MONTE CARLO METHODS FOR OPTION PRICING

Institute of Applied Mathematics (IAM)
METU
Term Project
MONTE CARLO
METHODS FOR OPTION
PRICING
Sibel KAPLAN
1509009
Submitted to: Coskun KÜÇÜKÖZMEN
June 2008
1

CONTENTS
1 Introduction
2 Background
2.1 Monte Carlo Integration …………………………………………………………...5
2.2 Generating Pseudorandom variates ………………………………………………..6
2.2.1 Inverse Transform Method ……………………………………………………..6
2.2.2 Acceptance-rejection method …………………………………………………..7
2.2.3 Ad hoc method …………………………………………………………………7
3 Variance Reduction Techniques
3.1 Antithetic sampling ………………………………………………………………..9
3.2 Common Random Numbers ……………………………………………………….9
3.3 Control variates …………………………………………………………………..10
3.4 Variance reduction by conditioning ……………………………………………...10
3.5 Stratified sampling ……………………………………………………………….11
3.6 Importance sampling ……………………………………………………………..11
4 Quasi Monte Carlo Methods
4.1 The Halton sequence ……………………………………………………………..12
4.2. The Sobol sequence ……………………………………………………………...12
5 Numerical Results
5.1 European Call Option …………………………………………………………….17
5.2 As-you-like-it (Chooser) Option …………………………………………………18
5.3 Call On The Maximum (Rainbow option) ……………………………………….19
5.4 Down-And-Out Put Option (Barrier options) ……………………………………21
6 Interpretations of Findings and Conclusions
References
2

ABSTRACT
Monte Carlo simulation has proved to be a valuable tool for estimating security prices for
which closed form solution does not exist. This project evaluates variance reduction
techniques and the Quasi-Monte Carlo methods that have attractive properties for the
numerical valuation of derivatives and examines the use of Monte Carlo simulation with low
discrepancy sequences for valuing derivatives versus the traditional Monte Carlo method
using pseudo-random sequences. The relative performance of the methods is evaluated based
on European call options and some types of exotic options.
1 INTRODUCTION
Simulation is the imitation of a real world process of system. In finance, a basic model for the
evolution of stock prices, interest rates, exchange rates, and other factors would be necessary
to determine a fair price of a derivative security. Simulations make assumptions about the
behavior of the system being modeled. This model requires inputs, called the parameters of
the model and outputs, a result that might measure the performance of a system. We usually
construct the model in such a way that inputs are easily changed over a given set of values.
This way allows for a more complete picture of the possible outcomes.
Simulation is used because it transfers work to the computer. We use a simulation to provide a
numerical answer to a question, assign a price to a given asset and also to verify the results
obtained from an analytic solution.
A Monte Carlo method is a technique that involves using random numbers and probability to
solve problems. The term Monte Carlo was coined by S. Ulam and Nicholas Metropolis in
reference to games of chance, a popular attraction in Monte Carlo, Monaco. It is accepted that
the basis of Monte Carlo Methods arisen with Buffon’s needle problem. In 1777 G.Comte de
Buffon made an experiment by drawing parallel strips with d distance on a floor. He put
needle with L length on this floor at random. He then calculate analytically the probability of
needle’s intersection with strips as p=2L/pd. p is the probability of intersection. He repeated
N time this experiment and count how many times the needle intersected with strips. n is the
number of intersections. He found that the proportion n/N is close to the analytic result p. The
proportion n/N converges to p as N gets greater. G.Comte de Buffon’s experiment contributes
much to the probability theory in the 20 th century.
The Monte Carlo method is just one of many methods for analyzing uncertainty propagation,
where the goal is to determine how random variation, lack of knowledge, or error affects the
sensitivity, performance, or reliability of the system that is being modeled. Monte Carlo
3

simulation is categorized as a sampling method because the inputs are randomly generated
from probability distributions to simulate the process of sampling from an actual population.
The Monte Carlo method was first introduced to the finance literature by Boyle (1977). Even
it can be applied to a wide range of complex problems, there are problems for which the
standard Monte Carlo approach is computationally burdensome. Therefore, various
techniques like variance reductions and quasi-Monte Carlo methods have been proposed to
speed up the convergence. Quasi-Monte Carlo methods use the low discrepancy sequences
that are dispersed throughout the domain of integration. Niederreiter (1992) provides a
comprehensive overview of low discrepancy sequences and their properties and shows that
low discrepancy sequences outperform standard Monte Carlo for high dimensions. However
Tan and Boyle (1997) indicate “…standard Monte Carlo is more efficient for higher
dimensional problems unless one uses a very large number of points.” Boyle, Broadie and
Glasserman (1996), Caflisch and Morokoff (1996), Ninomiva and Tezuka(1996) and Paskov
and Traub (1995) have used low discrepancy sequences in finance applications. The
consensus is that for most finance applications the superiority of the low discrepancy methods
over standard Monte Carlo methods persists for a much higher range of dimensions than is the
case for more general integrals.
The layout of the remainder of the paper is as follows. In the next section, I briefly review the
standard Monte Carlo approach. I discuss the variance reduction techniques in Section 3 and
low discrepancy sequences in Section 4. These variance reduction techniques and low
discrepancy sequences can be used to tackle problems in computational finance and I review
some results in Section 5. The MATLAB software is used for drawing figures in Section 4
and 5 and numerical calculations in Section 5. Then I conclude the research and results.
4

2 BACKGROUND
2.1 Monte Carlo Integration
Computing the definite integral of a function is deterministic. It involves no randomness.
However, it is made a stochastic problem by interpreting the integral as an expected value.
Consider an integral on the unit interval [0,1]:
I =
1
∫
0
g(
x)
dx
This integral may be thought as the expected value E[g(U)], where U is a uniform random
variable on the interval (0,1), i.e., U~(0,1). The expected value is estimated by a sample mean
that is a random variable. A sequence {Ui} of independent random samples from the uniform
distribution is generated and then the sample mean is calculated:
I
m
1
=
m
m
∑
i=
1
g ( U
i
)
The strong law of large numbers implies that, with probability 1, lim = I .
5
I m
m→∞
Random sampling, which is where Monte Carlo comes from, is not really possible with a
computer. However, a sequence of pseudo-random numbers can be generated using
generators.
Monte Carlo is the only viable option to compute a multidimensional integral. For example, if
an integral is like
∫
I = φ ( x)
dx
A
n
where A ⊂ R , we may estimate I by randomly sampling a sequence of points
x i
∈ A,
i = 1,...,
m,
and building the estimator
where (A)
m vol(
A)
I m = φ ( xi
)
m
∑
i=
1
vol denotes the volume of the region A . The ratio ( 1/
m) φ ( x ) estimates the
average value of the function and the integral is get by multiplying this ratio with volume of
the integration region that is usually taken as unit hypercube. Therefore, Monte Carlo is viable
solution to price exotic options that have multidimensional integral in pricing.
The price of a European style option is the expected value, under the risk-neutral measure, of
the discounted payoff of the option:
m
∑
i=
1
i

f = e
−rT
E[
fT
],
where f T is the payoff at the maturity date T and r is constant risk-free rate. If we assume
geometric Brownian motion, f T is calculated by generating a standard normal random
variable ε ~ N ( 0,
1)
:
f
T
= max{ 0,
S(
0)
e
2
( r−σ
2
) T + σ Tε
6
− K}.
Depending on the nature of the option at hand, the full sample paths or the terminal asset price
can be generated.
While there is no need to resort to Monte Carlo simulation to price a vanilla European style
option, we use it to compare with Black-Scholes formula.
2.2 Generating Pseudorandom variates
The generation of pseudorandom numbers are variates from the uniform distribution on the
interval (0,1). Then, suitable transformations are applied in order to obtain samples from the
desired distribution. I explain the inverse transform method, the acceptance-rejection
approach and ad hoc strategies for transformation.
The standard way to generate pseudorandom numbers is linear congruential generators
(LCGs). A LCG generates a sequence of non-negative integer numbers Z i , given an integer
number i−1
Z :
Zi i
= ( aZ − 1 + c)
mod m,
where a is the multiplier, c is the shift and m is the modulus. To generate a uniform variate on
the unit interval the number ( / m)
.
Z i
Actually, there is nothing random in the sequence generated by a LCG. LCG must start from
an initial number Z 0 that is called the seed of the sequence. I use MATLAB for numerical
part that sets the seed to a precise value.
2.2.1 Inverse Transform Method
The distribution function F( x)
= P{
X ≤ x}
is given to generate random variates. If F is
invertible, the following inverse transform method can be used:
1. Generate a random number U~U(0,1).
−1
2. Return X = F ( U ) .
−1
P { X ≤ x}
= P{
F ( U ) ≤ x}
= P{
U ≤ F ( x)}
= F ( x)

where F is monotonic and U is uniformly distributed.
2.2.2 Acceptance-rejection method
If it is difficult to invert a probability density f(x), the inverse transform method is
unattractive. At that point assume a function t(x) such that
t( x)
≥ f ( x)
∀ x ∈ I
The function t(x) is not a probability density function, however the related function r(x)=t(x)/c
is,
∫
c = t(
x)
dx
I
If the distribution r(x) is can be simulated, the following method is used to generate a random
variate X according to probability density f:
1. Generate Y~r
2. Generate U~U(0,1), independent of Y
3. If U ≤ f ( Y ) / t(
Y ) , return X=Y; otherwise reject and repeat the procedure.
2.2.3 Ad hoc method
The Box-Muller approach can be used for generation of normal variates. Let X,Y~N(0,1) be
two independent variables and ( R , θ)
be polar coordinates of the point of Cartesian
coordinates (X,Y) in the plane.
2 2
d = R = X +
The joint density of X and Y is
and
f ( x,
y)
=
1 1 −d
2
f ( d , θ ) = e .
2 2π
Y
1
e
2π
2
2
−x
2
1
tan Y −
θ =
1
e
2π
2
−y
2
7
X
1
= e
2π
2 2
−(
x + y )
2
Then, the Box-Muller algorithm may be conducted as follows:
1
= e
2π
1. Generate two independent uniform variates U U ~ U(
0,
1)
.
2
2. Set R = −2logU1
and θ = 2πU 2 .
3. Set X = Rcosθ
, Y = Rsin
θ .
1,
2
To avoid trigonometric function, Box-Muller approach is integrated with the rejection
approach:
1. U
U ~ U(
0,
1)
1,
2
−d
2

2. V = U −1,
V = U −1,
S = V + V
1
2 1
2
2 2
3. if S>1 return to step 1,
− 2InS
S
2
1
− 2InS
S
o.w X = V1
, Y = V2
I use inverse transformation of normal distribution and Box-Muller algorithm in numeric part.
The crude Monte Carlo, as explained, is to randomly sample from uniform distribution within
(0,1) interval and the average the values of ( i)
U g to obtain ) (
m 1
I m = g U i and lim I m = I .
m
m→∞
This average is an unbiased estimator of the integral and the variance of the estimator is
var(
I m
8
2
2
var( g(
U1))
) = . To improve the accuracy of estimation the number of simulation must
m
be increased. However, it is infeasible. One way to overcome this issue is to use variance
reduction techniques. Another one is to adopt a quasi-Monte Carlo approach.
∑
i=
1

3 VARIANCE REDUCTION TECHNIQUES
To improve the accuracy of estimation, the following variance reduction techniques work on
the numerator of
directly.
var(
I m
3.1 Antithetic sampling
var( g(
U1))
) = and reduce the variance of the samples X i = g(
U i)
m
The simplest method is antithetic sampling. Consider the sequence of paired replications
(X1 (i) , X2 (i) ), i=1,…,n:
X1 (1) X1 (2) … X1 (n)
X2 (1) X2 (2) … X2 (n)
i ( i)
( i)
These samples are horizontally independent. The pair-averaged samples X = ( X + X ) / 2
are independent. However, vertical independence is not required since for fixed i, X1 (i) and
X2 (i) may be dependent.
( i)
i)
( i)
( i)
( i)
Var(
X ) Var(
X1
) + Var(
X 2 ) + 2Cov(
X1
, X 2 ) Var(
X )
Var[
X ( n)]
= =
= ( 1+
ρ(
X 1,
X 2))
n
4n
2n
In order to reduce the variance of the sample mean, negatively correlated replications within
each pair can be taken. Hence, to induce a negative correlation, a random number sequence
{Uk} is used in the first replication in each pair, and then {1-Uk} in the second one.
3.2 Common Random Numbers
Suppose that we use Monte Carlo simulation to estimate a value depending on a parameter a
and try to estimate something like h( α ) = Ew[
f ( α;
w)]
by evaluating the sensitivity of this
value on the parameter a, dh(a)/da. The simplest idea is to simulate to estimate the value of
the finite difference,
of the difference
h( α + δα)
− h(
α )
δα
f ( α + δα;
w)
− f ( α;
w)
δα
for a small value of increment da. Actually, samples
is generated and its expected value is estimated.
Due to random noise and variation in the parameter, the expected value of the difference
between two random variables Z = X1
− X 2 is estimated where the expected values of these
two random variables are not equal. By Monte Carlo a sequence of independent samples
9
1
2

Z j X1,
j X 2,
j −
= is simulated. To improve the estimation, the variance of the samples should
be reduced:
Var( X j − X 2 j ) = Var(
X1
j ) + Var(
X 2 j ) − 2Cov(
X1
j , X 2
1 j
To reduced the variance of the samples, positive correlated samples must be induced. The
common random numbers technique is very similar to antithetic sampling and the same
monotonicity assumption is required.
3.3 Control variates
Let θ = E(X
) be estimated and there is another random variable Y, with a known expected
value ?, which is correlated to X. The variable Y is called the control variate.
X c
= X + c(
Y − v)
where c is a parameter we must chose.
E(
X ) =θ
c
2
Var(
X ) = Var(
X)
+ c Var(
Y)
+ 2cCov(
X,
Y)
c
The variance of the estimator can be reduced by a suitable choice of c,
c
*
Cov(
X , Y )
= −
Var(
Y )
*
Var(
X c )
= 1 − ρ
Var(
X )
2
xy
where ?xy is the correlation between X and Y. The sign of c depends on the sign of this
correlation.
The control variates technique is used when the analytical formula is not known for X.
3.4 Variance reduction by conditioning
When the expected value of a random variable X is be computed, it is useful to condition with
respect to another random variable Y by conditioning as following:
E [ X ] = E[
E[
X / Y ]]
Variances may be computed by conditioning, too:
Var ( X ) = E[
Var(
X / Y )] + Var(
E[
X / Y ])
This formula implies two consequences:
1_Var(
X ) ≥ E[
Var(
X / Y )]
2 _Var(
X ) ≥ Var(
E[
X / Y ])
Using the first inequality to reduce the variance of an estimator is variance reduction by
stratification explained following. The second one is variance reduction by conditioning.
10
)

3.5 Stratified sampling
Let estimate E[X] that X is dependent on the value of another random variable Y. Let Y to
take a finite set of values yj with known probability. P Y = y j}
= Pj
conditioning, it is seen that
= ∑
=
m
j 1
E[
X]
E[
X / Y = y ] .
j Pj
11
{ , j=1,…,m. Using
Therefore, it is stratified sampling to simulate to estimate the values E[X | Y=yj], for j=1,…,m
and use the formula above to put results together. This approach differs from variance
reduction by conditioning by sampling X with selecting a value for Y. In variance reduction
by conditioning, Y is sampled.
3.6 Importance sampling
Importance sampling approach is useful when simulating rare events of sampling from the
tails of a distribution. Consider the following estimation:
θ = E [ h(
X )] = h(
x)
f ( x)
dx
∫
where X is a random vector with joint density f(x). Let f(x)=0 whenever g(x)=0,
h(
x)
f ( x)
⎡h(
x)
f ( x)
⎤
*
θ = ∫ g(
x)
dx = Eg
= Eg
( h ( x))
g(
x)
⎢
g(
x)
⎥
⎣ ⎦
g must be chosen with care, otherwise, the method may backfire. Using the well-known
properties of the variance, it is obtained;
var [ h(
x)]
=
f
Δ var =
2
h ( x)
f ( x)
dx −θ
*
2 f ( x)
2
varg
[ h ( x)]
= ∫h
( x)
f ( x)
dx −θ
g(
x)
∫
∫
2 ⎡ f ( x)
⎤
h ( x)
⎢1
− f ( x)
dx
g(
x)
⎥
⎣ ⎦
From the last expression, the following selection must be done to reduce variance:
g(x)>f(x) when the term h 2 (x)f(x) is large
g(x)

4 QUASI-MONTE CARLO METHODS
The variance reduction techniques rely that random sampling is really random. However, the
random numbers generated by a LCG are not random at all. Therefore alternative
deterministic sequences of numbers that are obtained by the experience in Monte Carlo
simulation work well in generating samples. This ides may be made more precise by defining
the discrepancy of a sequence numbers. Low-discrepancy sequences are used to compute a
multidimensional integral on the unit hypercube. An alternative name for a low-discrepancy
sequence is quasi-random sequence where the name quasi-Monte Carlo comes from.
4.1 The Halton sequence
Halton (1960) generates low-discrepancy sequences based on the following algorithm:
• Representing an integer number n in a base b, where b is a prime number:
n=(…d4d3d2d1d0)b.
• Reflecting the digits and adding a radix point to obtain a number within the unit
interval:
h=(0. d0d1d2d3d4…)b.
the n th number in the Halton sequence with base b is formulated as:
h(
n,
b)
=
m
∑
k = 0
dkb
−(
k+
1)
For example, the number 13 with base 2 is;
• 13=1(8)+1(4)+0(2)+1(1)=1101
• Reverse the order of the digits; 1101 becomes 1011 and determine the number that this
is the binary decimal expansion for; 1011=1(1/2)+0(1/4)+1(1/8)+1(1/16)=11/16.
The one-dimensional sequence with base 2 is called Van der Corput. Halton sequences are
obtained in multiple dimensions, making sure different prime numbers are used for each base
which is associated to each dimension.
12

Fig. 1. Pseudorandom sample ((0,1)x(0,1)) in two
dimensions n=100
13
Fig. 2. Covering the bidimensional unit square with
Halton sequences base=2,3
Figure 1 is drawn by pseudorandom sample in two dimensions with 100 times simulation and
figure 2 is drawn by Halton sequences in two dimensions with bases 2 and 3 for 100 integers.
It could be argued from these figures that the covering of the Halton sequence is more even.
The result is more precise with more simulation as shown in figure 3 and 4.
Fig.3. Pseudorandom sample ((0,1)x(0,1)) in two
dimensions n=10000
Fig.4. Covering the bidimensional unit square with
Halton sequences base=2,3 n=10000

Fig.5. Covering the bidimensional unit square with
Halton sequences base=2,7
14
Fig.6. Covering the bidimensional unit square with
Halton sequences base=2,4(non-prime)
Fig.7. Covering the bidimensional unit square with Halton sequences base=14,15(non-prime)
The figures 5, 6 and 7 compare the prime bases with non-prime and larger bases. It results that
larger holes and gaps in non-prime bases figures. Poor coverage of the unit square is obtained
when large bases are used in Halton sequences.

4.2. The Sobol sequence
Sobol (1967) uses only base 2. In order to generate multidimensional sequences, the Van der
Corput sequence with base 2 is permuted by a mechanism linked to polynomials in a binary
arithmetic. Also Faure(1982) and Niederreiter (1987) generates sequences to overcome
problems of large prime numbers base. I will explain and use Sobol sequence in numerical
part.
Unlike the Halton sequence, the multidimensional Sobol sequence eliminates the problems
caused by large prime numbers by using the same base m=2. So, there is some computational
time advantage due the shorter cycle length. This project uses the faster algorithm of Antonov
and Saleev (1979).
The construction of a multidimensional Sobol sequence follows a three-step procedure.
Because the steps are identical for each dimension, the procedure for one dimension is
illustrated.
Step1:
Generate a set of odd integers mi, for i 1,2, ,log [ N]
0 2 i
< m < ,
i
= L , that satisfy the condition:
where N is the number number of price paths, and [ log 2 N ] is the smallest integer larger than
log N . [ ]
2
2
15
2
log N is the maximum number of digits in the expansion of N in base 2. To
generate the Sobol sequence, a primitive polynomial is needed for each dimension. Let the
polynomial be
P = x + ax + L + a x+
a ∈ {0,1} .
q q−1
1 q−1
1,
Our interest in the primitive polynomial is only in its coefficients. For each polynomial a set
of q initial odd integers mi is also required. Then, we generate the set of mi for all i > q using
the coefficients of the primitive polynomial and the recursive relationship:
Step2:
m = 2am ⊕2 am ⊕L⊕2 a m ⊕2 m ⊕m
2 q−1 q
i 1 i−1 2 i−2 q−1 i− q+ 1 i−q i−q Convert the odd integer mi into a binary fraction in the base 2 number system to gain the
direction number vi.
mi
vi= , i = 1,2 L ,log
i [ 2 N]
in base 2.
2
i
.

Step3:
Use the Antonov and Saleev recursive algorithm to calculate the n th Sobol number ∅ ( n)
:
∅ ( n) =∅( n−1) ⊕ vc () ,
where ∅ (0) = 0,
vc () is the c th direction number, and c is the rightmost zero-bit in the base 2
expansion of n − 1.
Fig.8. Bidimensional Sobol sequence n=100
Fig.10. Bidimensional Sobol sequence n=10000.
16
Fig.9. Bidimensional Sobol sequence n=1000
Figures 8, 9 and 10 show that Sobol sequences are good at covering the unit square. Sobol
sequences cover more than Halton sequences.

5 NUMERICAL RESULTS
The stock price has Geometric Brownian Motion path:
dS = μ S dt + σS
dZ
t
t
The simplest discretization approach, Euler scheme,
St + δt
= ( 1+
μδt)
St
+ σS
t
t
t
δtε
With the properties of standard Wiener process
St + δt
= St
exp(( μ −σ
2
/ 2)
δt
+ σ
17
δtε
where ε ~ N(
0,
1)
The following figure shows sample paths of stock price generated by crude Monte Carlo.
Fig.11. Sample paths generated by crude Monte Carlo
5.1 European Call Option
The price of a European style option is the expected value, under the risk-neutral measure, of
the discounted payoff of the option:
f = e
−rT
E[
fT
],
where f T is the payoff at the maturity date T and r is constant risk-free rate. If we assume
geometric Brownian motion, f T is calculated by generating a standard normal random
variable ε ~ N ( 0,
1)
:
f
T
= max{ 0,
S(
0)
e
2
( r−σ
2
) T + σ Tε
− K}.

Depending on the nature of the option at hand, the full sample paths or the terminal asset price
can be generated.
The following table shows the call option price calculated by different approach and the
results are compared with Black-Scholes option pricing. Halton sequences with huge
simulations and control variates technique give the closest result to B-S. Halton sequences
with non-prime base give indeed poor result.
S0=50 K=60 r=0.05 T=1 sigma=0.2
B-S 1.6237
Crude MC, n=1000 1.2562
Crude MC, n=1000000 1.6292
Halton&Box-Muller, n=1000, base=2 and 7 1.6271
Halton&Box-Muller, n=1000, base=2 and 4 2.3594 (poor result with non-prime
18
number)
Halton inverse n=1000 base=2 1.5747
Halton inverse n=1000000 base=2 1.6227
Sobol n=100 1.4407
Sobol n=1000 1.6697
Crude MC + antithetic n=1000 1.7076
Crude MC + antithetic n=1,000,000 1.6228
Crude MC + control variates pilot=1000, n=1000 1.6228
Table 1. Numerical results for a vanilla European call option
5.2 As-you-like-it (Chooser) Option
Also known as “as you like it” options, chooser options allow the owner to choose whether
the option is a put or a call at a specific date. The underlying options in chooser options are
usually both European and have the same strike price. More complex chooser options do not
have the same strike price or time to maturity. These options will be more expensive than the
individual underlying put or call options, because the owner has the right to choose. Chooser
options are most useful in hedging against a future event that might not occur. They are used
often around major events such as elections and when large market movements are expected.
The date at which the option holder must choose between the put and the call is often a few
days after the major event takes place. For example, a new act of parliament is expected to
positively affect a particular company. However, if the act is not passed, the company might

e affected negatively. Buying a chooser option allows a stockholder in the company to hedge
against a possible loss.
In numerical calculation, the option that is European-style and has maturity T2 is priced. At
time T1

at maturity T. There are also call options on min(S1(T), S2(T)) and put options on the
maximum and minimum of two of more asset prices. I assume the two assets have constant
volatilities s i, and a constant correlation ?.
1 2
*
log S1
( T ) = log S1(
0)
+ ( r + σ1
) T + σ1B1
( T )
2
1 2
*
log S2
( T ) = log S 2 ( 0)
+ ( r + ρσ1σ
2 − σ 2 ) T + σ 2B2
( T )
2
The value of a call option on the maximum of two risky asset prices with no dividends and
volatilities s 1 and s 2 and correlation ? is found as follow after some calculations:
σ1
− ρσ 2
σ 2 − ρσ1
−
S1(
0)
M ( d11,
d1,
) + S2
( 0)
M ( d21,
−d
2,
) + e KM(
−d12,
−d
22,
ρ)
− e
σ
σ
where M denotes the bivariate normal distribution function,
σ = σ
+ ,
d
d
d
1
2
2
1 − 2ρσ1σ 2 σ 2
S
σ T
S
σ T
2
1(
0)
1
log( ) +
2(
0)
2
= , d 2 = d1
− σ T ,
11
21
S1(
0)
1 2
log( ) + ( r + σ ) T 1
=
K 2
, d12 = d11
− σ1
T
σ T
1
S 2(
0)
1 2
log( ) + ( r + σ ) T 2
=
K 2
, d 22 = d21
− σ 2 T .
σ T
2
20
rT −rT
The following table shows the price of call on the max calculated by different approach. The
Halton sequence gives the closest result to theoretical formula given in part 5.3. It can be said
that the Halton and Sobol sequences give similar results.
S1=100, S2=120, K=100, T=1, r=0.05, sigma1=0.1, sigma2=0.2 ?=-0.5 (paths=365)
Theoretical 17.3473
Crude MC n=1000 26.5903
Halton n=1000 25.0943
Sobol n=1000 25.1160
Antithetic+crude n=1000 26.3454
Table 3. Numerical results for maximum on two call option
K

Maximum on Five Assets Call Options:
The payoff of a call option on the maximum of five assets with maturity T is
( L )
Max⎡Max S , S , , S − K,0⎤
⎣ 1 2 5 ⎦
Parameters are S1=100, S2=100, S3=100, S4=100, S5=100, K=100, T=1, r1=5%, r2=5%, r3=5%,
r4=5%, r5=5%, q1=2%, q2=2%, q3=2%, q4=2%, q5=2%, s 1=0.1, s2=0.2, s3=0.2, s4=0.2,
s 5=0.2, ?12=0.5 and the correlation matrix is
⎡ 1
⎢
0.5
⎢
⎢0.4 0.5
1
0.4
0.4
0.4
1
0.3
0.3
0.3
0.2⎤
0.2
⎥
⎥
0.2⎥
⎢
⎢
0.3 0.3 0.3 1 0.2
⎥
⎥
⎢⎣0.2 0.2 0.2 0.2 1 ⎥⎦
Time intervals=12 n=50000
Crude MC 22.14372
Halton 22.06235
Sobol 22.15743
Antithetic+crude MC 22.58089
Antithetic+Halton 22.14969
Antithetic+Sobol 22.16016
Table 4. Numerical results for maximum on five call option
Closed-form formula is not feasible for this maximum on five call option. To gain the
approximations of the prices, I use 60 dimensional random numbers to simulate. The result
shows that, regardless of the speed, simulation is a feasible method to calculate the prices.
5.4 Down-And-Out Put Option (one of barrier options)
The payoff of a barrier option depends on whether the underlying asset’s price reaches a
barrier level during a certain period of time. There are many types of barrier options. Barrier
options are first classified into knock-in and knock-out options. The contract is canceled if the
barrier value, B, is crossed at any time during the whole life in knock-out options. However,
knock-in options are activated only if the barrier is crossed. The barrier B may be above or
below the initial asset price S0. If B > S0 it is an up option. If B < S0, it is a down option. This
21

gives us four types of barrier options: up-and-in, up-and-out, down-and-in, down-and-out.
Furthermore, each of these options can be either a call or a put.
A down-and-out put option, Pdo, is a put option that becomes useless if the asset price falls
below the barrier B; in this case B < S0 and B < K. By holding Pdo, the risk for the option
writer is reduced. Therefore, a down-and-out put option is cheaper than a vanilla one. For the
option holder, the potential payoff is reduced. However, a cheaper insurance is got.
A down-and-in put option is activated only if the barrier level B < S0 is crossed. Holding both
a down-and-out and a down-and-in put option is equivalent to holding a vanilla put option.
The following relation is hold;
P= price of the vanilla put
P=Pdi + Pdo
Pdi= price of down-and-in put option
Pdo=price of down-and-out put option.
The main concern for pricing barrier option is the monitoring frequency. Lower monitoring
frequency makes crossing the barrier less likely and this may affect the price. The barrier
might be monitored continuously in principle. However, periodic monitoring is applied in
practice, especially at the close of trading at each day or month. Willmott (2000) shows
analytical pricing formulas for certain barrier options by assuming that barrier monitoring is
continuous:
−rT
P = Ke ( N(
d4
) − N ( d2
) − a(
N(
d7
) − N(
d5
))) − S0
( N(
d3
) − N(
d1)
− b(
N(
ds
) − N(
d6
)))
where
2
−1+
2r
/ σ
⎛ B ⎞
a = ⎜
S ⎟
⎝ 0 ⎠
2
log( S0
/ K)
+ ( r + σ / 2)
T
d1
=
σ T
2
log( S0
/ B)
+ ( r + σ / 2)
T
d 3 =
σ T
2
log( S0
/ B)
−(
r −σ
/ 2)
T
d 5 =
σ T
2
2
log( S0
K / B ) − ( r −σ
/ 2)
T
d 7 =
σ T
22
2
1+
2r
/ σ
⎛ B ⎞
b = ⎜
S ⎟
⎝ 0 ⎠
2
log( S0
/ K)
+ ( r −σ
/ 2)
T
d 2 =
σ T
2
log( S0
/ B)
+ ( r −σ
/ 2)
T
d 4 =
σ T
2
log( S0
/ B)
− ( r + σ / 2)
T
d 6 =
σ T
2
2
log( S0K
/ B ) − ( r + σ / 2)
T
d8
=
σ T
When monitoring discrete, the price for a down-and-out option is increased, since crossing the
barrier is less likely. Broadie, Glasserman and Kou (1997) suggest an approximate correction
that is using the analytical formula above, correcting the barrier as follows:

B ⇒ Be
± 0.
5826σ
δt
where the term 0.5826 derives from the Riemann zeta function, dt is time elapsing between
two consecutive monitoring time instants and “-“ sign for down options (when B < S) and “+”
sign for up options (when B > S).
I will use the fact that pricing the knock-in option is equivalent to pricing the knock-out
option for the variance reduction by conditioning techniques since it is known that
Pdo = P – Pdi.
P
di
−rT
= e E[
I(
S)(
K − SM
)
+
]
where asset price path for days i, i=1,…,M (T=Mdt): S= {S1, S2,…,SM} and indicator function
I is I(S)= 1 if Sj < B for some j and =0 otherwise. Then,
*
−rt
*
Pdi = I ( S)
e Bp
( S * , K,
T − t )
j
where t* is crossing time and Bp is the Black-Scholes price for a vanilla put option.
The run results on the table 5 shows that variance reduction by conditioning is very helpful.
The fact that the barrier has been crossed in 7808 replications for 50000 is very good.
However, it can be improved and importance sampling may help. I used the method explained
by Brandimarte (2002) that changing the drift of the asset price may increase the likeliness of
crossing the barrier. The result on the Table 5 shows that importance sampling proves useful
by crossing the barrier in 46233 replications for 50000. Moreover, the price calculated by
using conditional Monte Carlo and importance sampling is closer to analytical formula’s
price. In other words, the quality of estimate is improved. Sobol Sequences do not perform as
well as variance reduction techniques with crude Monte Carlo. It can be concluded that
conditional Monte Carlo with importance sampling outperforms Sobol sequences.
S0=50 K=60 B=40 r=0.05 T=0.75 sigma=0.2
Call = 1.0473 Put = 8.8390
Analytical formula 6.2071
Crude MC, number of monitoring= 270 n=1000 5.7876 (CI=5.4367, 6.1386 Ratio=12.13%)
23
NCrossed=147
Crude MC, number of monitoring= 270 n=50000 5.8646 (CI=5.8147, 5.9145 Ratio=1.70%)
NCrossed=7808
Sobol inverse, monitoring=270 n=1000 10.5347 NCrossed=23
Sobol inverse, monitoring=270 n=50000 Matlab out of performance

Conditional MC, monitoring=270 n=1000 6.0337 (CI= 5.6157, 6.4517 Ratio=13.86%)
24
NCrossed=148
Conditional MC, monitoring=270 n=50000 5.8972 (CI= 5.8371, 5.9574 Ratio=2.04%)
Conditional Sobol inverse MC, monitoring=270
n=1000
Cond.MC and importance sampling,
monitoring=270 n=10000 bp=20
Cond.MC and importance sampling,
monitoring=270 n=50000 bp=20
Cond.MC with Sobol and importance sampling,
monitoring=270 n=1000 bp=50
Table 5. Numerical results for down-and-out put option
NCrossed=7763
8.839 NCrossed=58
5.8991 (CI= 5.8505, 5.9477 Ratio=1.65%)
NCrossed=9257
5.8966 (CI= 5.875, 5.9187 Ratio=0.73%)
NCrossed=46233
8.821
NCrossed=63

6 INTERPRETATIONS OF FINDINGS AND CONCLUSIONS
I test the efficiency of Quasi-Monte Carlo, variance reduction techniques and crude Monte
Carlo by using them to price European call option and exotic options like chooser, rainbow
and barrier options. Halton sequence, one of quasi-monte carlo methods, gives better solution.
However, Sobol’s results are very close to Halton’s. It can be accepted that the performance
of QMC is much better than standard MC and no obvious difference in the two quasi-random
number sequences. However, variance reduction techniques perform better than low-
discrepancy sequences to value down-and-out put option.
In literature, Sobol sequences perform better. This result may be observed in high-
dimensional integrals. Birge (1994) found that low discrepancy sequences gave very accurate
results for pricing simple option contracts. Paskov (1994) used Sobol' sequences to value
mortgage-backed securities and concluded that at least for this security the use of Sobol'
sequences significantly outperformed the standard Monte Carlo. In the case of more general
(non-finance related) integrals, numerical experiments by Bratley, Fox and Niederreiter
(1992) and van Rensburg and Torrie (1993) indicated that the superiority of low discrepancy
algorithms decreases with the dimensions of the problem. These authors suggest that for
general integrals the superiority of the low discrepancy algorithms for dimensions around 30.
Against this background, Paskov's results were surprisingly good. Berman (1996) investigates
the relative performance of Sobol sequences against standard Monte Carlo and Monte Carlo
with variance reduction by using a range of popular exotic options for this comparison. The
dimensions considered are 8, 32 and 128. He concludes that the Sobol' sequences outperform
the use of pseudo-random sequences but that the advantage decreases with dimension. When
a particular type of variance reduction based on stratified sampling is introduced, the
performance of the two approaches is similar.
This project can be improved by working on higher dimensional option pricing and other
asset pricing.
25

REFERENCES
Antonov, I.A., Saleev, V.M. (1979). An Economic Method of Computing LP-Sequences.
U.S.S.R. Comp. Math. Math. Phys., Vol. 19, No. 252.
Berman L.(1996). Comparison of Path Generation Methods for Monte Carlo Valuation of
Single Underlying Derivative Securities. Working Paper JBM Research Division, T. J.
Watson Research Center, Yorktown Heigths, New York.
Birge J. R.(1994). Quasi-Monte Carlo Approaches to Option Pricing. Technical Report,
Department of Industrial and Operations Engineering, University of Michigan.
Boyle, P., Broadie, M. and Glasserman P. (1996). Monte Carlo Methods for Security Pricing.
Journal of Economics Dynamics and Control.
Boyle, P. (1977). Options: A Monte Carlo Approach. Journal of Financial Economics, Vol. 4,
No. 3, pp.323-338.
Brandimarte, P. (2002). Numerical Methods in Finance and Economics. John Wiley & Sons,
UK.
Bratley, P., Fox B.L. and Niederreiter H.(1992). Implementation and Tests of Low-
discrepancy sequences. ACM Bansactions on Modeling and Computer Simulation, No. 2,
pp.195-213.
Broadie M., Glasserman P., and Kou S.G. (1997). A Continuity Correction for Discrete
Barrier Options. Mathematical Finance, 7, 325-349.
Carlish, R.E. and Morokoff, W.J.(1996). Valuation of Mortgage-Backed Securities Using the
Quasi-Monte Carlo Method. International Association of Financial Engineers First Annual
Computational Finance Conference, Stanford University.
Faure, H. (1982). Dlscr6pance de suites associ~es a un syst~me de num6ration (en
dimensions). Aacta Arzthmetzc 41,337-351.
26

Halton, J.H. (1960). On the efficiency of certain quasi-random sequences of points in
evaluating multi-dimensional integrals. Numerical Mathematics 2, 84-90.
Ninomiya, S. and Tezuka, S. (1996). Toward Real-Time Pricing of Complex Financial
Derivatives. Applied Mathematical Finance , No. 3, 1-20.
Niederreiter, H.(1992). Random Number Generation and Quasi-Monte Carlo Methods.
Philadelphia, PA: SIAM.
Niederreiter, H.(1987). A Statistical Analysis of Generalized Feedback Shift Register
Pseudorandom Number Generators. Journal of Scientific Computing, Vol. 8, No. 6, pp. 1035-
1051.
Paskov, S.H. and Traub, J.F. (1995). Faster Valuation of Financial Derivatives. Journal of
Portfolio Management, Vol. 22, No.l, pp. 113-120.
Paskov, S.H. (1994). New Methodologies for Valuing Derivatives, Technical Report.
Computer Science Department, Colombia University.
Rensburg, J., Torrie, G.M. (1993). Estimation of Multidimensional Integral: Is Monte Carlo
the Best Method? Journal of Physics A: Mathematical and General, Vol. 26, pp. 943-953.
Sobol, I.M. (1967). On the distribution of points in a cube and the approximate evaluation of
integrals. U.S.S.R. Comput. Math. Math. Phys, Vol. 7, pp. 86-112.
Tan, K.S. and Boyle, P. (1997). Applications of Scrambled Low Discrepancy Sequences to
Exotic Options. Working Paper, University of Waterloo.
Willmott P. (2000). Quantitative Finance (vols. I and II). Wiley, Chichester, West Sussex,
England.
27