University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Chapter 4: Monte-Carlo Methods
A Monte-Carlo method is a technique for the numerical realization of a stochastic process by
means of normally distributed random variables.
In financial mathematics, it is used for the computation of the expectation of option prices in
situations where explicit solutions like the Black-Scholes formula are not accessible (e.g., exotic options).
For simplicity, we consider a European plain-vanilla put option on a stock St whose value behaves according to the geometric Brownian motion
(⋆)
d St = r St dt + σ St dWt
with a risk-free interest rate r ≥ 0, constant volatility σ > 0, and a Wiener process Wt.
We recall that the option price V(St, t) at t = 0 is given by the discounted expectation
(◦)
V(S0, 0) = exp(−rT) E(V(ST, T)) .
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
The Four Basic Steps of the Monte-Carlo Simulation
The idea behind the Monte-Carlo method is based on the approximation of the expectation
E(V(ST, T)) in (◦) by the simulation of M pathes {St | 0 < t < T} of the value of the stock:
Step 1: Simulation of the pathes
(k)
For M independent pathes, compute the solution St , 1 ≤ k ≤ M, of the geometric Brownian
motion (⋆). This requires the simulation of M independent realizations of the Wiener process
Wt and the numerical solution of the stochastic differential equation for the respective pathes.
Step 2: Computation of the payoff function
(k)
(k)
(k)
For 1 ≤ k ≤ M, compute the payoff function associated with the path St : VT = (K − ST )+.
Step 3: Computation of an estimate of the expectation
According to the law of large numbers, the arithmetic mean of equally distributed, independent random variables converges almost surely to the expectation. Hence, an appropriate es(k)
(k)
timate of the expectation in (◦) is given by VT = (K − ST )+.
Step 4: Computation of an estimate of the option price
Compute an approximation of the option price by V̂ = exp(−rT) Ê(VT).
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.1 Generation of Equally Distributed Random Numbers
For the simulation of a Wiener process, we use standardized normally distributed random
√
numbers Z ∼ N(0, 1) in order to compute the increments ∆W = Z ∆t.
Definition 4.1 (Equally Distributed Random Variables)
(i) A random variable X is called equally distributed on [a, b] ⊂ lR, (notation: X ∼ U[a, b]), if
its density function is given by f(x) = 1/(b − a) , x ∈ [a, b].
(ii) A sequence {Xn}n∈lN of random variables is called F-distributed, if the Xn are independent realizations of random numbers with the distribution function F.
In the sequel, we will consider the following random number generators:
• Linear congruential generators
• The generator RANDU
• Fibonacci generators
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.1.1 Linear Congruential Generators
Linear congruential generators are obtained by the following algorithm:
Algorithm 4.1: Linear Congruential Generators
Given M ∈ lN, a, b ∈ {0, 1, ..., M − 1}, a 6= 0, and a seed X0 ∈ {0, 1, ..., M − 1}, compute
For i = 1, 2, ...
Xi := (a Xi−1 + b) mod M ,
Ui := Xi/M .
where a mod M := M · (a/M − ⌊a/M⌋) with ⌊a/M⌋: largest integer less or equal a/M.
Lemma 4.1 (Properties of Linear Congruential Generators)
The random numbers generated by Algorithm 4.1 have the properties
(i)
Xi ∈ {0, 1, ..., M − 1}.
(ii) The Xi are periodic with period less or equal M.
(iii) If b = 0 (multiplicative congruential generator), we must exclude the seed X0 = 0.
(iv) If a = 1, we obtain Xi = (X0 + ib) mod M (too easy to predict!).
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Analysis of Linear Congruential Generators
The random numbers Xi and Ui, i ≥ 1, obtained by the linear congruential generator, can be
written as m-tuples (Xi, Xi+1, ..., Xi+m−1) or points (Ui, ..., Ui+m−1) ∈ [0, 1)m. They are situated on
(m − 1)-dimensional hyperplanes which can be constructed as the following analysis in case
m = 2 reveals:
Xi = (a Xi−1 + b) mod M
⇐⇒
Xi = a Xi−1 + b − k M ,
k M ≤ a Xi−1 + b < (k + 1) M .
For an arbitrary tuple (z0, z1) ∈ Z2 we obtain:
(†)
z0 Xi−1 + z1 Xi = z0 Xi−1 + z1 (a Xi−1 + b − k M) =
= Xi−1 (z0 + a z1) + z1 b − z1 k M =
z0 + a z1
= M · (xi−1
− z1 k) + z1 b .
M
|
{z
}
=: c
Division by M yields the equation of a straight line in the (Ui−1, Ui)-plane:
(‡)
z0 Ui−1 + z1 Ui = c + z1 b M−1 .
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Analysis of Linear Congruential Generators
Question: Does there exist a tuple (z0, z1) ∈ Z2 such that
the intersection of the straight lines with the square [0, 1)2
is non-empty for only a few lines which would violate the
requirement of uniform distributions of the points?
If we choose (z0, z1) ∈ Z2 and z0 + a z1 = 0 mod M, then c ∈ Z.
Solving for c in (†) results in the maximum interval
cmin ≤ c ≤ cmax
such that the associated straight line (‡) has non-zero intersection with [0, 1)2.
Figure 4.1: M = 2048 , a = 1229 , b = 1 , X0 = 1. The figure displays the first 499 points
(Ui−1, Ui) obtained by the linear congruential generator. These points are lying on 5 parallel
straight lines.
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.1.2 The Random Number Generator RANDU
The generator RANDU is based on a multiplicative congruential method with the data
M = 231 , a = 216 + 3 , b = 0 .
Figure 4.2 suggests that the points (Ui−1, Ui) are randomly distributed in the plane. However,
Figure 4.3 displays the tuples (Ui−2, Ui−1, Ui), and we observe that they are lying on only 15 different planes in the unit cube [0, 1)3 which is a serious disadvantage in many applications.
Figure 4.2
Figure 4.3
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.1.3 Fibonacci Generators
Fibonacci generators use the recursively given Fibonacci sequence according to
Xi := (Xi−1 + Xi) mod M ,
Ui := Xi/M ,
i≥2 ,
where X1, X2 are given seeds.
Figure 4.4 displays the first 2000 points (Ui−1, Ui)
obtained by the Fibonacci generator for M = 2179
and X1 = X2 = 1. Significantly less than 2000 points
are seen in the unit square due to the fact that the
sequence {Ui}i∈lN repeats any 197 entries.
Therefore, this Fibonacci generator is less suited for
random number generation.
Figure 4.4
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Better and widely used random number generators are the so-called lagged Fibonacci generators:
Algorithm 4.2: Lagged Fibonacci Generators
Given M ∈ lN, µ, ν ∈ lN and initial values X1, ..., Xmax(µ,ν), for i ≥ max{µ, ν} compute
Xi := (Xi−µ + Xi−ν ) mod M ,
Ui := Xi/M .
For M = 2048 , µ = 5 , ν = 17, Figure 4.5 shows
the first 2000 points (Ui−1, Ui) obtained by the
lagged Fibonacci generator.
Figure 4.5
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.2 Transformation of Random Variables
Normally distributed random variables can be obtained by a transformation of equally distributed random variables using either the inversion of the distribution function or a transformation between random variables.
4.2.1 Inversion of the Distribution Function
Theorem 4.1 (Inversion of the Distribution Function)
Assume that U ∼ U[0, 1] is an equally distributed random variable and assume further that F
is a continuous, strongly monotone distribution function. Then, the random variable F−1(U)
is F-distributed.
Proof. Since P(U ≤ ξ) = ξ , ξ ∈ [0, 1], we have
P(F−1(U) ≤ x) = P(U ≤ F(x)) = F(x) .
Remark: For the normal distribution Φ there are no explicit expressions neither for Φ nor for
Φ−1 which means that the nonlinear equation Φ(x) = u must be inverted numerically by, e.g., the
application of a Newton-like method. However, for u ≈ 1 the problem is ill-conditioned.
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.3.2 Marsaglia’s Polar Method
Marsaglia’s variant of the Box-Muller algorithm avoids the evaluation of trigonometric functions:
For U1, U2 ∼ U[0, 1], we obviously have Wi := 2Ui − 1 ∼ [−1, +1], 1 ≤ i ≤ 2. The pair (W1, W2) defines
a point in the plane. Only points inside the unit disk
K := {(W1, W2) | w12 + W22 < 1}
are accepted. A Transformation K → S := [0, 1]2 is provided by means of polar coordinates




x1 

x2


W12 + W22 

=  1
W2  .
arctan(
2π
W1 )
Algorithm 4.4: Marsaglia’s Polar Method
Step 1: Generate U1, U2 ∼ U[0, 1] and Wi := 2Ui − 1 , 1 ≤ i ≤ 2, as long as W12 + W22 < 1.
r
−2ln(W12 + W22)//(W12 + W22)
r
Step 2: Z1 := W1
and Z2 := W2 −2ln(W12 + W22)//(W12 + W22) are
normally distributed random variables.
Remark: Marsaglia’s polar method is more efficient than the Box-Muller algorithm.
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Algorithm 4.3: Box-Muller Algorithm
Step 1: Generate U1 ∼ U[0, 1] and U2 ∼ U[0, 1].
r
Step 2: Set Θ := 2π U2 and ρ := −ln(U1).
Step 3: Z1 := ρ cos(Θ) and Z2 := ρ sin(Θ) are normally distributed random variables..
The histogram of Z1 in Figure 4.6 based on 200000
random numbers shows that the algorithm indeed
approximately produces normally distributed random variables.
Figure 4.6
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.2 Normally Distributed Random Variables
4.3.1 The Box-Muller Algorithm
We apply Theorem 4.2 in case n = 2 , A = S := [0, 1]2 , f ≡ 1 on S, and h : S → lR2 given by
r
r
y1 = h1(x1, x2) = −2 ln(x1) cos(2πx2) ,
y2 = h2(x1, x2) = −2 ln(x1) sin(2πx2) .
Since y12 + y22 = −2 ln(x1) and y2/y1 = tan(2πx2), the inverse function can be easily computed
1
x1 = exp(− (y12 + y22)) ,
2
x2 =
1
y2
arctan( )
2π
y1
with the determinant det Jh−1 (y) of the Jacobian Jh−1 (y) given by
det Jh−1 (y) =
1
1
exp(− (y12 + y22))
2π
2
which is the density of the standard normal distribution in lR2.
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.2.2 Transformation of Random Variables
Theorem 4.2 (Transformation of Random Variables)
Assume that X is a random variable with density function f on A := {x ∈ lRn | f(x) > 0}. Assume
further that the mapping h : A → B := h(A) is continuously invertible. Then, the random variable
Y = h(X) has the density function
g(y) := f(h−1(y)) |det Jh−1 (y)| ,
y∈B ,
where Jh−1 (y) is the Jacobian of h−1 in y.
Proof. Applying the transformation rules for multiple integrals, we obtain
−1
Z
P(Y = h(X) ∈ B) = P(X ∈ h (B)) =
Z
f(x) dx =
h−1 (B)
B
f(h−1(y)) |det Jh−1 (y)| dy .
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.4 Correlated Random Variables
We consider a random variable Z = (Z1, ..., Zn) with density function f. We further denote by
µ = E(Z) = (E(Z1), ..., E(Zn)) the expectation and by Σ = (Σij)ni,j=1 with
Σij = (Cov Z)ij := E((Zi − µi)(Zj − µj))
the associated covariance matrix which is symmetric, positive definite.
We want to construct an N(µ, Σ)-distributed random variable Y.
Theorem 4.3 (Correlated Random Variables)
Under the above notations let Σ = LLT be the Cholesky decomposition of Σ with a lower triangular matrix L ∈ lRn×n. Then, Y = µ + LZ is an N(µ, Σ)-distributed random variable with density function
1
1
1
(x − µ)T Σ−1 (x − µ)) .
f(x) =
exp(−
n/2
1/2
(2π) (det Σ)
2
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Radical-Inverse Function and the Halton Sequence
(ii) Radical-Inverse Function
The radical-inverse function ΦB allows to define van der Corput sequences {xi}Ni=1 with respect
to an arbitrary basis B. Using the same notation as in (i), they are defined by
ΦB(i) :=
j
X
k=0
dk B−(k+1) ,
xi := ΦB(i) .
(iii) Halton Sequence
Assume that p1, ..., pm are pairwise prime integers. The Halton sequence is defined as the sequence of vectors
xi = {Φp1 (i), ..., Φpm (i)} ,
i ∈ lN .
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Van der Corput Sequence
(N)
(N)
Remark: The sequence {Xi }Ni=1 with xi = i/N, 1 ≤ i ≤ N, obviously satisfies DN = 1/N and
thus is of low discrepancy. However, it is not suited for Monte Carlo integration, since for
each N a new sequence has to be set up. For increasing N, it is more efficient to use already
computed numbers.
Examples of sequences of low discrepancy:
(i) Van der Corput sequence
Let i = (dj...d0)2 := jk=0 dk2k be the representation of the integer i ∈ lN in fixed point arithmetics with respect to the basis B = 2. Then, the sequence {xi}lN obtained by bit inversion
according to
P
xi = (.d0...dj)2 :=
is called van der Corput sequence.
j
X
k=0
dk 2−(k+1)
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Sequences of Low Discrepancy
Definition 4.3 (Sequences of Low Discrepancy)
A sequence {xi}Ni=1 of points or numbers xi ∈ lRm, 1 ≤ i ≤ N, is called a sequence of low discrepancy, if there exists a constant Cm > 0 such that for all N ∈ lN:
DN
(ln(N))m
≤ Cm
.
N
Remarks:
(i) For not too large m ∈ lN we have that approximately DN ≈ O(N−1). Compare this with
DN = O(N−1/2) for sequences of random numbers in general.
(ii) Let Ui ∈ {0, 1/M, ..., (M − 1)/M} be random numbers obtained by a linear congruential generator. These are not even uniformly distributed, since with Q′ := [1/(2M + 2), 1/(M + 1)]
card({Ui | Ui ∈ Q′})
1
DN ≥ |
− vol(Q′)| = vol(Q′) =
.
N
2M + 2
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Properties of Discrepancy and Star Discrepancy
Theorem 4.4 (Properties of Discrepancy and Star Discrepancy)
Let DN and D∗N be the discrepancy and star discrepancy of a point set {xi}lN, xi ∈ lRm, 1 ≤ i ≤ N.
Then, there holds
D∗N ≤ DN ≤ 2m D∗N .
Proof. The proof is left as an exercise.
Upper Bound for Monte Carlo Integration
Theorem 4.5 (Theorem of Hlawka and Koksma)
Assume that Ω := [0, 1]m, m ∈ lN, and f : [0, 1]m → lR is a function of bounded variation BV(f).
Then, for the error εN in the Monte Carlo integration there holds
|εN| ≤ BV(f) D∗N .
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.5.2 Discrepancy
The idea behind discrepancy is that for a uniformly distributed point set {x1, ..., xN} ⊂ [0, 1]m
the fraction of the points lying within some quadrilateral Q ⊂ [0, 1]m should correspond to the
volume of the quadrilateral, i.e.,
vol(Q)
card({xi, 1 ≤ i ≤ N | xi ∈ Q})
≈
.
N
vol([0, 1]m)
Definition 4.2 (Discrepancy and Star Discrepancy)
The discrepancy of a point set {x1, ..., xN} ⊂ [0, 1]m is defined by
DN :=
sup |
Q⊂[0,1]m
card({xi, 1 ≤ i ≤ N | xi ∈ Q})
− vol(Q)| ,
N
where the supremum is taken with respect to all quadrilaterals Q ⊂ [0, 1]m.
The star discrepancy D∗N is defined in the same way, but with the supremum taken with respect
Q
to those quadrilaterals Q∗ ⊂ [0, 1]m for which one corner is in the origin, i.e., Q∗ = m
i=1 [0, yi ]
where (y1, ..., ym) denotes the diagonally opposite corner.
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
If the random variables x1, ..., xN ∈ Ω are independent and uniformly distributed, the law of
R
large numbers implies QN → vol(Ω) E(f) = Ω f(x) dx as N → ∞. Hence, for the variance of the
error εN := I − QN we obtain
Z
Z
σ2(f)
2
2
2
2
2
vol(Ω) , σ (f) = f(x) dx − ( f(x) dx)2 .
Var(εN) = E(ε ) − E(εN) =
N
Ω
Ω
The central limit theorem implies the following convergence result for the standard deviation
r
Var(εN) = O(N−1/2) as N → ∞ ,
which means slow convergence (improvement of the accuracy by a factor of 10 requires to increase the computational cost by a factor of 100). Another disadvantage is the lack of a strict
error bound. Before we discuss methods of variance reduction in section 4.6, we will address
the issue to choose the random points xi, 1 ≤ i ≤ N, in such a way that with increasing N
the accuracy gets better but clustering is avoided.
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.5 Sequences of Random Numbers with Low Discrepancy
As a measure for the quality of uniform distribution of random numbers we introduce the notion of discrepancy and give examples of sequences of random numbers with low discrepancy.
A related issue is that to improve convergence properties in relevant applications such as the
numerical integration of high-dimensional integrals.
4.5.1 Monte Carlo Integration
Given a bounded domain Ω ⊂ lRm and an integrable function f : Ω → lR, the Monte Carlo integration provides an approximation of the m-dimensional integral of f over Ω by the weighted sum
of the values of the integrand f at randomly chosen points xi, 1 ≤ i ≤ N, in the domain of integration Ω according to
Z
N
vol(Ω) X
I :=
f(x) dx ≈
f(xi) =: QN .
N i=1
m
Ω⊂lR
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Algorithm 4.5: Correlated Random Variables
Step 1: Compute the Cholesky decomposition Σ = LLT.
Step 2: Compute Z ∼ N(0, 1) component-wise, i.e., Zi ∼ N(0, 1) , 1 ≤ i ≤ n, e.g., by Marsaglia’s
polar algorithm.
Step 3: Y = µ + LZ is N(µ, Σ)-distributed.
MATLAB program correlated.m
randn(′state′, 1)
Sigma = [543; 454; 345]
mu = [−5010]′; N = 10000;
L = chol(Sigma);
X = zeros(3, N);
for i = 1 : N
X(: .i) = mu + L ∗ randn(3, 1);
end
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Numerical Methods for Option Pricing in Finance
Proof. Setting x = Lz and observing dx = |det L| dz, we obtain
1
1
zTz
(L−1x)TL−1x
) dz =
f(z) dz =
exp(− ) dz =
exp(−
(2π)n/2
2
(2π)n/2
2
xT(LLT)−1x
1
1
xTΣ−1x
exp(−
=
) dz =
exp(−
) dx =
(2π)n/2
2
(2π)n/2|det L|
2
xTΣ−1x
1
exp(−
=
) dx .
(2π)n/2(det Σ)1/2
2
This shows that X = LZ is N(0, Σ)-distributed which readily gives the assertion.
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Properties of van der Corput Sequences
Theorem 4.6 (Properties of van der Corput Sequences)
For the discrepancy of van der Corput sequences there holds
N DN
=
lim sup
N→∞ ln(N)

















B2
4(B+1) ln(B)
4
B−1
ln(B)
, B even
,
, B odd
.
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.6 Numerical Integration of Stochastic Differential Equations
4.6.1 The Euler-Maruyama Method
In case of an ordinary differential equation
x′(t) = a(t, x(t)) ,
T>0 ,
where a ∈ C(lR+, lR) satisfies a Lipschitz condition in the second argument, the explicit Euler
method with uniform step-size h > 0 is given by
yi+1 = yi + h a(ti, yi) ,
ti := i h , i ≥ 0 .
It is well known that it has the order of convergence 1, i.e.
sup |yi − x(ti)| ≤ C h .
i≥0
For the stochastic differential equation
dxt = a(t, xt) dt + b(t, xt) dWt
the counterpart of the explicit Euler method is the Euler-Maruyama method
yi+1 = yi + h a(ti, yi) + b(ti, yi) ∆Wi ,
∆Wi := Wti+1 − Wti .
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.6.2 Strong and weak convergence
Definition 4.4 (Strong and Weak Convergence)
Assume that xht is an approximation of the solution xt of the stochastic differential equation
dxt = a(t, xt) dt + b(t, xt) dWt .
(i) The approximation xhT is said to converge strongly to xT of order γ > 0, if there exists a constant C > 0, such that for all sufficiently small h > 0
E(|x − xh |) ≤ C hγ .
T
T
(ii) The approximation xhT is said to converge weakly to xT of order κ > 0, if there exists a constant C > 0, such that for all sufficiently small h > 0
|E(x ) − E(xh )| ≤ C hκ .
T
T
Theorem 4.7 (Strong and Weak Convergence of the Euler-Maruyama Method)
The Euler-Maruyama method is strongly convergent of order γ = 0.5 and weakly convergent
of order κ = 1.
Proof. We refer to P. Kloeden and E. Platen; Numerical Solution of Stochastic Differential
Equations. Springer, Berlin-Heidelberg-New York, 1999
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.6.3 The Method of Milstein
We recall Itô’s Lemma which states: If Xt is an Itô process, then for sufficiently smooth f we
have that f = f(t, Xt) is an Itô process as well which satisfies
∂f
1 2 ∂ 2f
∂f
∂f
+ b
dWt .
df = ( + a
)
dt
+
b
∂t
∂x
2
∂x2
∂x
The associated integral form is
Zt
Zt
1 ′′
′
2
(∗) f(xt) = f(xt0 ) + (f (xs) a(xs) + f (xs) b(xs) ) ds + f ′(xs) b(xs) dWs .
2
t0
T0
In particular, for f(x) = x we obtain
Zt
Zt
(∗∗) xt = xt0 + a(xs) ds + b(xs) dWs .
t0
t0
Choosing f = a and f = b in (∗) and inserting it into (∗∗) results in
Zt
Zs
1 ′′ 2
xt = xt0 + (a(xt0 ) + (a a + a b ) dz + a′b dWz) ds +
2
t0
t0
t0
t
s
s
Z
Z
Z
1 ′′ 2
′
+ (b(xt0 ) + (b a + b b ) dz + b′b dWz) dWs ,
2
t0
t0
t0
where a = a(xz) , b = b(xz) etc.
Zs
′
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
Modifying the double integral w.r.t. dWzdWs by replacing the integrand b′(xz)b(xz)
b′(xt0 )b(xt0 ), we can write
Zt
xt = xt0 + a(xt0 ) (t − t0) + b(xt0 )
t0
Zt Zs
′
dWs + b (xt0 ) b(xt0 )
3/2
where the remainder is of the (higher) order R = O(h ).
The double integral can be evaluated as follows:
Zt Zs
Zt
dWz dWs =
t0 t0
t0
Zt
(Ws − Wt0 ) dWs =
t0
dWz dWs + R ,
t0 t0
Zt
Ws dWs − Wt0 dWs =
t0
1
t − t0
1
(Wt2 − Wt20 ) −
− Wt0 (Wt − Wt0 ) = ((Wt − Wt0 )2 − (t − t0)) .
2
2
2
This leads to the method of Milstein
1
yi+1 = yi + h a(ti, yi) + b(ti, yi) ∆W + b′(ti, yi) b(ti, yi) ((∆W)2 − h) ,
2
√
where ∆W := Z h , Z ∼ N(0, 1).
Theorem 4.8 (Convergence of the Method of Milstein)
The method of Milstein is strongly convergent of order γ = 1.
=
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.7 Reduction of Variances
The slow convergence of Monte-Carlo simulations can be explained as follows:
P
We assume that θn = nk=1 Φ(Xk)/n is some stochastic approximation of a stochastic integral
θ = E(Φ(x)), where Φ(x) := g(x)/f(x) and Xk, 1 ≤ k ≤ n, are independent samples of an F-distributed random variable with F′ = f.
Under the assumption E(Φ(Xk)) = θ , Var(Φ(Xk)) = σ2 , 1 ≤ k ≤ n, it follows that
n
n
1 X
1 X
σ2
E(θn) =
E(Φ(Xk)) = θ , Var(θn) = 2
Var(Φ(Xk)) =
.
n k=1
n k=1
n
Using the Chebychev inequality for square integrable random variables Y
√
Var(Y)
, δ := σ/ ε n ,
P(|Y − E(Y)|) ≥ δ) ≤
δ2
we obtain the error estimate
σ
σ
P(|θn − θ| ≥ √ ) ≤ ε =⇒ P(|θn − θ| < √ ) ≥ 1 − ε .
εn
εn
This means that the error gets the smaller the larger the number n of samples is chosen. But
in order to reduce the error by a factor of 10, the number of samples must be increased by a
factor of 100.
University of Houston/Department of Mathematics
Dr. Ronald H.W. Hoppe
Numerical Methods for Option Pricing in Finance
4.7.1 Antithetic Variables
Theorem 4.9 (Variance reduction by an Antithetic Variable)
Assume that the approximation θn has been generated by a random variable Z ∼ N(0, 1) and
generate an approximation θ−n by means of −Z ∼ N(0, 1) such that Var(θn) = Var(θ−n ).
Introduce the antithetic variable θˆn according to θˆn := 1 (θn + θ−n ). Then, there holds
2
Var(θˆn) < Var(θn) .
Proof. The fundamental relation
Var(X ± Y) = Var(X) + Var(Y) ± 2 Cov(X, Y)
implies
1
1
Var(θˆn) = Var(θn + θ−n ) = (Var(θn) + Var(θ−n ) + 2 Cov(θn, θ−n )) .
4
4
Taking into account that
1
Cov(X, Y) ≤ (Var(X) + Var(Y)) ,
2
we thus obtain
1
Var(θˆn) ≤ (Var(θn) + Var(θ−n )) , if Cov(θn, θ−n ) ≤ 0 ,
4
1
Var(θˆn) < (Var(θn) + Var(θ−n )) , if Cov(θn, θ−n ) > 0 ,
4
which gives the assertion observing Var(θn) = Var(θ−n ).