In this section, we present some numerical tests designed for the typical multi-asset options— including the exchange options, basket options and quanto options—to emphasize the efficiency of our method.
4.1. Exchange Options
The exchange option, which was first studied by Margrabe [
70], empowers its holder with the right to exercise it by comparing the difference between the prices or the rates of return of two underlying assets. Its payoff is:
If the underlying assets evolute with constant volatilities
and
, the exchange option has a pricing formula at time
t, as shown by Margrabe [
70] and Jiang [
20] as follows:
where
and
is the cumulative distribution function of a standard normal variable. It is easy to derive the derivatives:
We assumed the stochastic volatilities obey the Heston model, for
:
The parameters should satisfy the Feller condition [
71] to guarantee the positiveness of variance, i.e.,
and
. We used a truncated Euler discrete scheme [
39,
72,
73] with equal time intervals to simulate the Heston process in our tests.
At first, we wanted to examine the acceleration effect of a CMC simulation compared with a traditional MC simulation. We fixed the parameters
,
,
, and
. Additionally, we took the number of time steps
, and the number of simulations
100,000 in all numerical simulations. Note that
should be satisfied from Equation (
17). Taking
for simplicity, the numerical results with different correlation coefficients are recorded in
Table 2 and
Table 3.
Table 2 records the estimated option values calculated by the MC and CMC simulations, which are denoted as
and
, respectively. The upper part of
Table 3 records the standard errors of an MC simulation, denoted as
. The standard errors are almost the same for various correlation coefficients, and increase slightly with correlation
while decreasing with
. The exchange option can be seen as a call option on asset
for fixed
; a higher correlation
implies a larger variation in the price of asset
, thus resulting in a larger value of the option price and a larger simulation variance. Similar analysis can be conducted with respect to correlation
by regrading the exchange option as a put option on asset
for fixed
.
The middle part of
Table 3 records the standard errors of a CMC simulation, denoted as
. Obviously, the standard CMC errors are always smaller than MC. It is interesting that a standard CMC error rapidly declines as correlation coefficient
or
tends to zero. Thus, the ratio of the standard errors of a CMC simulation to an MC simulation reduces. We denote this ratio as
and present its values at the bottom of
Table 3.
R becomes larger when the correlation coefficient is getting closer to the original point, and decays rapidly in the opposite direction. For example, for
and
, the reduction ratios of the standard error are 13.0340, 3.8805, 2.1170, and 1.4001, respectively. This can be explained by Equation (
11); the CMC simulation removes the randomness that is independent from the stochastic variances
, and its quantity is proportional to
or
. In other words, a larger variance reduction ratio is promised when the absolute value of
or
is smaller. This property indicates that a CMC simulation is more competitive when the correlation between the underlying asset and stochastic volatility is weak.
We also investigated the computational costs of the MC and CMC methods. The computational platform for this paper was an Intel i5-6200U CPU, 2.30 GHz, 8 GB memory, and the software environment was Matlab R2018a for Windows 10. It took 50.88 s to calculate all of the data in the upper part of
Table 3 and 25.85 s for the middle part, which means that the time cost of a CMC simulation is almost half that of an MC simulation. This is because the MC method needs to simulate four random variables,
and
, while the CMC method only needs to simulate two random variables,
and
.
Taking the variance reduction ratio into consideration, the speed up ratio of a CMC simulation to an MC simulation is defined as . Thus, when correlation , the speed up ratio of the CMC is . Even for the case of a larger correlation , the speed up ratio of the CMC is , which improves the efficiency of the MC simulation by roughly 75%. In summary, the CMC simulation enjoys the advantages of saving time and having a great variance reduction ratio, especially when the correlation coefficients are small.
We next tested the efficiency of our martingale CV method. As a contrast, we constructed another CV for the stochastic model, as suggested by Ma and Xu [
74]. Consider dummy assets whose prices
satisfy the following stochastic differential equations:
where
is a determined function. The covariance of
is given by Equation (
3). It can be computed by matching the first two moments of the underlying asset prices as
. In the case of a Heston stochastic volatility model:
We used the payoff
as a CV to the MC method, and we called this CV method a function CV method. The corresponding exchange option price can be computed using Equation (
35) by replacing
in Equations (
36) and (
37) with the average volatility on the interval
given by:
We changed the values of the correlation coefficients and kept the other parameters fixed as before. Remember that
. The detailed results are shown in
Table 4.
In
Table 4,
,
, and
are the standard errors from the MC simulation, the martingale CV method, and the function CV method, respectively.
is the standard error reduction ratio of the martingale CV method compared to the MC simulation, and
is the the standard error reduction ratio of the function CV method compared to the MC simulation.
It is obvious that the standard error reduction ratio of the CMC is much larger than that of the function CV method, the former falling in 9–20 while the latter being about 3 or 4.
Table 4 also shows that, for a fixed
, the standard errors of the MC simulation, martingale CV method, and function CV method decrease with the correlation value of
. For a fixed
, the standard errors of the MC simulation and function CV method increase with the value of
while the martingale CV method decreases with the absolute value of
, which is mainly caused by the properties of the CMC. Thus, the standard error reduction ratio of the martingale CV method also decreases with the absolute value of
.
The computing times for all values of the MC, the martingale CV, and the function CV methods are 22.33, 22.26, and 25.50 s, respectively. The time costs of the MC method and the martingale CV method are almost the same, while the function CV method is slightly slower. Thus, the martingale CV method proposed in our paper is superior to the function CV method, when considering the variance reduction ratio and the time cost.
Fixing the parameters
and
, we next examined the effects of the volatility parameters for the stochastic volatility. In the Heston stochastic volatility model, the Feller condition should be satisfied [
71]; thus,
and
. Numerical results of these tests are shown in
Table 5.
As shown in
Table 5, the standard errors of the three simulation methods all increase with increasing volatilities of stochastic volatilities. Standard error reduction ratios also decline with the volatility of the stochastic volatilities. However, our martingale CV method is much more efficient than the function CV method, especially in the case of large volatility.
4.2. Basket Options
The payoff of the basket option at maturity depends on the average price of the underlying assets. Since the basket option with arithmetic average price does not have a closed-form price, even with constant volatility, we considered the geometric average basket option whose payoff at time
T is:
where
n is the number of underlying assets,
are the weights of each underlying asset with
, and
K is the strike price.
The geometric average basket option has a closed-form solution if the underlying assets have constant volatilities as
. Denote:
The geometric average basket option price at time
t is given by Jiang [
20]:
where
Thus, the derivatives are:
For a basket option with
n underlying assets, we still used the Heston stochastic volatility model and function CV method as a comparison. The expectation of the corresponding CV can be calculated using Equation (
39) by substituting
with
We fixed the parameters
,
and
. We allocated equal weights for the underlying assets, which means that
. For the initial value of the stochastic volatility, we took a linear interpolation between
and
for the
n assets. In other words, the initial variance vector was
for
and
for
. We took the long-term mean of stochastic variance as
, which was
for
, for example. For the correlations between Brownian noises, we took
for simplicity. To guarantee the positive definiteness of the matrix
, the parameter
should satisfy
. In addition,
is needed for the proper definition of
. Thus, we set
at first. We fixed the number of time steps to
and the number of simulations to
100,000. We tested the acceleration effects of the CVs for different correlation coefficients
. Numerical results are shown in
Table 6.
Table 6 again shows that
, the standard error of the martingale CV method, decreases as the correlation coefficients
tends to zero, resulting in a greater standard error reduction ratio
in those cases. For example,
goes from 30 to 9 when
goes from 0 to 0.75. On the other hand, the simulation error
and, thus, the corresponding reduction ratio
of the function CV method are not sensitive to the correlation coefficient. The reduction ratio is around 5 in all cases. Considering the number of underlying assets, the reduction ratio of the martingale CV slightly decreases as the number of assets
n becomes larger, except for the
cases. For example, the reduction ratios of the martingale CV method are 9.2740, 7.6660, and 6.3077 for
, and 10, respectively, and for
. On the other hand, the ratios are 30.5615, 110.3248, and 291.4214 for the
case. As a contrast, the performance of the function CV method is more stable with different
n. It is obvious that our martingale CV is much more efficient than the function CV method.
Next, we fixed
and changed the value of
, the volatility of the stochastic volatility. For convenience, we took an equal
for every underlying asset. The results are recorded in
Table 7.
As shown in
Table 7, the standard errors of the three simulation methods increase with the volatilities of the stochastic volatility at fixed
n, and decrease with the number of underlying assets for a fixed
. The standard error reduction ratios of the two CV methods decrease with increasing volatility of the stochastic volatility and increasing number of assets. However, the martingale CV method is more robust for different volatilities compared to the function CV method. For example, for the case of
, the standard error reduction ratio of the martingale CV method decreases from 13.7479 to 12.2580 when
increases from 0.1 to 0.4, while that of the function CV method sharply decreases from 15.6830 to 4.1894. The results suggest that our martingale CV method is especially efficient in high volatility cases, while the function CV method has some advantages in a low volatility environment.
4.3. Quanto Options with Real Data
The quanto option is a contract written when someone invests money in foreign securities. Usually, its risk depends on the volatility of the securities’ prices and the change of the foreign currency rate. Its main purpose is to provide exposure to a foreign asset without taking the corresponding exchange rate risk. We applied our method to price a quanto option. Park et al. [
62] used a power series expansion method to obtain an analytic approximation value for the quanto option price under the Hull–White stochastic volatility model.
First, we give the quanto option pricing model with Hull–White stochastic volatility, as shown in Park et al. [
62]. Let
be a stock price in foreign currency, and
be a foreign exchange (FX) rate, that is the amount of domestic currency value per one foreign currency value. In a risk-neutral world, they are assumed to obey the following stochastic differential equations:
where
is a risk-free domestic interest rate and
is a risk-free foreign interest rate.
The correlations among the Brownian noises are given by
,
, and
. Additionally,
and
are the stochastic volatilities of the stock price and the FX rate, respectively. This form of the Hull–White stochastic volatility is a little different from that in
Table 1 (for more details, please see Park et al. [
62]). The parameters
and
are constants.
Park et al. [
62] considered a specific quanto option with payoff:
where
is a predetermined FX rate, and
K is the strike price. A more general quanto option payoff would be
(see Jiang et al. [
20]). When volatilities
and
take constant values
and
, respectively, the authors gave the Black–Scholes quanto option price as:
where
It is easy to obtain the derivative
The authors [
62] supposed a quanto European call option of the S&P500 index with 1200 strike and a predetermined FX rate of 1100 (KRW/USD). The model parameters shown in
Table 8 were observed on 13 October 2010. Furthermore, we assume that the contract multiplier of the S&P500 option is 100 and the maturity is 13 October 2011. Without loss of generality, we set the unobserved values as zeros.
We changed the values of the correlation between the S&P500 and the FX rate and fixed all other parameters. The number of time steps was set to
and the number of simulations was set to
100,000. The numerical results for these models are recorded in
Table 9.
In
Table 9,
stands for the correlation between the S&P500 and FX rate.
is the approximated value obtained by the series expansion method in Park et al. [
62].
,
, and
are the estimated values of the MC simulation, the martingale CV method, and the function CV method, respectively.
,
, and
are the standard errors of the MC simulation, the martingale CV method, and the function CV method, respectively.
is the standard error reduction ratio of the martingale CV method compared to the MC simulation, and
is the the standard error reduction ratio of the function CV method compared to the MC simulation. For the function CV method, the expectation of the corresponding CV can be calculated by using Equation (
41) and substituting
with
, where
. It is obvious that our martingale CV method has a larger standard reduction ratio than the function CV method. This, again, shows the efficiency and robustness of our method.