
α i dW i (t) (3.29)
As we are considering correlation that is characteristic of Browninan Motions, we can
neglect the drift terms since they are deterministic and do not affect randomness or
stochastic factors in any way. The coefficients α and β will be used for notational
convenience, instead of just α i as in (3.29):
i=1
dL 1 (t) = · · · + α 1 dW 1 (t) + β 1 dW 2 (t)
dL 2 (t) = · · · + α 2 dW 1 (t) + β 2 dW 2 (t)
(3.30)
We postulate that dW 1 (t) and dW 2 (t) are two independent Wiener processes that
follow N (0, dt). Covariance of dL 1 (t) with itself is its variance:
[ (α1
Cov(dL 1 (t), dL 1 (t)) = E dW 1 (t) + β 1 dW 2 (t) )( α 1 dW 1 (t) + β 1 dW 2 (t) )]
= E
[α1dW 2 1 (t) 2 + 2α 1 β 1 dW 1 (t)dW 2 (t) + β1dW 2 2 (t) 2]
= α1E[dW 2 1 (t) 2 ] + 2α 1 β 1 E[dW 1 (t)dW 2 (t)] +β 2
} {{ }
1E[dW 2 (t) 2 ]
(
= α1
2 Var ( dW 1 (t) )
) )
− E 2 [dW 1 (t)] + β
} {{ } } {{ }
1dt
2
dt
0
= α1dt 2 + β1dt 2 = (α1 2 + β1)dt 2 = Var ( dL 1 (t) )
With calibration to caplets approach, this means that (α 2 1 + β 2 1)dt = ||σ i || 2 dt where
||σ i || is Black implied volatility. With this result at hand, we can extract the volatility
4 Matrix A contains coefficients that determine to what extent the individual Brownian Motion term contributes
to the composite Wiener process that drives the stochastic process dL i (t)/L i (t)
0
26

and weight factor parts from the coefficients via scaling:
dL 1 (t) = . . . + ||σ 1 ||( α 1
||σ 1 || dW 1(t) + β 1
||σ 1 || dW 2(t) = ||σ 1 || ( a 11 dW 1 (t) + a 12 dW 2 (t) )
dL 2 (t) = . . . + ||σ 2 ||( α 2
||σ 2 || dW 1(t) + β 2
||σ 2 || dW 2(t) = ||σ 2 || ( a 21 dW 1 (t) + a 22 dW 2 (t) )
For Variance, we have:
Var(dL 1 (t)) = ||σ 1 || 2 (a 2 11 + a 2 12)dt = ||σ 1 || 2 BLACK (3.31)
For the equality above to hold, the correlation components in a matrix row should add
up to one : a 2 11 + a 2 12 = 1. So it follows:
√
a 22 = 1 − a 2 11
With all said above and for present moment, our coefficient matrix takes the following
form:
[ ] [ ] [ √ ] [ ]
a11 a 12 dW1 (t) a11 1 − a
2
≡ √ 11 dW1 (t)
a 21 a 22 dW 2 (t) a 22 1 − a
2
22
dW 2 (t)
A
Now, let’s look at covariance between two rates:
(
)
Cov(dL 1 (t), dL 2 (t)) = ||σ 1 || ||σ 2 || Cov a 11 dW 1 (t) + a 12 dW 2 (t), a 21 dW 1 (t) + a 22 dW 2 (t)
[ (a11
= ||σ 1 || ||σ 2 || E dW 1 (t) + a 12 dW 2 (t) )( a 21 dW 1 (t) + a 22 dW 2 (t) )]
= ||σ 1 || ||σ 2 || E
[a 11 a 21 dW 1 (t) 2 + a 12 a 22 dW 2 (t) 2]
= ||σ 1 || ||σ 2 || (a 11 a 21 + a 12 a 22 ) dt
Cor(dL 1 (t), dL 2 (t)) = ✟✟ ||σ 1
✟|| ✟✟ ✟ ||σ 2 ||(a 11 a 21 + a 12 a 22 )✚dt
✟||σ ✟ 1 ||✟ √ dt ✟ ✟ ||σ✟✟
2 ||✟ √ dt ✟ = ρ
Finally, lets denote: (
1 ρ
ρ 1
A
)
= ρ
Matrix ρ is symmetric and positive-definite, threfore we can apply Cholesky Decomposition
to find the matrix coefficient matrix:
( ) ( ) (
AA T a11 a
=
12 a11 a 21
a 2 11 + a 2 12 a 11 a 21 + a 12 a 22
a 21 a 22 a 12 a 22 a 21 a 11 + a 22 a 12 a 2 21 + a 2 22
A T =
)
= ρ
27

Correlation. Now one thing that is left to specify is the correlation values between
any pair of rates. We go along with the approach from [Rebonato, 1998] 5 , where it is
suggested that:
ρ ij = e −β|T i−T j |
(3.32)
Beta is estimated from market data. This form makes sense as the correlation between
two rates is more likely to decrease as they are located farther apart on the tenor structure.
3.5 The LIBOR Market Model PDE
Let’s consider an interest rate economy whose population is described by a set of forward
rates:
L(t) = (L 1 (t), L 2 (t), . . . , L N−1 (t))
Suppose we need to price an interest rate product U(t, L) ≡ U(t, L(t)). According
to risk-neutral valuation principle, the value of such a derivative at time t < T N in
LIBOR Market Model will be:
U(t, L)
D(t, T N ) = EQT N
[ ]
U(TN , L)
D(T N , T N ) ∣ F t
(3.33)
where Q T N
is the terminal martingale measure imposed by the forward rate with latest
maturity. Values of discount bonds D(t, T N ) and D(T N , T N ) are known at time t and
therefore:
U(t, L) = D(t, T N )E [ QT N
U(T N , L) ∣ ]
∣F t (3.34)
However, the closed-form solutions based on (3.34) are rarely available due to the
multi-asset feature of most LIBOR derivatives. One approach that readily offers itself
in such circumstances is the appplication of the Feynman-Kac theorem to link the
expectation (3.34) of the functionals of SDE solutions to the solution of the partial differential
equation. A detailed account of this methodology with application to LIBOR
Market Model can be found in [Pietersz, 2002]. Without proof, we now sketch the
derivation of the LIBOR Market PDE, proceeding along the following lines:
First, let’s denote Ũ = U(t, L)/D(t, T N ) at any given time t. The process dŨ(t, L)
can be obtained by application of multi-dimensional Itô’s Lemma:
N−1
∂Ũ
dŨ(t, L) =
∂t dt + ∑
1
2
i,j=1
N−1
∂ 2 Ũ
∂L i (t)∂L j (t) dL ∑
i(t)dL j (t) +
i=1
∂Ũ
∂L i (t) dL i(t)
The LIBOR processes dL i (t) under the terminal measure are known from (3.17), so
5 also used in [Blackham, 2004]
28

we can substitute into the equation above to get:
⎛
dŨ(t, L) = ⎝ ∂Ũ
N−1
∂t + ∑
1
2
ρ ij σ i (t)σ j (t)L i L j
i,j=1
N−1
∑
∂Ũ
+ µ i (t)L i ∂L i
dW QT N
i=1
∂ 2 Ũ
∂L i ∂L j
+
N−1
∑
i=1
⎞
∂Ũ
µ i (t)L i
⎠
∂L i
dt
(3.35)
It is important to remember, that in order to comply with no-arbitrage conditions
and (3.34), the numeraire-rebased process dŨ(t, L) has to be a martingale under terminal
measure Q T N
. To satisfy this requirement, the drift term dt in (3.35) must be
equal to zero. The final PDE takes the following form:
∂Ũ
∂t + 1 N−1
∑
ρ i,j σ i (t)σ j (t)L i L j
2
i,j=1
∂ 2 Ũ
+
∂L i ∂L j
N−1
∑
i=1
µ i (t)L i
∂Ũ
∂L i
= 0 (3.36)
where
µ i (t) =
{
−σ i (t) ∑ N−1 α k L k (t)
k=i+1 1+α k L k (t) ρ ikσ k (t) if i < N − 1
0 if i = N − 1
and terminal Dirichlet boundary conditions given by the option payoff function p(·) at
time T N :
Ũ(T N , L) = p(L(T N )) (3.37)
As a result, the derivative price at time t is given by (3.34), where Ũ satisfies (3.36).
This LMM PDE imposes a relation between the time derivative (the manner in which
our price changes in time) and ”space”-derivatives (that describe how numeraire-rebased
price is affected by the change in one of the underlying rates).
29

Chapter 4
Computational Methods
In order to price derivatives, whose values are not accessible analytically, we need to
employ numerical methods to approximate the solution. First, we look at Monte Carlo
(MC) simulation and provide a general overview of its distinctive features. Second, we
address the Finite Difference (FD) approach and derive a discrete form of the LMM
PDE. The main focus of this thesis is on the FD approach, while Monte Carlo simulation
is used as a benchmark.
4.1 Monte Carlo
Monte Carlo Simulation approach takes advantage of the Law of Large Numbers from
Probability theory. This result states that, given n independent, identically distributed
random variables with mean µ, the average of these variables will converge to µ, as
n → ∞. Let’s see how this can be applied to derivative pricing in the framework of
LMM.
Simulation The risk-neutral valuation ansatz defines the price of a derivative to be
the discounted expectation of a future payoff taken under the equivalent martingale
measure. As before, the process for a simulated forward rate is assumed to be given
by (3.5). If the lifetime of a derivative is divided into N small intervals of length ∆t, a
continuous path of (3.5) can be approximated more accurately 1 if we operate in terms
of log L i (t). The LIBOR process for log L i (t) can be obtained by application of Itô’s
lemma to (3.5) and the discretization follows:
log L i (t + ∆t) − log L i (t) = − 1 2 σ(t)2 ∆t + σ(t) ( W (t + ∆t) − W (t) ) . (4.1)
From the property 1 of Brownian Motion in section (2.2), we know that W (t + ∆t) −
W (t) ∼ N (0, ∆t). Then, given the discrete process (4.1), we can easily go back to
operating in terms of the underlying rate:
{
L i (t + ∆t) = L i (t) exp − 1 2 σ i(t) 2 ∆t + σ i (t) √ }
∆t ɛ i . (4.2)
1 since the increment in the simulated process does not depend on the original process that we are trying
to approximate.
30

with ɛ i being independent standard normal random variables. In practice, we need to
model the dynamics of a tenor structure of N forward rates. Pricing under terminal
measure in LMM introduces a drift-correction term into an individual LIBOR process.
So, discretization of (3.17) yields:
L i (T n+1 )
L i (T n )
= exp
{(
−σ i (T n )
N∑
k=i+1
α k ρ ik σ k (T n )L k (T n )
− σ i(T n ) 2 )∆t+σ i (T n ) √ }
∆tɛ i
1 + α k L k (T n ) 2
(4.3)
Having simulated the relevant LIBOR paths for the underlying rates with (4.3), we
can compute sample payoff value for a derivative at maturity. In other words, we get a
sample value of an independent random variable V (T ) that represents an option payoff.
Next step, by recipe, is to repeat the sampling procedure n number of times and take the
average over the all sampled payoff values. In the limit, by the Law of Large Numbers,
this average will converge to the expectation value.
Recall that, in order for the risk-neutral valuation principle to hold, we must operate
in terms of numeraire-rebased quantities:
[ ]
V (t) V (T )
N(t) = EQ N(T ) ∣ F t
where N is the numeraire asset and Q is the unique equivalent martingale measure
imposed by the choice of N. Thus, we have to make sure that every sample payoff
V (T ), is denominated in terms of the terminal bond D, representing our choice of
numeraire in LMM. Then, by averaging over all D-rebased sample payoff values, we
shall obtain the value of the Q-expectation and, equivalently, V (t)/D(t, T ).
Standard Error If we are able to obtain the approximate value for expectation, it is
easy to get a number for standard deviation ω. As in [Hull, 1999], the expression for a
standard error of Monte Carlo is:
ω
√ n
A convenient fact from [Hull, 1999] also states, that a 95% interval of trust for the
derivative lies within approximately 2 standard errors:
µ − 1.96ω √ n
< c t < µ + 1.96ω √ n
Sometimes the standard error of MC simulation can be reduced by means of variance
reduction techniques, such as antithetic variables or control variates. Yet, this subject is
outside the scope of present work. A useful reference to such techniques is [Rebonato,
1998].
Correlated Brownian Motions If we recall the discussion from the previous chapter
on calibration to correlation between the rates, an individual LIBOR process can be
driven by a composite Brownian motion ˆɛ i such, that:
ˆɛ i =
i∑
α ik W k
k=1
31

Values for α ik are obtained by application of the Cholesky decomposition to a correlation
matrix 2 ρ.
Early Exercise Options with early-exercise feature are quite a hard nut for Monte
Carlo simulation due to so-called “Monte Carlo on Monte Carlo” effect. Indeed, at
every point of time T i , where early-exercise is allowed, one has to find:
V (T i ) = (max[E(T i ), H(T i )]) +
with E(T i ) and H(T i ) denoting immediate exercise value and hold value at time T i ,
respectively. In order to find H(T i ) (i.e. the expectation of the future payoff), canonical
Monte Carlo method would have to run an entire simulation run at every such timepoint
for all sample paths. In contrast, recombinant tree and PDE methods have the value of
H(T i ) available at each time-step, working by backward induction.
The authors of [Longstaff and Schwartz, 2001] have proposed a method which
approximates the conditional expectation value H(T i ) on the basis of cross-sectional
information stored for each simulated path. The idea of such approximation is to use a
least-square fit method to regress the estimated value with a polynomial basis function
of a chosen type and order.
Greeks The weakest point of MC simulation
appears to be the computation of
the ’Greeks’, the quantities that relate to
parameter-specific risk in an option position.
We have already encountered one
such sensitivity, ∆, earlier in the text. In
order to compute a ∆ value with MC,
the entire simulation procedure has to be
repeated for a small shift ɛ in the respected
underlying asset value. The same
methodology is to be applied with respect
to other possible factors of risk, for instance,
volatility. The pointwise valuation
nature of Monte Carlo drastically increases
the computational costs in case of
the overall sensitivity profile, characteristic
of a derivative instrument.
Delta value
0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
10
FD solution. Mesh 256x256
100000 MC trials
400000 MC trials
1
0.1
0.01
Shift in the underlying rate in bp. (1bp = 0.01%)
Figure 4.1: Monte Carlo. Delta of the option
with a digital feature.
Even greater difficulties arise when the ’Greeks’ of the products with discontinuous
payoff are considered, e.g. digital options. By a simple argument, if ɛ is not large
enough, the proportion of sample paths ending up in-the-money vs. out-of-the-money
is likely to remain unchanged. Even though there will more paths approaching the
digital barrier, a sensitivity (e.g. ∆ or V) will show zero. On the other hand, the value
of the ’Greek’ letter could explode if the small change in price is divided by an even
smaller ɛ, as in Figure (4.1). In general, a very large number of simulation trials and
2 which, for example, could be given by (3.32).
0.001
32

a good compromise on the size of ɛ are required to capture the reaction of MC option
value to small parameter shifts.
4.2 Finite Difference Method
While Monte Carlo simulation remains the industry’s tool of choice for pricing interest
rate derivatives within LMM framework, the difficulties mentioned above motivate
researchers appeal to alternatives approaches.
d=2
4
3
2
1
0
0
1
2
h
1
}
3
4
} h 2
with:
d=1
h=
L=
m
U h , L
( h 1,h
) t
2
( 2,3) t
Figure 4.2: 2D Solution mesh.
A numerical technique where the issues with
early exercise and the greeks are solved quite
naturally is the Finite Differences Method. This
approach works by approximating a PDE solution
on a discrete mesh of points. An original
PDE is rewritten into a difference equation (or
a system of such) to be solved either directly or
iteratively. Discretization can be done according
to one of the several discretization schemes
that vary in the degree of accuracy, stability
and speed of convergence [Travella and Randal,
2000].
Solution space. The number of subsequent interest
rates, that underlie a derivative product, determines the “Space”-dimensionality
of our LIBOR Market PDE. In other words, an i-th dimension in our solution mesh
is assigned to an [L min
i . . . L max
i ] range in an i-th forward rate. The step size in an
underlying rate L i is defined by a number of mesh points M i in the corresponding
dimension. Let’s denote: 3
• h = (h 1 , h 2 , . . . , h d ) t to be a mesh width vector of a d-dimensional mesh, where
h i = Lmax i
−L min
i
M i −1
.
• L i±k = (L 1 , . . . , L i ± kh i , . . . , L d ) t to be a vector of coordinates of the point
on the mesh.
• Uh,L m to be the solution at time level m at coordinate L on a grid with space
discretization h.
Time discretisation. Relation (3.37) stipulates that the boundary condition for our
equation should correspond to the value of an option payoff at T N ≡ T d+1 . In reality,
an eventual cash inflow from such product fixes at T d and stays unchanged for [T d , T d +
α d ]. This fact allows us to restrict the time horizon to that part of the tenor structure
where the dynamics of working forward rates do affect the outcome. Discounting with
D(t, T d+1 ), as in (3.34), ensures that extra α d time of position existence is accounted
3 In this section we shall take advantage of convenient notation used in[Blackham, 2004]
33

for. With all said above, it is natural for the LMM PDE to be solved backwards in time.
We set τ = T d − t, ∆τ = and discretize the time derivative as:
T d
M−1
∂u
∂t = −∂u ∂τ = −Um+1 h,L
− Uh,L
m
∆τ
+ O(∆τ) (4.4)
Space discretisation. For the time being, we have chosen second-order approximations
of spatial derivatives.
• First derivative (by central differences):
• For cross-derivative term with i ≠ j:
∂U
∂L i
≈ U m h,L i+1
− U m h,L i−1
2h i
+ O(h 2 ) (4.5)
∂ 2 U
∂L i ∂L j
≈ U m h,L i+1,j+1
+ U m h,L i−1,j−1
− U m h,L i+1,j−1
− U m h,L i−1,j+1
4h i h j
+ O(h 2 )
• For a second derivative term:
∂ 2 U
= ∂2 U
∂L i ∂L i ∂L 2 i
≈ U m h,L i+1
− 2U m h,L + U m h,L i−1
h 2 i
(4.6)
+ O(h 2 ) (4.7)
• The discretized µ ∆ i drift-correction term is as follows:
{
µ ∆ −σi (m∆τ) ∑ N−1 α k ρ ik L k
i (m∆τ) =
k=i+1 1+α k L k
σ k (m∆τ) if i < N − 1
0 if i = N − 1
(4.8)
LMM PDE discretized. Note, that the time difference equation (4.4) is first-order
accurate, while approximations of spatial derivatives (4.5), (4.6) and (4.7) are secondorder
accurate. An important aspect of rewriting a PDE into a difference form is the
choice of an accurate discretization scheme, such as Crank-Nicolson scheme, which is
second-order accurate both in space and time [Crank and Nicolson, 1949].
The general, so-called θ-scheme of PDE discretization into Finite Difference form
is:
U m+1
h,L
− Uh,L
m = θW m+1
h,L
+ (1 − θ)Wh,L m (4.9)
∆τ
where W m+1
h,L
and Wh,L m are implicit-in-time and explicit-in-time discretizations of remaining
PDE terms.
W m h,L = 1 2
N−1
∑
i,j=1
i≠j
N−1
1 ∑
σi 2 (m∆τ)L 2 i
2
i=1
ρ ij σ i (m∆τ)σ j (m∆τ)L i L j
U m h,L i+1,j+1
+U m h,L i−1,j−1
−U m h,L i+1,j−1
−U m h,L i+1,j−1
4h ih j
U m h,L i+1
−2U m h,L +U m h,L i−1
h 2 i
+ ∑ N−1
i=1 µ i(m∆τ)L i
U m h,L i+1
−U m h,L i−1
2h i
(4.10)
34

Three different θ values represent three canonical discretization schemes. By setting
θ = 0.5 for Crank-Nicolson, we shall first average and then discriminate explicit and
implicit parts as follows:
U m+1
h,L
− ∆τ
2 W m+1
h,L
= Uh,L m + ∆τ
2 W h,L m (4.11)
As a result of such discretization we should arrive at Ax = b system of discrete stencil
equations, where A is a M i ×· · ·×M d band matrix of known coefficients, x is a vector
of unknown solutions U m+1 and b is a vector of known values that correpond to the
right hand side of (4.11). The necessity to simultaneously solve a large system of
discrete equations poses the dilemma of the choice of an efficient iterative solver, such
as BiCGSTAB [van der Vorst, 1992] or Multigrid [Wesseling, 2004].
Stencil equation for 2 forward rates. Given the discrete equation for 2 forward
rates, we can rearrange the terms of (4.11) to arrive at the following stencil equality:
−ψU m+1
i−1,j−1 −βU m+1
i−1,j ψU m+1
i−1,j+1
−ζU m+1
i,j−1 (γ + 2)U m+1
i,j −ηU m+1
i,j+1
ψU m+1
i+1,j−1 −αU m+1
i+1,j −ψU m+1
i+1,j+1
} {{ }
unknowns
where the known coefficients are:
=
ψUi−1,j−1 m βUi−1,j m −ψUi−1,j+1
m
ζUi,j−1 m −γUi,j m ηUi,j+1
m
−ψUi+1,j−1 m }
αUi+1,j m {{
ψUi+1,j+1
m }
knowns
α = ∆τσ2 i (m∆τ)L2 i
4h 2 i
β = ∆τσ2 i (m∆τ)L2 i
4h 2 i
γ = ∆τσ2 i (m∆τ)L2 i
2h 2 i
+ ∆τµi(m∆τ)Li
4h i
− ∆τµ i(m∆τ)L i
4h i
+ ∆τσ2 j (m∆τ)L2 j
2h 2 j
ψ = ∆τρ ijσ i (m∆τ)σ j (m∆τ)L i L j
8h ih j
ζ = ∆τσ2 j (m∆τ)L2 j
4h 2 j
η = ∆τσ2 j (m∆τ)L2 j
4h 2 j
− ∆τµ j(m∆τ)L j
4h j
+ ∆τµ j(m∆τ)L j
4h j
− 1
and the band coefficient matrix A having the same pattern as Figure (4.3).
Forward rate expiry. While solving (4.9), we should keep in mind that the lifetime
of any given forward rate is bounded by its fixing date on the tenor structure. Has such
maturity of an underlying rate been reached, the dimensionality of original problem
reduces by one. The simpliest and most natural way to introduce such behavior into
valuation process is to reset forward rate’s volatility value to zero after it has matured.
Therefore, by solving LMM PDE backwards in time, we gradually increase the number
of rates that are setting in motion the numerical mechanism of option valuation.
Notation σ(m∆τ) and µ(m∆τ) is meant to serve as a reminder of this feature.
35

Greeks. Opposed to MC simulation, computing
solutions for the Greeks with Finite Difference
Method comes at much lesser cost in terms
of both time and accuracy. Approximations to
’spatial’ derivatives are implicitly provided with
the discrete PDE solution. The profiles for ∆
and Γ can be extracted from the solution mesh
in the form of discrete first and second derivatives,
i.e (4.5) and (4.6). In order to acquire the
solution for a V, the numerical procedure needs
to be repeated with a small shift in volatility of
an underlying rate.
Figure 4.3: Band coefficient matrix.
36

Chapter 5
Sparse Grids
In the previous chapter we discussed two computational methods that are available
for pricing of the claims within the LMM framework. In practice, Monte Carlo is almost
always in favour, since its computational costs are only linearly dependant on
the dimensionality of the problem. Finite Difference Method suffers from the ’curse
of dimensionality’, the number of points grows exponentially with the increase in the
number of underlying variables. This only fact undermines all the nice features FD has
to offer, since a great number of realistic products are contingent on up to 30 LIBOR
rates. Sparse Grid methods, suggested by Zenger [Zenger, 1991], operate on spatial
discretizations that rely on far fewer points and, at the same time, offer reasonable
accuracy. In this chapter we shall look at the common issues related to function approximation,
introduce Sparse Grid technique and postulate results for error analysis.
At the end we shall discuss the combination technique used for function interpolation
on a sparse grid. Presentation of the subject in this chapter relies on exposition and
notation of [Garcke, 2005].
5.1 Full grid approximations.
Consider a PDE of the form Lu = f, subject to certain boundary conditions, defined
on a d-dimensional solution space [0, 1] d . An equidistant d-dimensional grid Ω l could
then be taken as a discrete approximation of such solution space, with :
• l = (l 1 , . . . , l d ) ∈ N d as a multi-index, i.e. discretization resolution of Ω l
• h l := (h l1 , . . . , h ld ) := (2 −l 1
, . . . , 2 −l d
) as a mesh width
in each coordinate direction t ∈ [1 . . . d].
Given Ω l , then the discretized PDE, denoted as L l u l = f l , can be approximated
with either standard basis or hierarchical basis functions.
37

(j-1)h (j+1)h
0
1
Figure 5.1: Standard Basis functions S 3
Standard basis functions. Let’s define a corresponding space S l of piecewise d-
linear basis functions as:
S l := span{f l,j | j t = 0, . . . , 2 l t
, t = 1, . . . , d} (5.1)
where the d-linear piecewise function f l,j is obtained by a tensor product operation
over simple linear ’pagoda’ functions:
f l,j :=
d⊗
f l,j (5.2)
Such one-dimensional functions have support [x j − h l , x j + h l ] ∩ [0, 1], as illustrated
in Figure (5.1) above, and have the following definition:
t=1
{
1 − |x/hl − j| x ∈ [(j − 1)h
f l,j (x) =
l , (j + 1)h l ] ∩ [0, 1]
0 otherwise
(5.3)
Hierarchical basis functions. One of the main advantages of hierarchical bases over
the standard nodal point bases described above is the fact that the approximating function
space acquires a multi-level structure. By using the hierarchy of basis functions
we can distinguish between high-level and low-level representations, i.e., bases that
contain a significant part of the information and those where approximation is quite
rough [Bungartz and Dornseifer, 1998], respectively.
Let’s denote :
• |l| 1 := ∑ d
t=1 l t to be a discrete multi-index L 1 -norm.
• |l| ∞ := max 1≤t≤d l t to be a discrete multi-index L ∞ -norm.
The first step on the way to hierachical representation is obtaining a hierarchical difference
space T l from the stardard base of the same level:
T l = S l \
d⊕
S l−et (5.4)
where e t is the unit vector in t-th dimension. The resulting function space consists
of basis functions that are characteristic of the current level l and are not a part of the
t=1
38

T
0 , 0
T
0 , 1
T
0 , 2
T
0 , 3
|i|
1 = 0
T
1 ,0
T
1 , 1
T
1 ,2
· · ·
|i|
1
= 1
T
1 ,3
T
2 ,0
T
2 , 1
·
·
T
2 ,2
T
2 , 3
|i|
1 = 2
T
3 ,0
|i|
1 = 3
|i|
1 = 4
T
3 , 1
|i|
1
= 5
T
3 ,2
|i|
1
= 6
T
3 , 3
Figure 5.2: Hierarchical representation.
spaces with a smaller |l| 1 value. For example, applying (5.4) to S 3 should yield linear
images of spaces labeled T 3,0 or, T 0,3 on Figure (5.2). Note that, unlike S 3 , the support
of the basis functions of T 3 is not overlapping.
Now we can use (5.4) to obtain the final n-level hierarchy of subspaces S ∗ n:
S ∗ n =
n⊕
· · ·
l 1=0
n⊕
T l =
l d =0
⊕
|l| ∞ ≤n
T l (5.5)
’≤’ corresponds to index-wise relation in the subscript of the direct sum operator. Figure
(5.2) illustrates such multi-level subspace decompostion S3 ∗ in two dimensions.
Figure (5.3) shows how hierarchical subspaces T l1,l 2
reconstruct Ω 3,3 , where each tuple
relates to the subspace index. As stated in the beginning of this chapter, our original
intent is to approximate a PDE solution space. The equation of our interest, given
by (3.36), is subject to Dirichlet boundary condition, which implies no degrees of freedom
exist on the boundaries of solution domain. For this specific reason, subspaces
with l = 0 index components have not been included in the hiearchical representation
of (5.3).
39

33 23 33 13 33 23 33
32 22 32 12 32 22 32
33 23 33 13 33 23 33
31 21 31 11 31 21 31
33 23 33 13 33 23 33
32 22 32 12 32 22 32
33 23 33 13 33 23 33
Figure 5.3: Mapping of hierarchical representation S ∗ 3 onto a full grid.
At this point, a reasonable question to ask in the context of our PDE problem would
be: how accurate is the Sl ∗ -approximation of a solution u, given that u is a smooth,
continuous function on Ω?
Here we shall provide the result for such error analysis in case of a two dimensional
discretized PDE L i,j u i,j = f i,j . Results for d-dimensional case can be easily derived.
The following definitions will apply:
• The approximate solution of u on Si,j ∗ , denoted with u∗ i,j , can be decomposed as:
u ∗ i,j =
i⊕
j⊕
s=1 t=1
u s,t
• As in [Zenger, 1991], we shall define a semi-norm:
∂4 u
|u| = ∥ ∥ ∥∞
∂x 2 ∂y 2
∂
assuming that a mixed derivative
4 u
∂x 2 ∂y
of u exists and is a continuous function
2
itself.
• In the original paper [Zenger, 1991], the following a property of the representation
in the product-type basis has been shown:
∥
∥u i,j
∥
∥∞ ≤ 4 −i−j−1 |u|
Now, by using properties of infinite series and matrix norms, it can be shown that the
error of approximation is
∥
∥u − u ∗ ∥
i,j ∞
= · · · = 1 (
)
36 |u| h 2 i + h 2 j + h 2 i h 2 j
(5.6)
giving the convergence order of interpolation of the function with hierachical basis
function space as O(h 2 i + h2 j ), which is second-order convergence. For the derivation
of this result an interested reader is referred to [Griebel et al., 1992] or [Zenger, 1991].
However, representation of a function in a hiearchical basis still suffers from the
’curse of dimensionality’. As argued in [Garcke, 2005], the number of basis functions,
which describe a u ∈ Sl ∗, in hierarchical basis is (2l + 1) d , where l is the level of
discretization in each of the d dimensions. For instance, taking l = 4 and d = 10, the
number of basis functions is 2 × 10 12 !
40

00 03 02 03 01 03 02 03 00
30 30
20 12 20
30 30
10 21 11 21 10
30 30
20 12 20
30 30
00 03 02 03 01 03 02 03 00
Figure 5.4: Mapping of hierarchical representation Ŝ∗ 3 onto a sparse grid.
5.2 Sparse grid approximations.
In order to decrease the amount of points involved in the approximation, we turn to the
sparse discretization suggested in [Zenger, 1991]. Our solution space is chosen such
that the hierarchical basis functions with the small support and, therefore, a small contribution
to the final approximation, are not included in the resulting discrete space. The
authors of the original paper suggested the following combination of basis functions:
Ŝn ∗ = ⊕
T l (5.7)
|l| 1≤n
The required set of difference spaces is acquired by substitution of L ∞ -norm of (5.5)
for L 1 -norm in the direct sum condition of (5.7). Figure (5.2) shows difference spaces
T i,j comprising Ŝ∗ 3 outlined with the bold frame. The lower right triangular section
provides discretization details smaller than required tolerance and, thus, is excluded.
The set of spaces T i,j which meet the |l| 1 ≤ n condition is called a sparse grid. According
to [Garcke, 2005], its number of degrees of freedom equals:
|Ŝ∗ n| = O(h −1
n
log(h −1
n ) d−1 ) (5.8)
while the interpolation error, by similar analysis as in (5.6), shows
∥
∥u − û ∗ ∥
n,n ∞
= 1 (
48 |u| h2 n log(h −1
n ) + 4 )
3
(5.9)
For the proof, an interested reader is referenced to [Zenger, 1991].
As one can tell from (5.9) and (5.8), the drastic reduction in the number of degrees
of freedom is complemented by an insignificant increase in the interpolation error. For
the 2D case, the use of Ŝ∗ n instead of Sn ∗ decreases the number of points from O(h −2
n ) to
O(h −1
n log(h −1
n )). At the same time, the accuracy of representation stays at reasonable
O(h 2 nlog(h −1
n )), as compared to O(h 2 n).
Figure (5.4) shows the resulting sparse grid space, obtained by application of (5.7)
to Sn ∗ of Figure (5.2). Boundary spaces have been kept in order to populate a very
sparse space.
41

5.3 A Combination technique
So far, we’ve had a d-linear basis function as the unit of approximation on a discrete
function space. The sparse grid combination technique, introduced in [Griebel et al.,
1992], employes a different technique leading to the function representation on a sparse
grid. Similar to Richardson extrapolation, it is based on the principle of multi-variate
extrapolation and relies on the fact that leading-order discretization errors cancel out as
the result of linear combination of grids. The combination technique uses a node-based
discretization, as opposed to function-based one.
The grids are combined as follows 1 :
• 2D case :
• 3D case :
• dD case:
û c n,n =
∑
u i,j − ∑
u i,j (5.10)
i+j=n+1 i+j=n
û c n,n =
∑
∑
u i,j − 2 u i,j + ∑
u i,j (5.11)
i+j=n+2
i+j=n+1 i+j=n
∑d−1
( ) d − 1 ∑
û c n,n = (−1) k k
k=0
|l| 1=n−k
u |l| (5.12)
First, PDE is solved on the individual grids by means of a numerical method (e.g.,
finite difference method, as in Chapter 4). Second, solutions are linearly combined to
produce a interpolated approximation to u.
Essencially, the combined approximation of a solution to the PDE u c l
is not the
same as the u ∗ l
. However, it can be shown [Griebel et al., 1992] via asymptotic error
expansion for 2D, that the estimation is of the same order of accuracy:
∥
∥ u − û
c ∥∞ n,n ≤ Kh 2 l (1 + 5 4 log 2(h −1
l
)) = O(h −1
n log(h −1
n )) (5.13)
where K is some constant and h l = 2 −l . Consider referring to [Blackham, 2004] for a
d-dimensional error analysis.
Recently, advances have been made in the subject of combination techniques for
sparse grids. Novel combination approaches have been introduced in [Leentvaar and
Oosterlee, 2006] or [Reisinger, 2004].
1 here we usePnotation for a direct sum operator to emphasize the node-based nature of discretization,
used in combination technique, instead of the commonLdirect sum notation
42

Chapter 6
Numerical Results. Valuation
and the Greeks
In Chapter 1 we have outlined the list of objectives for the practical part of this thesis.
According to the first objective, our intention is to revisit valuation results of [Blackham,
2004]. Therefore, we choose to price a chooser option and a bermudan swaption
products in the setting specified in [Blackham, 2004]. Even though real-world LMM
derivatives are contingent on a larger number of forward rates, it has been decided to
constrain the scope of work by looking into two and three rate cases.
As a starting point for each case, the convergence behaviour of the price is studied
by means of Monte Carlo simulation. The following step involves implementation
of a finite difference solver to obtain a PDE solution on a full mesh of points, while
earlier Monte Carlo results serve as a useful benchmark. Convergence of a sparse grid
solution, obtained with the combination technique of Chapter 5, is then compared with
a full grid approximation. As required by our original motivation, further tests include
solving for different strike and spot rates and taking a closer look at the ’Greeks’ (∆
and V).
Finally, we address the case of a discontinuous payoff and present results for valuation
and the ’Greeks’ of a 2D digital option.
Setup of Pricing Framework Before any valuation can take place, one needs to calibrate
the pricer code to the real market data. The same calibration dataset is used
as in [Blackham, 2004]. Movements in the underlying rates are restricted to the domain
1 of [0.0, 0.15] d . Volatility term structure is assumed flat, meaning the value of an
instantaneous volatility function remains constant throughout the tenor period. Oneyear
LIBOR forward rates and corresponding caplet volatilities are provided in (6.2).
The amount of cash that underlies gains or losses in interest rate movements (a notional
amount) is chosen to be 10000. The correlation matrix is given by formula (3.32). We
assume strong correlation between the neighbouring rates, which can be achieved by
setting β = 0.1. The following two tables give a summary of our setup:
1 unless stated otherwise
43

Currency EUR
Tenor size α i 1 year
Strike K 5.5%
Volatility Constant
Correlation parameter β 0.1
Forward rate range L min − L max 0%-15%
Notional amount 10000 e
Table 6.1: Setup of LMM. Input data for LMM
Start date End date Forward LIBOR Rate Volatility %
T 0 29.07.04 29.07.05 2.423306 0
T 1 29.07.05 29.07.06 3.281384 24.73
T 2 29.07.06 29.07.07 3.931690 22.45
T 3 29.07.07 29.07.08 4.364818 19.36
T 4 29.07.08 29.07.09 4.680236 17.43
T 5 29.07.09 29.07.10 4.933085 16.15
T 6 29.07.10 29.07.11 5.135066 15.02
T 7 29.07.11 29.07.12 5.273314 14.24
Table 6.2: Setup of LMM. LIBOR rates and caplet volatilies
6.1 Two Dimensional Case. 2D Chooser Option
A chooser option is a European-style derivative product, whose payoff is determined
by (3.28). A two-rate chooser payoff follows:
V Cho2D
3 = α 2 (max(L 1 , L 2 ) − K) +
All action takes place on a T 0 , ..., T 3 subset of a tenor structure. The relevant discount
bond prices can be calculated according to (3.4).
A few words need to be said about imposing boundary conditions on 2D solution
space for FD. Two of the four boundaries, corresponding to maximum values in underlying
rates, are always fixed to the maximum possible payoff value: L max − K. As
of the remaining boundaries, with one of the rates being at its minimum L min , there
are two possible scenarios. In the first case, solution can be considered fixed to the
intrinsic value of the option. The second case proceeds by solving a lower dimensional
problem with respect to the non-L min rate and current time step. This solution is conveniently
given by the analytical Black formula. Even though the second approach
seems more logical intuitively, experience so far shows that accuracy of final solutions
wasn’t significantly affected by preference of one method over another. In order to
remain confident in this fact, we continued to use them interchangably.
It was convenient to start with an out-of-the-money setting (strike of 5.5% and
time-zero forward rates as provided in Table (6.2)), since there are results available for
this specific case in [Blackham, 2004].
44

6.1.1 Convergence of MC, Full and Sparse Grid solutions.
MC convergence and its standard error are shown in Figure (6.1). Simulation was
performed for the maximum of 4, 000, 000 trials, reaching minimum standard error of
0.022. FD solution for full grid discretization was computed for up to discretization
level L = 9. Figure (6.1) shows convincing convergence behavior of simulated solution
Level of Discretization
6 6.5 7 7.5 8 8.5 9
10.2
MC Simulation Value+Std.Error
Full-Grid FD value
10.15
10.1
Option price
10.05
10
9.95
9.9
9.85
500000 1e+006 1.5e+006 2e+006 2.5e+006 3e+006 3.5e+006 4e+006
Number of MC simulations
Figure 6.1: 2D Chooser Option. MC vs. FD Convergence.
and full-grid FD, providing accurate reference values for further tests. For our sparse
approximation experiments we have used a classical sparse grid (a.k.a ”maximumnorm-based”
type) and a basic combination technique, both described in the previous
chapter. Table (6.3) and Figure (6.2) give idea of FD convergence on full and sparse
grids 2 :
Level
Full grid Execution Sparse Grid Execution Full grid Sparse grid
solution time(sec.) solution time(sec.) points count points count
5 10.5969 1.5 11.351 0.36 1089 257
6 10.1755 6.6 11.477 0.88 4225 577
7 10.0964 30.1 9.8731 2.16 16641 1281
8 10.0762 176.4 10.005 5.18 66049 2817
9 10.0762 651.4 10.165 12.89 263169 6145
10 10.153 31.64 1050625 13313
11 9.998 74.86 4198401 28673
Table 6.3: 2D Chooser. Convergence values for full and sparse grid solutions.
2 The execution time for sparse grid technique does not include interpolation procedure
45

11.8
11.6
11.4
Black-Scholes Boundary
Fixed Boundary
Full Grid, Level 9 Value
11.2
Option price
11
10.8
10.6
10.4
10.2
10
9.8
6 7 8 9 10 11
Discretization Level
Figure 6.2: 2D Chooser. Convergence of Full and Sparse Grid.
Level
L 1 =L 2 =5.5%, K=5.5% K = 3.9% L 1 =L 2 =7.5%, K=5.5%
Full Sparse Full Sparse Full Sparse
5 83.669 71.9602 49.325 37.3126 251.755 262.1945
6 82.947 83.0537 48.869 37.9287 251.535 258.5324
7 82.547 81.1169 48.818 41.3625 251.449 253.7469
8 82.551 79.0752 48.790 46.9370 251.438 253.1529
9 82.4895 48.0288 251.9835
10 82.5755 48.0717 251.8426
MC
Price Std.error Price Std.error Price Std.error
82.399 0.01298 48.653 0.0089 251.468 0.0218
Table 6.4: 2D Chooser. At-the-money and in-the-money option values.
Level
L 1 =L 2 =2%, K=5.5% L 1 =L 2 =2%, K=2%
Full Sparse Full Sparse
5 0.0572 0.5587 32.7686 23.4880
6 0.0269 0.0239 31.2000 28.7051
7 0.0202 0.0701 30.1360 31.3129
8 0.0193 0.1163 30.1523 28.5730
9 0.0383 28.5550
10 0.0246 29.8075
MC
Price Std.error Price Std.error
0.0194 0.000153 30.0825 0.00472
Table 6.5: 2D Chooser. Deep out-of-the-money and at-the-money with low strike.
In addition to the out-of-the money case from [Blackham, 2004], we were interested
in the convergence of sparse solution in case of different strike prices and different
values of spot forward rates. The results are presented in Tables (6.4) and (6.5), where
MC prices have been calculated for 500000 simulation trials.
46

Full Solution. Level 6
Option Price
900
800
700
600
500
400
300
200
100
0
900
800
700
600
500
400
300
0 100
200
0.16
0.14
0.12
0.1
0.08
0.06
Forward Rate L2 0.04
0.02
0
0
0.120.140.16
Forward Rate L1
0.020.040.060.080.1
(a) Full Grid Surface.
Sparse Grid. Level 6
Option Price
900
800
700
600
500
400
300
200
100
0
900
800
700
600
500
400
300
200
100
0
0.16
0.14
0.12
0.1
0.08
0.06
Forward Rate L2 0.04
0.02
0
0
0.120.140.16
Forward Rate L1
0.020.040.060.080.1
(b) Sparse Grid Surface.
Figure 6.3: 2D Chooser. Solution Surface for level 6.
47

900
800
Full Solution
Sparse Solution
450
400
Full Solution
Sparse Solution
700
350
600
300
Option value
500
400
Option value
250
200
300
150
200
100
100
50
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
(a) Option Values for L 2 =3.75%
(b) Plot (a) zoomed in.
900
800
Full Solution
Sparse Solution
450
400
Full Solution
Sparse Solution
700
350
600
300
Option value
500
400
Option value
250
200
300
150
200
100
100
50
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
(c) Option Values for L 2 =7.5%
(d) Plot (c) zoomed in.
Figure 6.4: 2D Chooser. Sparse vs. full solution. Slices.
The Figure (6.4) above shows how Level 6 sparse-grid approximated solution corresponds to
the full grid counterpart as we freeze one of the forward rates.
48

6.1.2 Discussion: Convergence of Pricing.
Results from Tables (6.3), (6.4) and (6.5) prove that the use of BiCGSTAB+ILU iterative
solver with second order FD method manages to produce promising results in
case of sparse discretization. In addition to impressive improvement in speed, the “per
grid point“ convergence of sparse grid PDE solution is by far better than that of a full
grid. This feature of sparse grid technique proves to be crucial when approximating
functions in higher dimensional spaces.
In order to gain a better insight on accuracy of the solution we have plotted the
slices of solution surface, where full and sparse approximations were compared against
each other. Figure (6.4) presents two of such slices, corresponding to L 2 = 7.5% and
L 2 = 3.75% fixings. On a broader range both plots show relatively strong agreement
of both full and sparse approximations. Yet, as we zoom closer, the plot(d) exhibits
inaccuracies that form a distinguishable “wave” pattern, and less so plot(b).
The origin of this pattern is better illustrated by Figure (6.5), that provides an insight
into the absolute error distribution, characteristic of a sparse grid solution on level 8.
It shows that the largest differences between solutions appear along the center of the
domain and in the region of discontinuities of initial boundary conditions.
Error Profile. Level 8
25
20
15
10
-10
-505
-15
Error
25
20
15
10
5
0
-5
-10
-15
0.14
0.12
0.1
0.08
0.06
Forward Rate L2 0.04
0.02
0.14
0.12
0.1
0.08
0.06
0.04 Forward Rate L1
0.02
Figure 6.5: 2D Chooser. Absolute error surface on level 8.
Since all grids of sparse discretization are treated independently by our iterative
solver, a discretization error is to be introduced while solving subproblems on ”narrower”
and borderline grids of sparse space (as in Figure (5.2)), where spatial step h i
in one dimension is significantly larger than in the other. In the process of combination,
some points remain less-influenced or unaffected at all by addition and subtraction
of “sparser” grids, and therefore introducing “spikes” into solution. For instance, the
volatile center of the solution is such due to the points contributed by the “narrowest”
49

grids with the largest multi-index value.
6.1.3 Greeks: Delta
In Chapter 2, we have discussed setting up a hedge portfolio Π t . It has been made
clear that finding a process ∆ t for holdings in an underlying is vital for being able to
replicate an option payoff. Derivatives in LIBOR Market Model, in this sense, belong
to a “multi-asset” type of products. Thus, with LIBOR rates as underlying assets, we
can denote:
∆ i = ∂U
∂L i
(6.1)
where ∆ i is option’s sensitivity to the change in a i-th forward rate. Has the final solution
been obtained, finding deltas with respect to one of the forward rates is straightforward.
With full-grid Finite Differences, delta values can be extracted directly from
the final mesh of points in the form of first derivatives. In case of sparse grids, it
is not possible to apply the same procedure to the combination result, due to the its
non-uniformity. Instead, we can approximate delta surface by recomputing the solution
with a small shift in one of the underlyings. The Figure (6.6) below presents the
surface of delta solution with respect to the forward rate L 1 .
Interpolated Delta surface
Delta Value
1.2
0.8 1
0.6
0.4
0.2
-0.2 0
0.14
0.12
0.1
0.08
0.06
Forward Rate L1
0.04
0.02
0.02
0.04
0.06
0.08
0.1
0.12
0.14
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
Forward Rate L2
Figure 6.6: 2D Chooser. Sparse delta surface on level 7.
50

1
Full Delta Value
Sparse Delta Value
1
Full Delta Value
Sparse Delta Value
0.8
0.8
Delta Value for L2 = 3.2%
0.6
0.4
Delta Value for L2 = 4.6%
0.6
0.4
0.2
0.2
0
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
(a) Delta values for L2=3.2%
(b) Delta values for L2=4.6%
1
Full Delta Value
Sparse Delta Value
1
Full Delta Value
Sparse Delta Value
0.8
0.8
Delta Value for L2 = 5.6%
0.6
0.4
Delta Value for L2 = 6.7%
0.6
0.4
0.2
0.2
0
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
(c) Delta values for L2=5.6%
(d) Delta values for L2=6.7%
1
0.9
Full Delta Value
Sparse Delta Value
1
Full Delta Value
Sparse Delta Value
Delta Value for L2 = 7.5%
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Delta Value for L2 = 11.2%
0.8
0.6
0.4
0.2
0
-0.1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
(e) Delta values for L2=7.5%
(f) Delta values for L2=11.2%
Figure 6.7: 2D Chooser. Full vs. sparse delta profile on level 7
51

Remark. The way we impose boundary conditions, that is keeping them fixed or
setting them to Black-Scholes value, does make a difference when we look at extracted
Delta profile. Yet, in both cases, the approximation is still not accurate enough to
provide smooth surfaces on the Delta plot. Taking advantage of the fact that people
are mostly interested in the ∆’s in at-the-money or close to at-the-money states, the
disturbed boundaries were excluded from the plots above.
6.1.4 Greeks: Vega
∆ is just one of the several sensitivities that help explore exposure to risk in the short
position as a function of different model parameters. Other ’Greek letters’ include Θ
with respect to time changes and Γ as the second derivative with respect to changes in
underlying. In this part of the text we turn to vega, V, a sensitivity of a product with
respect to the volatility in one of the underlying forward rates:
V = ∂U
∂σ i
(6.2)
Information about V is not encapsulated in the solution explicitly. Here, whether full
and sparse, the solution has to be recomputed with a small shift in volatility value of
one of the underlying forward rates. Figure (6.8) below shows the sparse solutions
for vega with respect to the σ 1 on Level 8. Plots in Figure (6.9) illustrate slices of a
resulting vega profile with fixed L 2 rate.
Sparse Vega points. Level 8
Vega Value
0.03
0.025
0.015
0.02
0.005
0.01
-0.005 0
-0.01
0
0.02 0.04 0.06 0.08
0.1 0.12 0.14 0.16
0
Forward Rate L1
0.04
0.02
0.16
0.14
0.12
0.03
0.025
0.02
0.015
0.01
0.005
0
-0.005
-0.01
0.1
0.08
Forward Rate L2
0.06
Figure 6.8: 2D Chooser. Sparse vega surface on level 8.
52

0.03
0.025
Full vega Value
Sparse vega Value
0.03
0.025
Full vega Value
Sparse vega Value
0.02
0.02
vega Value for L2 = 3.2%
0.015
0.01
0.005
0
vega Value for L2 = 4.6%
0.015
0.01
0.005
0
-0.005
-0.005
-0.01
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
-0.01
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
(a) Vega values for L2=3.2%
(b) Vega values for L2=4.6%
0.03
0.025
Full vega Value
Sparse vega Value
0.03
0.025
Full vega Value
Sparse vega Value
0.02
0.02
vega Value for L2 = 5.6%
0.015
0.01
0.005
0
vega Value for L2 = 6.7%
0.015
0.01
0.005
0
-0.005
-0.005
-0.01
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
-0.01
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
(c) Vega values for L2=5.6%
(d) Vega values for L2=6.7%
0.03
0.025
Full vega Value
Sparse vega Value
0.03
0.025
Full vega Value
Sparse vega Value
0.02
0.02
vega Value for L2 = 7.5%
0.015
0.01
0.005
0
vega Value for L2 = 11.2%
0.015
0.01
0.005
0
-0.005
-0.005
-0.01
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
-0.01
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L1
(e) Vega values for L2=7.5%
(f) Vega values for L2=11.2%
Figure 6.9: 2D Chooser. Full vs. sparse vega slices on level 8.
53

Naturally, the characteristic feature of ’spikes’ of a sparse combination solution is
to be expected in the higher dimensional problems, where the contribution of coarse
grids is bigger. The following section summarizes results for 3D Bermudan Swaption
product.
6.2 Three Dimensional Case. 3D Bermudan Swaption
As discussed in Section (3.3), a Bermudan swaption is an interest rate derivative with
the ’early-exercise’ feature. It allows the holder to enter into a swap contract on a set
of prescribed dates and its general payoff is given by (3.27). Since we are computing
the price of a swaption in the framework of a LIBOR Market Model, the terminal bond
price D(t, T N+1 ) will play a role of a numeraire asset and Q will be its associated
martingale measure. In case of 3 underlying rates, one can engage in a swap at the
following set of fixing dates (T 0 ,T 1 ,T 2 ,T 3 ). Denote Vt
BSw (T t , . . . , T N : T N+1 ) to be
a Bermudan Swaption value at time t with expiry time T N+1 and set of fixing dates
(T t , . . . , T N ). Then, its value at T 0 follows:
V BSw
0 (T 0 , T 1 , T 2 , T 3 : T 4 ) = max ( H 0 (T 0 ), E 0 (T 0 ) ) (6.3)
with a hold value H 0 (T 0 ) recursively defined from T 3 :
[ (
max
H 2 (T 2 ) = D(2, 4) E Q H3 (T 3 ), E 3 (T 3 ) )
]
D(3, 4) ∣ F 2 = E Q[ ]
(L 3 − K) + |F 2
[ (
max
H 1 (T 1 ) = D(1, 4) E Q H2 (T 2 ), E 2 (T 2 ) )
]
D(2, 4) ∣ F 1
[ (
max
H 0 (T 0 ) = D(0, 4) E Q H1 (T 1 ), E 1 (T 1 ) )
]
D(1, 4) ∣ F 0
where the first relation holds since:
E 3 (T 3 ) = D(3, 4) (L 3 − K) +
and H 3 (T 3 ) = 0
(6.4)
Figure (6.10) shows solution surfaces for 3D Bermudan Swaption with L 1 =5.62% fixing:
54

L1=5.62% Sparse Solution
Swaption Price
1800
1600
1400
1200
1000
800
600
400
200
0
1800
1600
1400
1200
1000
800
600
0 200
400
0.14
0.12
0.1
0.08
Forward Rate L3 0.06
0.04
0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Forward Rate L2
(a) Full Solution Surface for L 1 =5.62%
L1=5.62% Sparse Solution
Swaption Price
1800
1600
1400
1200
1000
800
600
400
200
0
1800
1600
1400
1200
1000
800
600
400
200
0
0.14
0.12
0.1
0.08
Forward Rate L3 0.06
0.04
0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Forward Rate L2
(b) Sparse Solution Surface for L 1 =5.62%
Figure 6.10: 3D Bermudan Swaption. Full and Sparse solution surfaces.
55

6.2.1 Convergence Study
Table (6.6) shows steady convergence of full grid values. On the other hand, convergence
of a sparse grid solution is not very smooth. However, it soon becomes clear
that such convergence behaviour should be attributed to a large interpolation error.
Level
Full Exec. Time Sparse Exec. Time Full point Sparse point
solution (in sec.) solution (in sec.) count count
4 31.639 11.94 4913 593
5 29.802 121.11 39.691 1.51 35937 1505
6 29.212 1107.16 38.494 5.27 274625 3713
7 33.341 17.48 2146689 8961
8 36.259 54.39 16974593 21249
9 32.532 158.14 135005697 49665
Table 6.6: 3D Bermudan Swaption. Spot Rates as Table (6.2)
Figure (6.11) shows the discrepancy
between the calculated price and
the benchmark provided in [Blackham,
2004]. 500000 simulation
runs of MC showed the value of
29.064 with standard error 0.133.
As suggested in [Andersen, 1999],
numeraire-rebased swap value has
been chosen as the explanatory variable
for regression. Conditional expectations
were fitted with 4th-degree
Legendre polynomial basis function.
Tables (6.7) and (6.8) below show
good convergence of on-grid points
for a set of different at-the-money and
out-of-the-money forward rates.
Option price
38
37
36
35
34
33
32
31
30
29
Level of Discretization
3 3.5 4 4.5 5 5.5 6
MC Simulation Value+Std.Error
Full-Grid FD value
Full-Grid J.Blackham value
28
100000 150000 200000 250000 300000 350000 400000 450000 500000
Number of MC simulations
Figure 6.11: 3D Bermudan Swaption. Benchmarking
against [Blackham, 2004] and Monte
Carlo.
Level
L 2 =5.62% L 2 =1.875% L 2 =3.75%
L 3 =3.75% L 3 =3.75% L 3 =2.81%
Full Sparse Full Sparse Full Sparse
4 51.335 8.189 4.684
5 51.828 50.361 8.243 10.985 4.337
6 51.897 51.694 8.232 9.472 4.246 9.546
7 52.605 8.772 4.823
8 51.499 8.272 4.348
9 51.711 8.107 4.245
MC
Price Std.error Price Std.error Price Std.error
52.2593 0.2053 8.3734 0.0601 4.294 0.0493
Table 6.7: 3D Bermudan Swaption. Full and sparse on-grid solution values for
L 1 =3.75%. 500000 MC simulation trials.
56

Level
L 2 =1.87% L 2 =1.875% L 2 =5.62%
L 3 =7.5% L 3 =3.75% L 3 =5.62%
Full Sparse Full Sparse Full Sparse
4 196.542 10.945 176.422
5 196.766 204.006 10.903 178.124
6 196.628 207.901 10.875 13.080 178.474
7 194.337 11.263 165.244
8 196.054 11.067 178.232
9 196.378 10.796 177.958
MC
Price Std.error Price Std.error Price Std.error
196.514 0.337 11.062 0.075 177.208 0.383
Table 6.8: 3D Bermudan Swaption. Full and sparse on-grid solution values for
L 1 =5.62%. 500000 MC simulation trials.
400
350
Full Solution L6
Sparse Solution L6
Sparse Solution L8
L1=3.75% L2=3.75%
200
L1=3.75% L2=3.75%
Full Solution L6
Sparse Solution L6
Sparse Solution L8
300
150
Option value
250
200
150
Option value
100
100
50
50
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
(a) L 1 and L 2 out-of-the-money
(b) Plot (a) Zoomed.
400
350
Full Solution L6
Sparse Solution L6
Sparse Solution L8
L1=5.62% L2=5.62%
200
L1=5.62% L2=5.62%
Full Solution L6
Sparse Solution L6
Sparse Solution L8
300
150
Option value
250
200
150
Option value
100
100
50
50
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
(c) L 1 and L 2 in-the-money
(d) Plot (b) Zoomed.
Figure 6.12: 3D Bermudan Swaption. Solution slices corresponding to fixings in L 1
and L 2 .
57

6.2.2 Greeks. Delta and Vega.
In this section we provide some results and comments on the results for the delta and
vega of the 3D Bermudan Swaption. Figure (6.13) shows the corresponding surface
plots in case of a full grid approximation.
L1=5% Full Delta L6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.16
0.14
0.12
0.1
0.08
Forward Rate L3 0.06
0.04
0.02
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Forward Rate L2
(a) Delta
L1=5.62% Full Vega Solution
0.04
0.03
0.02
0.01
-0.01 0
-0.02
-0.03
0.14
0.12 0.1 0.08 0.06
Forward Rate L3
0.04
0.02
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0.04
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
Forward Rate L2
(b) Vega
Figure 6.13: 3D Bermudan Swaption. Delta and Vega Surfaces for L 1 =5.62%. Full
grid.
58

Figures (6.14) and (6.15) show the slices of a sparse grid delta and vega profiles, respectively,
plotted against a full grid solution. Basis point (bp) is equal to 0.01%.
1.4
L1=5% L2=1.25%
1.4
L1=5% L2=1.25%
1.2
1.2
1
1
Delta value
0.8
0.6
Delta value
0.8
0.6
0.4
0.4
0.2
0.2
Full Delta L6
Sparse Delta L8. dL3 = 7bp
Sparse Delta L8. dL3 =2bp
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
0
Full Delta L6
Sparse Delta L10. dL = 2bp
-0.2
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
(a) Full L6 vs. Sparse L8. L 2 = 1.25%
(b) Full L6 vs. Sparse L10. L 2 = 1.25%
1.4
L1=L2=5%
1.4
L1=5% L2=5%
1.2
1.2
1
1
Delta value
0.8
0.6
Delta value
0.8
0.6
0.4
0.4
0.2
Full Delta L6
Sparse Delta L6. dL3 = 30bp
Sparse Delta L6. dL3 = 7bp
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
0.2
Full Delta L6
Sparse Delta L8. dL8 = 7bp
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
(c) Full L6 vs. Sparse L6. L 2 = 5%
(d) Full L6 vs. Sparse L8. L 2 = 5%
1.4
L1=5% L2=5.62%
1.4
L1=5% L2=5.62%
1.2
1.2
1
1
Delta value
0.8
0.6
Delta value
0.8
0.6
0.4
0.4
0.2
Full Delta L6
Sparse Delta L8. dL3 = 7bp
Sparse Delta L8. dL3 = 2bp
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
0.2
Full Delta L6
Sparse Delta L10. dL10 = 2bp
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
(e) Full L6 vs. Sparse L8. L 2 = 5.62%
(f) Full L6 vs. Sparse L10. L 2 = 5.62%
Figure 6.14: 3D Bermudan Swaption. L 3 -Delta. Slices corresponding to a fixing in L 1 = 5%
59

0.035
0.03
L1=3.75% L2=3.75% Shift in L3 = 10bp
Full Vega L6
Sparse Vega L7
Sparse Vega L9
0.035
0.03
L1=3.75% L2=3.75% Shift in L3 = 0.1bp
Full Vega L6
Sparse Vega L7
Sparse Vega L9
0.025
0.025
Vega value
0.02
0.015
0.01
Vega value
0.02
0.015
0.01
0.005
0.005
0
0
-0.005
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
-0.005
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
(a) L 2 = 3.75% Step in σ 3 = 10bp
(b) L 2 = 3.75% Step in σ 3 = 0.1bp
0.04
0.03
L1=5.62% L2=5.62% Shift in L3 = 10bp
Full Vega L6
Sparse Vega L7
Sparse Vega L9
0.04
0.03
L1=5.62% L2=5.62% Shift in L3 = 0.1bp
Full Vega L6
Sparse Vega L7
Sparse Vega L9
0.02
0.02
Vega value
0.01
Vega value
0.01
0
0
-0.01
-0.01
-0.02
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
-0.02
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
(c) L 2 = 5.62% Step in σ 3 = 10bp
(d) L 2 = 5.62% Step in σ 3 = 0.1bp
Figure 6.15: 3D Bermudan Swaption. L 3 -Vega. Slices corresponding to a fixing in L 1 = 5.62%
Remarks. One can make several observations when looking at the plots above. First,
sparse solution for the greeks has proved to be stable with respect to the size of the shift
in the underlying and volatility step. Second, despite the clear presence of well-defined
’wiggles’, sparse grid approximation is capable of outlining the shape of the delta and vega
slopes.
60

6.3 Discontinuous Payoff. 2D Digital option.
The third and final task case that we have considered is the case of a discontinuous
payoff, characterictic of so-called digital options. Figures (6.16) and (6.5) show the
digital payoff that describes our specific product:
{
V Digital 0.02 L1 > K OR L
(T ) =
2 > K
0 otherwise
(6.5)
With full grid of level 6 and K=5.5%, such payoff can be demonstrated visually:
Discontinuous Payoff
0.02
0.015
0.01
0.005
0
0.16
0.14
0.12
0.1
0.08
Forward Rate L2 0.06
0.04
0.02
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Forward Rate L1
0.02
0.015
0.01
0.005
0
Figure 6.16: 2D Digital Option. Terminal payoff discretized on a L6 full grid.
The stripe of different color in the corner hints that in such setting the strike rate K
is an off-grid point. Indeed, in case of level 6 discretization, K falls roughly between
values 0.05391 and 0.05625.
6.3.1 Valuation Results.
Level of Discretization
7 7.5 8 8.5 9 9.5 10
21.2
MC Simulation Value+Std.Error
Full-Grid FD value
21
20.8
Option price
20.6
20.4
20.2
20
19.8
19.6
500000 1e+006 1.5e+006 2e+006 2.5e+006 3e+006
Number of MC simulations
Figure 6.17: 2D Digital Option. Convergence of Monte Carlo and full grid FD.
61

Figure (6.17) above shows Monte Carlo and Full grid FD solutions for increasing number
of simulation trials and different levels of discretization, respectively. The FD convergence
requires discretisation level 10.
Plots in Figure (6.18) illustrate the solution surfaces for full and sparse grid discretizations:
2D Digital Solution.
2D Digital Solution. Sparse Grid.
200
180
160
140
120
100
80
60
40
20
0
200
180
160
140
120
100
80
60
40
20
0
200
180
160
140
120
100
80
60
40
20
0
200
180
160
140
120
100
80
60
40
20
0
0.14
0.12
0.1
0.08
0.06
Forward Rate L2 0.04
0.02
0.14
0.12
0.1
0.08
0.06
0.04
Forward Rate L1
0.02
0.14
0.12
0.1
0.08
Forward Rate L2
0.06
0.04
0.02
0.14
0.12
0.1
0.08
0.06Forward Rate L1
0.04
0.02
(a) Full Solution
(b) Sparse Solution
Figure 6.18: 2D Digital Option. Full and sparse grid solution surfaces. Level 7.
The bumps on the sparse solution surface and oscillations on the boundaries reveal the
presence of ’spikes’. Still, Tables (6.9) and (6.10) show relatively satisfactory convergence
in case of on-grid points:
Level
L 1 =1.87% L 1 =1.87% L 1 =3.75% L 1 =3.75%
L 2 =3.75% L 2 =5.62% L 2 =3.75% L 2 =4.68%
Full Sparse Full Sparse Full Sparse Full Sparse
5 17.092 23.628 89.300 130.156 22.065 29.124 51.215 51.094
6 15.436 22.188 84.271 97.389 19.716 23.805 47.414 54.958
7 16.490 17.813 86.700 85.402 21.138 20.356 49.646 50.124
8 16.015 15.403 85.473 82.088 20.481 21.968 48.617 49.894
9 15.776 16.443 84.863 87.704 20.153 20.470 48.101 48.572
10 16.031 84.883 19.991 20.163 48.341
11 15.769 84.665 20.042 48.044
12 15.653 84.462 20.160 48.049
MC
Price Std.error Price Std.error Price Std.error Price Std.error
15.661 0.0793 84.758 0.141 20.013 0.0886 47.735 0.124
Table 6.9: 2D Digital. Full and sparse grid convergence of on-grid points. 500000 MC simulation trials.
62

Level
L 1 =3.75% L 1 =4.68% L 1 =5.62% L 1 =5.62% L 1 =5.62%
L 2 =5.62% L 2 =3.75% L 2 =1.87% L 2 =3.75% L 2 =5.62%
Full Sparse Full Sparse Full Sparse Full Sparse Full Sparse
5 89.456 93.811 48.683 48.707 94.433 144.257 95.101 105.22 117.35 158.660
6 84.664 85.297 44.155 53.857 87.891 108.154 88.791 92.075 111.236 137.341
7 87.192 82.942 46.758 47.944 91.012 91.672 91.977 87.894 114.009 123.701
8 85.965 88.521 45.535 47.775 89.430 86.233 90.408 93.333 112.555 115.810
9 85.352 85.157 44.925 45.498 88.644 89.739 89.625 91.783 111.835 113.733
10 85.229 45.319 88.988 89.533 113.023
11 85.055 44.913 88.480 89.292 112.396
12 85.340 44.921 88.171 89.592 111.965
MC
Price Std.error Price Std.error Price Std.error Price Std.error Price Std.error
84.601 0.141 44.492 0.122 87.453 0.1413 88.572 0.141 111.124 0.137
Table 6.10: 2D Digital Continued. Full and sparse grid convergence of on-grid points. 500000 MC simulation trials.
Remarks. From results presented above, it can be concluded that both 2D Digital full
grid and, more so, sparse grid have less distinct on-grid convergence than the cases of 2D
Chooser and 3D Bermudan. As mentioned before, our strike price happens to lie between
grid points, which is, in fact, a typical problem of a finite difference space discretization
relative to pricing of the digital payoff. Consequently, the discontinuity of the option value
is approximated with a large quantification error and the accuracy of a FD solution is deteriorated.
This issue, specific of a canonical spacial discretization, shows itself to a much
larger extent in case of sparse grid delta calculation.
63

6.3.2 Greeks. Delta and Vega.
Delta Figures (6.19) and (6.20) show the delta profile characteristic of a 2D digital
option and how full approximation compares with sparse grid representation.
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
0.02 0.04 0.06 0.08 0.1
Forward Rate L1 0.12 0.14
Forward Rate L2
0.02 0.040.060.080.10.120.14
Figure 6.19: 2D Digital Option. Full grid Delta on L7.
L1=4.68%
L1=4.68%
0.8
0.7
Full Delta L7
Sparse Delta L7
Sparse Delta L9
0.8
0.7
Full Delta L7
Sparse Delta L7
Sparse Delta L9
0.6
0.6
0.5
0.5
Delta value
0.4
0.3
Delta value
0.4
0.3
0.2
0.2
0.1
0.1
0
0
-0.1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
-0.1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L2
Forward Rate L2
(a) L 1 = 4.68%. Shift in L 1 = 10bp
(b) L 1 = 4.68%. Shift in L 1 = 1bp
L1=5.62%
L1=5.62%
0.6
0.5
Full Delta L7
Sparse Delta L7
Sparse Delta L9
0.6
0.5
Full Vega L7
Sparse Vega L7
Sparse Vega L9
0.4
0.4
Delta value
0.3
0.2
Vega value
0.3
0.2
0.1
0.1
0
0
-0.1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
-0.1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L3
Forward Rate L2
(c) L 1 = 5.62%. Shift in L 1 = 10bp
(d) L 1 = 5.62%. Shift in L 1 = 1bp
Figure 6.20: 2D Digital Option. Sparse Delta L7 and L9 slices.
64

Figures (6.21) and (6.22) show the vega profile characteristic of a 2D digital option
and how full approximation compares with sparse grid representation.
0.015
0.01
0.005
-0.005 0
-0.01
-0.015
-0.02
-0.025
0 0.02 0.04 0.06 0.08 0.1 0.12
Forward Rate L1 0.14 0.16
0.14
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
-0.02
-0.025
0.02
0.04
0.06
0.08
0.10.12
Forward Rate L2
Figure 6.21: 2D Digital Option. Full grid vega on L7.
L1=4.68%
L1=4.68%
0.014
0.012
Full Vega L7
Sparse Vega L7
Sparse Vega L9
0.014
0.012
Full Vega L7
Sparse Vega L7
Sparse Vega L9
0.01
0.01
0.008
0.008
Vega value
0.006
0.004
Vega value
0.006
0.004
0.002
0.002
0
0
-0.002
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
-0.002
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L2
Forward Rate L2
(a) L 1 = 4.68%. Shift in σ 1 = 10bp
(b) L 1 = 4.68%. Shift in σ 1 = 0.1bp
L1=5.62%
L1=5.62%
0.005
0
Full Vega L7
Sparse Vega L7
Sparse Vega L9
0.005
0
Full Vega L7
Sparse Vega L7
Sparse Vega L9
-0.005
-0.005
Vega value
-0.01
Vega value
-0.01
-0.015
-0.015
-0.02
-0.02
-0.025
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
-0.025
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Forward Rate L2
Forward Rate L2
(c) L 1 = 5.62%. Shift in σ 1 = 10bp
(d) L 1 = 5.62%. Shift in σ 1 = 0.1bp
Figure 6.22: 2D Digital Option. Sparse Vega L7 and L9 slices.
Remarks. As anticipated, sparse grid delta approximation appeared to be very demanding
to the size of the shift. It becomes obvious from Figure (6.20), plot(b), that
many of the less accurate solution points become irresponsive to the shift as large as
1%. When the step size is reduced to one-tenth of a percent, the differential of the value
with respect to step in the underlying becomes zero. Such behavior of the delta solution
is, again, the magnified effect of quantification error mentioned earlier. Discontinuity
65

of the payoff is also accountable for the severe spikes in sparse vega solution near the
at-the-money position.
6.4 Discussion and Recommendations.
In financial community, the quality of a computational method is, to a large extent,
determined by its ability to show accurate approximations of risk sensitivities, such as
∆ and V. We can conclude from ∆ and V slice plots of each test case that the impact of
the spikes effect is being magnified as we start operating on smaller ranges of values.
What didn’t show in quality of price estimates, shows distinctly in the values for the
Greeks. This fact makes immediate use of a sparse grid method in a discussed setting
quite problematic. Steps need to be taken to improve the accuracy of approximation.
There are several possible recipes one could follow:
• Iterative Solver. When benchmarking against the results from[Blackham, 2004],
the latter shows smoother convergence in both discretizations, and, moreover,
surprisingly close results on earlier levels. Most likely, such behaviour is inflicted
by the choice of an iterative solver. The author of [Blackham, 2004]
reports to have used second-order finite differences with a multigrid iterative
solver. The multigrid technique is designed to operate on a hierarchy of increasingly
“coarser” meshes where, after desired accuracy has been reached, every
individual mesh represents a part of the common solution.
However, despite smoother convergence, the author of [Blackham, 2004] reports
the presence of spikes in the solution. Thus, similar inaccuracies are to be expected
in the results for the Greeks.
• Order of FD. A possible way of with dealing discretization errors could be refining
spatial discretization to a fourth-order finite differences, as explored in
[Leentvaar and Oosterlee, 2006]. Again, such discretization poses difficulties
for ”narrow” grids and solutions adjacent to boundary conditions, that appear to
rely on ”ghost” outlier points. These issues can be addressed with application of
extrapolation or rewriting of PDE with one-sided differences on the borders of
solution domain.
• Combination technique and sparse grid type. It would be worthwhile investigating
the effect of the choice of an alternative sparse grid combination technique.
Some of the possible alternatives to the “classical” sparse grid and combination
technique are discussed in [Klimke, 2003] and [Leentvaar and Oosterlee,
2006], respectively.
• Quantification Error. In practice, the quantification error of discontinuous payoff
can be significantly reduced by peforming coordinate transformation. As described
in [Travella and Randal, 2000], by rewriting the PDE in different terms,
one can transform the grid in such a way that a larger amount of points is concentrated
in the discontinuity region. Since this procedure does not degrade the
accuracy of discretization, it could be an employed by the sparse grid method to
battle the quantification error in case of digital option.
66

Chapter 7
Conclusion and Future Work
In this thesis, we have investigated the application of sparse grid combination technique
to the 2D and 3D interest rate products in the LIBOR Market Model framework.
The selected test cases have been solved with second-order finite differences
and BiCGSTAB iterative solver.
In two dimensions, a chooser option and a digital option show good agreement
of MC, full and sparse solutions in case of on-grid points for higher levels of discretization.
For the chooser option, the price has converged to the value provided in
[Blackham, 2004]. In addition, we have included results for different sets of strike and
spot rates in case of on-grid solution points, where good convergence in both full and
sparse discretizations was recorded.
In three dimensions, the valuation of a bermudan swaption demonstrated convergence
behaviour on a full grid. However, the obtained FD and MC values were different
from the benchmark provided in [Blackham, 2004]. While convergence of a benchmark
problem solution on a sparse grid was not smooth due to a large interpolation
error, results for on-grid values were by far more convincing.
The accuracy of approximations of the ’Greeks’ has been shown to suffer seriously
from the ’spikes’, introduced into the final combination by approximations on
the coarser grids in both 2D and 3D. Moreover, the delta of the digital option becomes
a victim of a large quantification error.
Our suggestions for the future work are the following:
• refining discretization scheme of Finite Differences to the fourth order could help
to obtain more accurate solutions on the coarse grids of sparse discretization.
• it would be interesting to see how results of BiCGSTAB compare with those
of a different iterative solver. In our view, a study of different combinations
of discretization order and iterative solver could make a positive change on the
quality of the ’Greeks’.
• a study of the effect of grid stretching and coordinate transformation on the quality
of digital ’Greeks’ with sparse grid approach.
• combination technique is inherently well-fit for parallel computing. When solving
problems in higher than 3D, parallel computation of solutions on individual
grids will be required.
67

• reduction in the dimension of the PDE could be attempted by fast-drift approximation,
as in [Pietersz et al., 2005].
At the end we conclude that sparse grid combination technique is a promising
method to help revive Finite Difference approach in the interest-rate option pricing
field in higher dimensions.
68

Appendix A
Implementation Details
In this appendix, we present a brief list of software tools we employed throughout the
project:
1. C++, as the language of choice, allowed us to take advantage of the power of
templates, operator overloading and generic STL containers and algorithms.
2. Subgrids have been implemented on top of Blitz++ generic array library, described
in [Veldhuizen].
3. LMM PDE solver made use of the BiCGSTAB routine and supplimentary data
structures of the free-source iterative methods library, described in [Dongarra
et al.].
4. As in case of full grid, off-grid points of sparse grid solution need to be found
with interpolation. The current implementation of sparse grid solver doesn’t
provide a routine to interpolating non-uniform meshes, so this procedure has
been outsourced to Matlab. Interpolation of off-grid values has been peformed
as a post-processing step with griddata and griddata3 routines.
5. GnuPlot software came very handy in drawing pretty graphs and color-mapped
surfaces.
6. The text of the thesis has been typeset in L A TEX.
69

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