MC/QMC Methods for Option Pricing under Stochastic Volatility Models
Chuan-Hsiang Han
Department of Quantitative Finance, National Tsing-Hua University, Hsinchu,Taiwan 30013, ROC.
E-mail: [email protected]
Yongzeng Lai
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada
E-mail: [email protected]
Abstract
In the context of multi-factor stochastic volatility models, which contain the widely used Heston model,
we present variance reduction techniques to price European options by Monte Carlo (MC) and QuasiMonte Carlo (QMC) methods. We formulate a stochastic integral as a martingale control for the payoffs
to be evaluated. That control corresponds to the cost of an approximate delta hedging strategy. Putting
together this control variate method and randomized quasi-Monte Carlo methods, including the Sobol’
sequence and L’Ecuyer type good lattice points, we find that the variance reduction ratios can reach
up to 300 times for European options under multi-factor stochastic volatility models.
Keywords: Option pricing; Multi-factor stochastic volatility models; control variate method; Monte
Carlo and quasi-Monte Carlo methods.
1
Introduction
Stochastic volatility models have been an important class of diffusions as an extend of the BlackScholes models. Under stochastic volatility models, closed-form solutions for typical option pricing
problems such as European and barrier options barely exist. Numerical methods become essential.
Along the development of Monte Carlo methods, there have been some efficient and robust numerical methods proposed within last decade in order to reduce variances. In particular, many of these
methods bridge out the concept of derivative pricing and imperfect hedging under incomplete market
models. Therefore, reduced variances obtained from these Monte Carlo methods are not only measurements of computing efficiency but also represent risks associated to particular hedging strategy
adopted in the algorithms.
To the best of authors’ knowledge, Clewlow and Carverhill [4] are the first to utilize the value
of some delta hedging portfolio and possibly other Greeks hedge as controls for discounted payoffs to
evaluate option prices. They found that the variance obtained by this control variate method under
an one-factor stochastic volatility model can be significantly reduced. By means of the least-squares
method [22], Potters, Bouchaud, and Sestovic [25] proposed an optimal hedged Monte Carlo method
to price options under the ”historical” probability measure. Later, Pochart and Bouchaud [27] extended the same idea to problems when shortfall risk and transaction cost constraints are considered.
Heath and Platen [13] and Fouque and Han [6] used different approximate European option prices to
approximate delta portfolio under stochastic volatility models and showed significant variance reduction performance. Moreover, in [7], Fouque and Han obtained an asymptotic result to characterize
1
the variance of a martingale control variate under stochastic volatility models when the time scales of
driving volatility processes are well-separated. Applications of Barrier options and American options
are also presented.
Some other variance reduction techniques with less or no financial implication include conditional
Monte Carlo [31], importance sampling [11, 5], direct sampling [3], etc. All Monte Carlo methods
mentioned so far are fundamentally related to psudo random sequences. However integration methods
using the alternative quasi-random sequences (or called low-discrepancy sequences) have drawn lots
of attentions in recent years because its theoretical rate of convergence is O(1/n1−ε ) for all ε > 0
subjected to the dimensionality and the regularity of the integrand. A brief introduction about quasiMonte Carlo (QMC in short) methods, or low-discrepancy methods in Section 5. Also, we refer to [12],
[17] and [18] for further discussions on Monte Carlo and Quasi-Monte Carlo methods in applications
of computational finance.
In this paper, we study several possible combinations of control variate methods and quasi-Monte
Carlo methods for the pricing problems of European options under multi-factor stochastic volatility
models. We conduct numerical experiments to compare the variance-reduction performance for the
martingale control variate method mentioned above with or without randomized quasi-Monte Carlo
methods. The organization of this paper is the following. In Section 2, we review the martingale control
variate method for European options under multi-factor stochastic volatility models. These models
are adequate to capture long-time and short-time volatility effects as documented in [1, 10, 23] and
references therein. Additionally we take Heston’s model as a simplified example and obtain a dramatic
variance reduction performance. In Section 3. Numerical experiments by Monte Carlo method and
quasi-Monte Carlo methods are presented. We test several combinations of martingale control variate
methods with or without quasi-Monte Carlo methods, including the Sobol’ sequence and L’Ecuyer
type good lattice points together with the Brownian bridge sampling technique.
2
Monte Carlo Pricing under Multi-factor Stochastic Volatility Models
Under a risk-neutral probability measure IP ? parametrized by the combined market price volatility
premium (Λ1 , Λ2 ) , we consider the following multi-factor stochastic volatility models defined by
(0)∗
dSt = rSt dt + σt St dWt ,
σt = f (Yt , Zt ),
q
g1 (Yt )
g1 (Yt )
1
(0)∗
(1)∗
ρ1 dWt + 1 − ρ21 dWt
,
dYt = c1 (Yt ) + √ Λ1 (Yt , Zt ) dt + √
ε
ε
ε
h
i
√
dZt = δc2 (Zt ) + δg2 (Zt )Λ2 (Yt , Zt ) dt
q
√
(0)∗
(1)∗
(2)∗
2
2
+ δg2 (Zt ) ρ2 dWt + ρ12 dWt + 1 − ρ2 − ρ12 dWt
,
(1)
where St is the underlying asset price process with a constant risk-free interest rate r. Its random
volatility σt is drivenby two stochastic processes
Yt and Zt varying on the time scales ε and 1/δ, respec
(0)∗
(1)∗
(2)∗
tively. The vector Wt , Wt , Wt
consists of three independent standard Brownian motions.
2
Instant correlation coefficients ρ1 , ρ2 , and ρ12 satisfy |ρ1 | < 1 and |ρ22 + ρ212 | < 1. The volatility function
f is assumed to be smooth bounded. Coefficient functions of processes Yt and Zt , namely (c1 , g1 , Λ1 )
and (c2 , g2 , Λ2 ) are assumed to be smooth such that they satisfy the existence and uniqueness conditions for stochastic differential equations. Mean-reverting processes such as Ornstein-Uhlenbeck (OU)
processes or square-root processes are typical examples to model driving volatility processes [1, 10, 14].
Under this setup, the joint process (St , Yt , Zt ) is Markovian.
Given the multi-factor stochastic volatility model (1), the price of a plain European option with the
integrable payoff function H and expiry T is defined by
P ε,δ (t, x, y, z) = IE ? t,x,y,z e−r(T −t) H(ST ) ,
where IE ? t,x,y,z is a short notation for the expectation with respect to IP ? conditioning on the current states St = x, Yt = y, Zt = z. A basic Monte Carlo simulation estimates the option price
P ε,δ (0, S0 , Y0 , Z0 ) at time 0 by the sample mean
N
1 X −rT
(i)
e H(ST )
N i=1
(2)
(i)
where N is the total number of sample paths and ST denotes the i-th simulated stock price at time
T . Variance reduction techniques are particularly important to accelerate the computing efficiency
of the basic Monte Carlo pricing estimator (2). Next we briefly review the construction of a generic
algorithm, i.e. martingale control variate method, recently proposed by Fouque and Han [6, 7].
2.1
Construction of Martingale Control Variates
Assuming that the European option price P ε,δ (t, St , Yt , Zt ) is twice differentiable in state space and
once differentiable in time, we apply Ito’s lemma to its discounted price e−rt P ε,δ , then integrate from
time 0 to the maturity T . After canceling out the pricing partial differential equation in some nonmartingale terms, the following martingale representation can be obtained
√
1
P ε,δ (0, S0 , Y0 , Z0 ) = e−rT H(ST ) − M0 (P ε,δ ) − √ M1 (P ε,δ ) − δM2 (P ε,δ ),
ε
where centered martingales are given by
Z T
∂P ε,δ
ε,δ
M0 (P ) =
e−rs
(s, Ss , Ys , Zs )f (Ys , Zs )Ss dWs(0)∗ ,
∂x
0
Z T
∂P ε,δ
(s, Ss , Ys , Zs )g1 (Ys )dW̃s(1)∗ ,
M1 (P ε,δ ) =
e−rs
∂y
0
Z T
∂P ε,δ
(s, Ss , Ys , Zs )g2 (Zs )dW̃s(2)∗ ,
M2 (P ε,δ ) =
e−rs
∂z
0
where the Brownian motions are
q
1 − ρ21 Ws(1)∗ ,
q
(0)∗
(1)∗
= ρ2 Ws + ρ12 Ws + 1 − ρ21 − ρ212 Ws(2)∗ .
W̃s(1)∗ = ρ1 Ws(0)∗ +
W̃s(2)∗
3
(3)
(4)
(5)
(6)
These martingales play the role of “perfect” controls for Monte Carlo simulations. Namely, if the martingales (4), (5), and(6) can be exactly computed, then one can just generate one sample
ε,δpathε,δto evalu∂P
∂P ε,δ
∂P
ate the option price through Equation (3). Unfortunately the gradient components ∂x , ∂y , ∂z
of the option price appearing in the martingales is not possibly known in advance when the option
price P ε,δ itself is exactly what we want to estimate. However, one can choose an approximate option
price to substitute P ε,δ used in the martingales (4, 5, 6) and still retain martingale properties.
When time scales ε and 1/δ are well separate; namely, 0 < ε, δ 1, an approximation of the BlackScholes type is derived in [10]
P ε,δ (t, x, y, z) ≈ PBS (t, x; σ̄(z))
(7)
√ √ ε, δ for continuous piecewise payoffs. We denote by
with its pointwise accuracy of order O
PBS (t, x; σ̄(z)) the solution of the Black-Scholes partial differential equation with the constant volatility
σ̄(z) and the terminal condition PBS (T, x) = H(x). The z-dependent effective volatility σ̄(z) is defined
as the square root of an averaging of the variance function f 2 with respect to a limiting distribution
of Yt :
Z
2
(8)
σ̄ (z) = f 2 (y, z)dΦ(y) = f 2 (y, z) ,
where Φ(y) denotes the invariant distribution of the fast varying process Yt while setting the volatility
premium Λ1 as zero. We use the
√ bracket to represent such average. In the OU case, we choose that
c1 (y) = m1 − y and g1 (y) = ν1 2 with Λ1 = 0 such that 1/ε is the rate of mean reversion, m1 is the
long run mean, and ν1 is the long run standard deviation. Its invariant distribution Φ is normal with
mean m1 and variance ν12 . We refer to [10] for detailed discussions of such models.
Note that the approximate option price PBS (t, x; σ̄(z)) is independent of the variable y such that √
the
(i)
term M1 (PBS ) diminishes. Since the approximate martingale M2 (PBS ) for (6) is small of order δ.
Intuitively we can neglect this term as well. We then select the stochastic integral M0 (PBS ) as a
control for the estimator (2) and formulate the following martingale control variate estimator:
N
i
1 X h −rT
(i)
(i)
e H(ST ) − M0 (PBS ) .
N i=1
(9)
This is the approach taken by Fouque and Han [6], in which the proposed martingale control variate
method is numerically superior to an importance sampling method in [5] for pricing European options.
Under OU-type processes to model (Yt , Zt ) in (1) with 0 < ε, δ 1, the variance of the martingale
control variate for European options is small of order ε and δ. Recall the asymptotic result shown in
[7]
Theorem 1 Given the payoff function H being a call or put, the driving volatility processes Yt and
Zt are of OU type, and the volatility function f (y, z) are smooth bounded, for any fixed initial state
(0, x, y, z), there exist ε > 0 and δ > 0 small enough and a positive constant C such that
V ar e−rT H (ST ) − M0 (PBS ) ≤ C max{ε, δ}.
Moreover the financial interpretation of the martingale control term
Z T
∂PBS
M0 (PBS ) =
e−rs
(s, Ss ; σ̄(Zs ))f (Ys , Zs )Ss dWs(0)∗
∂x
0
corresponds to the cumulative cost of a delta hedging strategy. This martingale control variate method
can be easily applied to hitting time problems like barrier options and optimal stopping time problems
like American options as discussed in [7].
4
2.2
An Example: Martingale Control Variate Method for Heston’s Model
Another widely applied mean-reverting process is Heston model [14] which incorporates a square-root
diffusion to model the random volatility process. Heston model is an one-factor stochastic volatility
model so that it can be viewed as a special case of our multi-factor SV model (1) after the square-root
process is rescaled in the following way:
q
2κ p
1
(1)∗
(0)∗
2
(10)
Yt ρ1 dWt + 1 − ρ1 dWt
dYt = (θ − Yt ) dt + √
ε
ε
√
and Zt = 0 for all t ≥ 0. The volatility function is defined by f (y, z) = y. The driving volatility (10)
admits a limiting distribution while ε approaches zero. It is known as the central Chi-square random
variable
κ2 χν (0)
(11)
with the degree of freedom ν = θ/κ2 . Its density function is denoted by
x ν/2−1
−x
1
1
exp
φ(x) = 2 ν/2 ν
κ 2 Γ( 2 ) κ2
2κ2
(12)
for x > 0, where Γ(·) is the Gamma function.
Under the risk-neutral pricing probability measure, the European option price is
P ε (t, x, y) = IE ? t,x,y e−r(T −t) H(ST ) .
Following the singular perturbation expansion [9], one can derive an asymptotic result
P ε (t, x, y) ≈ PBS (t, x; σ̄)
(13)
when Rε 1. As defined in (8) but without the z-component, the effective volatility σ̄ is defined from
σ̄ 2 = f (y)2 φ(y)dy. By change of variable z = κy2 , the effective variance
σ̄
2
κ2
= ν/2 ν
2 Γ( 2 )
Z
∞
z ν/2 e−z/2 dz
0
2
= κν
= θ
(14)
is deduced. We readily observe that the effective variance σ̄ 2 is exactly the long-run mean θ of the
square root process (10).
Under the one-factor Heston model specified in Table 1 and 2, the effective variance is σ̄ 2 = θ = 0.3.
Several variance ratios are illustrated in Table 3 and show significant reduction gained from the
martingale control variate estimator (9) versus the basic Monte Carlo estimator (2). Notice that the
variance reduction ratios performs better when ε are either small (close to 1/50) or large (close to
1/0.1). In the small ε regime, one can use singular perturbation analysis to argue that the approximate
option price PBS (t, x; σ̄) is close to the true option price. Hence the variance reduction is effective. In
the large ε regime, the process Yt is slowly moving
√
√ around Y0 = 0.3. This initial value is particularly
chosen so that the driving volatility process Y t is close to the effective volatility σ̄ = θ when
0 ≤ t ε. Therefore, one can apply regular perturbation argument to explain the accuracy of the
option price approximation and the variance reduction effect. Both singular and regular perturbation
analysis can be found in [10].
5
Table 1: Parameters used in the rescaled Heston’s model ( 10).
r
5%
θ
0.09
κ
0.1
ρ
-0.3
f (y)
√
y
Table 2: Initial conditions and European call option parameters.
$S0
50
Y0
0.09
$K
50
T years
1
Table 3: Comparison of simulated European call option prices and their standard errors defined in
(23) for different time scale ε. Let “MC price” denote the sample mean obtained from the basic
Monte Carlo simulation; “ MC+ CV price” denote the sample mean obtained from the Monte Carlo
simulation using martingale control variates. Their standard errors are shown in parenthesis next to
the option price estimates. Let V M C denote the sample variance obtained from the basic Monte Carlo
method, and V M C+CV (PBS (σ̄)) denote the sample variance computed from the martingale control
variates method with the option price approximation PBS (σ̄) in (13).
ε
1/50
1/25
1/10
1
5
10
25
50
MC price (std err)
7.0771 (0.1080)
6.9862 (0.1072)
7.1530 (0.1085)
7.1588 (0.1065)
7.0064 (0.1059)
7.0785 (0.1090)
7.1145 (0.1101)
6.9737 (0.1095)
MC+CV price (std err)
7.0883 (0.0064)
7.0646 (0.0069)
7.0466 (0.0079)
7.0244 (0.0096)
7.0236 (0.0092)
7.0577 (0.0080)
7.0729 (0.0068)
7.0737 (0.0063)
6
V M C /V M C+CV (PBS (σ̄))
287
244
187
122
132
185
266
298
3
Numerical Test Results
In this section, we will compare efficiencies for pricing European call option using control variates
technique developed in the above, combined with Monte Carlo and quasi-Monte Carlo methods.
3.1
Quasi-Monte Carlo Methods
We briefly describe the QMC methods here, and refer details to the literature. Consider the problem
of estimating an s-dimensional integral
Z
f (x)dx,
(15)
µ=
Cs
where C s = [0, 1]s is the s-dimensional unit hypercube, f (x) is such a function that (at least) the
integral (15) exists. The integral (15) can be expressed as the expectation
Z
f (x)dx =E[f (X)],
(16)
µ=
[0,1]s
where X is a random vector uniformly distributed on [0, 1]s . Thus, µ can be approximated by MC
simulation. In fact, if X1 , X2 , · · ·, XN are iid and uniformly distributed on [0, 1]s , then the (sample)
mean
N
1 X
µ
bN =
f (Xj ),
(17)
N j=1
is an unbiased estimator of µ, i.e., E(b
µN ) = µ. The corresponding (sample) variance (assuming N > 1)

" N
#2 
N
N


X
X
1 X
1
2
2
2
σ
bN =
(18)
[f (Xj ) − µ
bN ] =
[f (Xj )] −
f (Xj )

N −1
(N − 1)N 
j=1
j=1
j=1
is an unbiased estimator of the variance of f (X):
2
2
2
2
Z
σ = E [f (X)−µ] = E [f (X)] − µ =
2
Z
f (x)dx−
Cs
2
f (x)dx .
(19)
Cs
Thus,
lim µ
bN = µ
N →∞
(20)
with probability one by the Strong Law of Large Numbers. Notice that the asymptotic convergence
rate O( √1N ) of the MC method is independent of the dimension of the problem, the most important
advantage of this method. Based on the Central Limit Theorem, we can construct the (1−p) confidence
interval (CI) for a given confidence level p ∈ (0, 1):
CI(p) = [b
µN − zp/2 δN , µ
bN + zp/2 δN ]
(21)
where δN = √σbNN is the (sample) standard error, and zp/2 > 0 is the quantile of the (one-dimensional)
standard normal distribution such that
Z zp/2
1
2
√ e−y /2 dy = 1 − p.
2π
−zp/2
7
The main drawback of the MC method is that it converges slowly. Thus, various variance
reduction methods were developed. Furthermore, the so called quasi-Monte Carlo methods were also
developed. A QMC method to estimate integral (15) is a quadrature rule having the form
µ≈
N
1 X
f (xj )
N j=1
(22)
where {x1 , · · ·, xN } ⊂ [0, 1]s is a deterministic sequence, instead of iid and uniformly distributed on
[0, 1]s . Such a deterministic sequence, called a quasi-random sequence or low discrepancy sequence
(LDS), satisfies certain properties. An LDS is usually more uniformly distributed over [0, 1]s than a
pseudo-random sequence does.
There are two classes of low-discrepancy sequences. One is called the digital net sequences,
such as Halton’s sequence, Sobol’s sequence, Faure’s sequence, and Niederreiter’s (t, s)−sequence, etc.
s
A QMC method using this kind of LDS has convergence rate O( (logNN ) ) when estimating integral
(15). The other class is the integration lattice rule points. This type of LDS is especially efficient for
estimating
multivariate integrals with periodic and smooth integrands, and it has convergence rate
(log N )αs
O( N α ), where α > 1 is a parameter related to the smoothness of the integrand. The monograph
by Niederreiter [24] gives detailed information on digital net sequences and lattice rule points, while
the monographs by Hua and Wang [15], and Sloan and Joe [28] describe lattice rules.
Since their early successful applications in financial problems (see [18] and [26]), the quasi-Monte
Carlo (QMC) Methods have been used more and more popular. For a class of low dimensional (say,
less than 10) European style options, the QMC methods using lattice rules are especially efficient when
estimating both the option prices and sensitivities, the lattice rules are much more efficient than the
digital net sequences, while the digital net sequences are still more efficient than the MC method. For
details, see [2] and [19].
L’Ecuyer also made contributions to lattice rules based on linear congruential generator. One of
the features of this type of lattice rule points (referred to L’Ecuyer’s type lattice rule points, LTLRP,
thereafter) is that it is easy to generate high dimensional LTLRP point sets with convergence rate
comparable to digital net sequences. Details of LTLRP can be found in [21] and references therein.
We will apply the LTLRP as well since our problem is high dimensional. Besides the above LDS, we
also apply the Brownian bridge (BB) sampling technique to our test problems. Detailed information
about Brownian bridge sampling can be found in [12].
To compare the efficiencies of different methods, we need a benchmark for fair comparison. If
the exact value of the quantity to be estimated can be found, then we use the absolute error or relative
error for comparison. Otherwise, we use the unbiased sample variance σ
bn2 for comparison. For LDS
sequences, we define the unbiased sample variance σ
bn2 by introducing random shift as follows. Assume
that we estimate
µ = E[f (X)],
s
s
where X is an s−dimensional random vector. Let {xi }m
i=1 ⊂ I = [0, 1] be a finite LDS sequence and
(j)
{rj }nj=1 ⊂ I s be a finite sequence of random vectors. For each fixed j, we have a sequence {yi }m
i=1 ,
(j)
m
where yi = xi + rj . Such a sequence still has the same convergence rate as {xi }i=1 does as proved in
[29]. Denote
8
m
1 X
(j)
f (yi ),
µj =
m i=1
and
n
1X
µ
bn =
µj .
n j=1
The unbiased sample variance is (assuming n > 1)
Pn
σ
bn2 =
2
−µ
bn )
=
n−1
j=1 (µj
n
Pn
2
j=1 µj −
P
(n − 1)n
n
j=1
µj
2
.
Then the standard error of the randomized LDS is defined as
σ
bn
stderr = √ .
n
(23)
And the variance of the MC method is calculated as usual based on the pseudo random point set
{xi + rj } with mn points. Thus, we can easily construct the confidence interval for a given confidence
level. The variance reduction ratio of a QMC method over the MC method is defined by
variance reduction ratio =
The sample variance of the MC method
.
The sample variance of the randomized QMC method
(24)
Here we do not include the CPU times for different methods. The main reason is that all the programs
were run on a main frame using C++ under the UNIX operating system. There were many other jobs
were also running when our programs were run on the main frame machine. Besides, we noticed that
the differences of CPU times are not significant.
We assume that the underlying asset S is given by (1). In our computations, we use C++ on
Unix as our programming language. The pseudo random number generator we used is ran2() in [30].
In our comparisons, the sample sizes for MC method are 10240, 20480, 40960, 81920, 163840, and
327680, respectively; and those for Sobol’ sequence related methods are 1024, 2048, 4096, 8192, 16384,
and 32768, respectively, each with 10 random shifts; and the sample sizes for LTLRP related methods
are 1021, 2039, 4093, 8191, 16381, and 32749, respectively, and again, each with 10 random shifts.
In the following examples, we divide the time interval [0, T ] into m = 128 subintervals. In the
following Tables 7, 8 and 9, numbers without parentheses in columns (except column 1, which contains
the numbers of points) are option values, and numbers within parentheses in the quasi-Monte Carlo
methods are variance reduction ratios.
Example 1: The European call option with payoff f (ST ) = max(ST − K, 0) = (ST − K)+ , where K is
the strike price. We take input parameters as follows: S0 = $55, K = $50, r = 0.1 = 10%, T = 1 year,
m1 = m2 = −0.8, ν1 = 0.5, ν2 = 0.8, ρ1 = ρ2 = −0.2, ρ12 = 0.0, y0 = z0 = −1.0. For ε and δ, we take
two groups of values: ε = 1/5, δ = 1.0 and ε = 1/50, δ = 0.5. The results are listed in Tables 7 and 8,
where MC+CV stands for Monte Carlo method using control variate technique, Sobol+BB means the
quasi-Monte Carlo method using Sobol’ sequence with Brownian bridge sampling technique, LTLRP
for QMC method using L’Ecuyer type lattice rule points, etc.
Table 7: Comparison of simulated European call option values and variance reduction ratios for
ε = 1/5, δ = 1.0
9
N
1024
2048
4096
8192
16384
32768
MC
12.034
12.216
12.166
12.145
12.144
12.206
MC+CV
12.217(16.7)
12.165(22.1)
12.135(20.3)
12.165(19.1)
12.169(26.3)
12.166(21.1)
Sobol’
12.240(1.6)
12.201(4.6)
12.190(1.1)
12.169(1.5)
12.255(0.2)
12.150(3.1)
Sobol+CV
12.154(35.1)
12.197(127.2)
12.156(27.7)
12.160(30.2)
12.157(64.6)
12.167(78.5)
Sobol+BB
12.030(2.2)
12.235(2.4)
12.141(2.2)
12.183(8.0)
12.179(3.9)
12.133(2.6)
Sobol+CV+BB
12.159(55.3)
12.180(112.4)
12.156(31.1)
12.173(85.3)
12.155(161.0)
12.157(44.8)
Table 7: Comparison of simulated European call option values and variance reduction ratios for
ε = 1/5, δ = 1.0 (continued).
N
1024
2048
4096
8192
16384
32768
LTLRP
12.376(1.2)
12.179(2.1)
12.251(3.2)
12.173(1.9)
12.156(1.6)
12.170(2.6)
LTLRP+CV
12.201(25.4)
12.175(41.1)
12.155(60.8)
12.163(40.5)
12.170(180.1)
12.167(57.0)
LTLRP+BB
12.100(4.7)
12.160(2.9)
12.083(2.5)
12.201(4.0)
12.138(2.6)
12.144(3.2)
LTLRP+CV+BB
12.129(112.1)
12.196(5.1)
12.153(50.5)
12.168(135.9)
12.154(77.0)
12.152(137.3)
Table 8: Comparison of time (in seconds) used in the simulation of the above European option
N
1024
2048
4096
8192
16384
32768
MC
6
11
23
45
91
182
MC+CV
8
17
34
68
136
271
Sobol’
6
11
22
45
88
177
Sobol+CV
7
15
29
61
120
240
Sobol+BB
6
11
23
47
93
186
Sobol+CV+BB
8
17
34
68
134
273
Table 8: Comparison of time (in seconds) used in simulation of the above European option
(continued).
N
1024
2048
4096
8192
16384
32768
LGLP
5
10
19
39
77
155
LGLP+CV
8
15
30
62
121
244
LGLP+BB
8
17
35
69
137
273
LGLP+CV+BB
6
12
23
48
95
191
From Tables 7, we observed the following facts. Using the control variate technique, the variance
reduction ratios are around 20 for pseudo-random sequences. Without control variate, both Sobol’
sequence and L’Ecuyer type LATTICE RULE POINTS, even combined with Brownian bridge sampling
technique, the variance reduction ratios are only a few times better than the MC sampling at most.
However, when combined with control variate, the variance reduction ratios for the Sobol’ sequence
increase from about 28 to 128; and the variance reduction ratios for the L’Ecuyer type LATTICE
RULE POINTS increase from about 25 to 180. This implicitly indicates that the new controlled
payoff e−rT (ST − K)+ − M0 (PBS ) is smoother than the original call payoff e−rT (ST − K)+ . It can
be easily seen that under the Black-Scholes model with the constant volatility σ, the controlled payoff
is exactly equal to the Black-Scholes option price PBS (0, S0 ; σ), which is C ∞ , while the original call
payoff function is only continuous but not differentible.
10
Another interesting observation is that the variance reduction ratios do not always increase when
the two low-discrepancy sequences are combined with control variate and Brownian bridge sampling,
compared with when they are combined with control variate without Brownian bridge sampling. For
this option, the Sobol’ sequence and the LTLRP are “comparable”: sometimes the Sobol’ sequence
achieves better efficiency, and sometimes the other way around.
Regarding time used in simulations, from Table 8 we observed that the time differences among
methods without control variates are not significant, but the time differences between methods with
and without control variates are not ignorable. Similar conclusions are true regarding time used in
simulations for other cases.
Table 9: Comparison of simulated plain vanilla European call option values and variance reduction ratios for 1/ε = 50.0, δ = 0.5
N
1024
2048
4096
8192
16384
32768
MC
11.97
11.97
12.03
11.97
11.99
12.04
MC+CV
12.03(30.3)
12.00(32.7)
11.97(34.6)
11.99(34.4)
12.00(33.2)
11.99(34.0)
Sobol’
12.24(2.2)
12.02(1.6)
12.03(1.3)
12.03(2.0)
12.02(1.4)
11.99(1.7)
Sobol+CV
11.99(219.2)
12.00(300.4)
11.99(107.8)
11.99(104.9)
11.99(92.8)
12.00(156.6)
Sobol+BB
11.75(2.9)
12.00(4.0)
11.97(2.7)
11.98(3.3)
11.99(6.5)
11.99(3.9)
Sobol+CV+BB
11.98(66.6)
12.01(69.9)
11.98(133.5)
11.99(99.9)
11.99(119.4)
11.99(133.4)
Table 9: Comparison of simulated plain vanilla European call option values and variance reduction ratios for 1/ε = 50.0, δ = 0.5 (continued).
N
1024
2048
4096
8192
16384
32768
LTLRP
12.15(1.8)
12.06(2.7)
12.06(2.2)
11.99(3.2)
11.97(2.9)
11.99(6.7)
LTLRP+CV
11.99(39.0)
11.99(75.4)
11.99(65.9)
12.00(173.9)
12.00(173.7)
11.99(227.1)
LTLRP+BB
11.85(8.1)
11.99(3.9)
11.95(2.4)
12.04(2.0)
11.97(5.2)
12.00(13.2)
LTLRP+CV+BB
11.96(172.8)
11.96(159.6)
11.99(37.0)
12.00(25.3)
11.98(45.6)
11.98(87.1)
The conclusion is similar to that of Table 8, but variances are further reduced in this case.
4
Conclusion
In this paper, we study the option pricing problems for European options under multi-factor stochastic
volatility model. Based on the construction of martingale control variates by Fouque and Han [7], we
implemented this method with Monte Carlo and randomized quasi-Monte Carlo methods. For complex
stochastic volatility models, our tests show that quasi-Monte Carlo methods are still more efficient
than Monte Carlo methods. Control variates combined with quasi-Monte Carlo methods are the most
efficient ways in dealing with such option pricing problems. A theoretical problem induced from these
numerical experiments is an error analysis to quantify the error of quasi-Monte Carlo methods under
martingale control variate. This analysis can be used to explain the smooth effect by martingale
control variate. We leave this problem in our future research.
Acknowledgement: We are grateful to a financial support by the Institute for Mathematics and its
Applications (IMA) at the University of Minnesota, USA. The work of C-H Han is supported by NSC
grant 95-2115-M-007-017-MY2, Taiwan; and the work of Y. Lai was partially supported by an Natural
11
Sciences and Engineering Research Council (NSERC) of Canada grant.
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