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

More documents

Recommendations

Info

$Learning LATEX by Doing$

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
Page 1 and 2: SPARSE GRID METHOD IN THE LIBOR MAR
Page 3 and 4: Acknowledgements This project would
Page 5 and 6: 6.3.1 Valuation Results. . . . . .
Page 7 and 8: 6.18 2D Digital Option. Full and sp
Page 9: Chapter 1 Introduction The modern f
Page 13 and 14: Risk-Neutral Pricing Another import
Page 15 and 16: S 0 u d u S 0 S u 2 0 Su 3 0 S du 2
Page 17 and 18: or log(S T ) d −→ log(S 0 ) +
Page 19 and 20: There is an important difference be
Page 21 and 22: Delta. In order to set up a riskfre
Page 23 and 24: Theorem 2.3.2. Let W P (t) = (W1 P
Page 25 and 26: Chapter 3 LIBOR Market Model Having
Page 27 and 28: The right-hand side of the equality
Page 29 and 30: At this point of the text all neces
Page 31 and 32: pays off at expiry. Conveniently, t
Page 33 and 34: 1. Tenor structure of N interest ra
Page 35 and 36: and weight factor parts from the co
Page 37 and 38: we can substitute into the equation
Page 39 and 40: with ɛ i being independent standar
Page 41 and 42: a good compromise on the size of ɛ
Page 43 and 44: Three different θ values represent
Page 45 and 46: Chapter 5 Sparse Grids In the previ
Page 47 and 48: T 0 , 0 T 0 , 1 T 0 , 2 T 0 , 3 |i|
Page 49 and 50: 00 03 02 03 01 03 02 03 00 30 30 20
Page 51 and 52: Chapter 6 Numerical Results. Valuat
Page 53 and 54: 6.1.1 Convergence of MC, Full and S
Page 55 and 56: Full Solution. Level 6 Option Price
Page 57 and 58: 6.1.2 Discussion: Convergence of Pr
Page 59 and 60: 1 Full Delta Value Sparse Delta Val
Page 61 and 62:
0.03 0.025 Full vega Value Sparse v
Page 63 and 64:
L1=5.62% Sparse Solution Swaption P
Page 65 and 66:
Level L 2 =1.87% L 2 =1.875% L 2 =5
Page 67 and 68:
Figures (6.14) and (6.15) show the
Page 69 and 70:
6.3 Discontinuous Payoff. 2D Digita
Page 71 and 72:
Level L 1 =3.75% L 1 =4.68% L 1 =5.
Page 73 and 74:
Figures (6.21) and (6.22) show the
Page 75 and 76:
Chapter 7 Conclusion and Future Wor
Page 77 and 78:
Appendix A Implementation Details I
Page 79:
A. Klimke. Piecewise multilinear sp
show all

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

Create successful ePaper yourself

Delete template?

Save as template?