4.2 Variance reduction methods from option pricing
4.2.1 Asian call options
At first, we consider the pricing problem of an Asian call option on a single stock in a complete market in two different models. The terminal condition in this case is of the form
φ µ
ST, Z T
0 Sudu
¶ , and consequently the forward Markov process is chosen as
Xt= µ
St, Z t
0
Sudu
¶ .
For approximation purposes we replace the terminal condition by some discrete-time counterpart. Here we choose the simplest approximation
φ µ
SπtN,
N−1
∑
j=0
Sπtj∆j
¶ .
Corollary 4.4 of Zhang [46] completes the error analysis of Theorem 1.1.1 and Corollary 2.1.2: There is a constantCsuch that
E
·¯¯
¯¯φ µ
ST, Z T
0
Sudu
¶
−φ µ
SπtN,
N−1
∑
j=0
Sπtj∆j
¶¯¯
¯¯
2¸
≤C|π|.
The first model which we consider is the standard Black-Scholes model with constant coefficients. Hence the FBSDE becomes, after the measure transformation,
dSth = ³
bSth+σSthht´
dt+σSthdWt, dYth =
µ
rYth+b−r
σ Zth+Zthht
¶
dt+ZthdWt, S0h = s0, YTh=
µ1 T
Z T
0 Sthdt−K
¶
+
.
88 4.2. Variance reduction methods from option pricing
Moreover, we consider the model introduced by Bergman [6], where we have different interest rates for lendingrand borrowingR, resulting in a nonlinear driver of the backward part:
dSth = ³
bSth+σSthht´
dt+σSthdWt, dYth =
µ
rYth+b−r
σ Zth−(R−r) µ
Yth−Zth σ
¶
−
+Zthht
¶
dt+ZthdWt, S0h = s0, YTh=
µ1 T
Z T
0 Sthdt−K
¶
+
,
where we used the notationx−=min{x, 0}for any real numberx.
We now have to specify the discrete processhti for the importance sampling. In this section we con-centrate on the variance induced by the terminal condition and neglect the variance stemming from the driver. Therefore we can make use of the rich literature concerning the choice ofh, which is available in the context of option pricing. We pick out a rather heuristic approach by Glasserman [20] for this task.
Glasserman et al. [19] show that this method is equivalent to another asymptotically optimal approach for the linear problem.
We restrict ourselves to equidistant time increments, i.e.∆i =|π|,i=0, . . . ,N−1, and alsohti will have the simple structurehti = √hi
|π| forhi ∈ R. Hence, using the Euler-Maruyama scheme for the forward equation under the measureQh,π, we receive
Sti+1 =Sti+bSti|π|+σSti
hi
p|π||π|+σSti∆Wi=Sti+bSti|π|+σSti
p|π|(ξi+hi)
for standard normal random variablesξi. Definingξ= (ξ0, . . . ,ξN−1)>andh= (h0, . . . ,hN−1)>we also obtainΨ0tN =exp{−h>ξ−12h>h}.
As approximation for the terminal condition we chooseφ(XtN) = ³
N+11 ∑i=0N Sti −K´
+, which is in this form only available in the equidistant time discretization. In abuse of notation we considerφas function ofξso that we can write
E£
φ(ξ)1φ(ξ)>0¤
= Eh
exp{ln(φ(ξ))}1φ(ξ)>0i
= E
·
exp{ln(φ(ξ+h))}exp
½
−h>ξ−1 2h>h
¾ 1φ(ξ)>0
¸
≈ E
· exp
½
ln(φ(h)) +∇ln(φ(h))ξ−h>ξ−1 2h>h
¾ 1φ(ξ)>0
¸ .
If one can findhwhich satisfies∇ln(φ(h)) =h>, we would approximatively end up with the expectation of a constant random variable independent of ξ and consequently get a low-variance estimator. The condition∇ln(φ(h)) =h>is the necessary condition for a local maximum of the functionh7→ln(φ(h))−
12h>h. However, maximizing this function is equivalent to maximizingφ(h)exp(−12h>h). This can be done numerically as described by Glasserman [20].
In our simulations we use 20 time steps and a function basis of bivariate monomialsxα1·x2β forα,β = 0, . . . , 3 for every point in the time grid. That is we chooseK0,i =K1,i = 16 andp0,i(·) = p1,i(·)fori = 1, . . . ,N−1. Choosing the same basis for theY- and theZ-component, we can use the same matrices for several computations thereby reducing considerably the computation time. In full detail, the projection matrices withLrows and 16 columns have the following form:
A0,iL =A1,iL = 1
√L
1Ψ0ti1 1Ψ0ti1Sti . . . 1Ψ0ti(1Sti)3(i+11 ∑ij=0 1Stj)3
... ... ... ...
LΨ0ti1 LΨ0tiLSti . . . LΨ0ti(LSti)3(i+11 ∑ij=0 LStj)3
.
4.2. Variance reduction methods from option pricing 89
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 104 6
6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
number of paths Y0
Mean and std of Y as a function of the number of paths
Importance sampling for Asian call Crude least−squares Monte Carlo
At the money option, linear BSDE.
Figure 4.1: Convergence ofYbtn0stop,Lfor Asian call option.
As indicated in the former subsection the Picard-iteration is executed until the difference of two subse-quent estimators of the initial price is smaller than the stop condition 0.001 that is the iteration is stopped, if |Ybtn,L0 −Ybtn−1,L0 | < 0.001 for the first time. In the option pricing problems of this section the number of executed Picard iterations is rather low. There are no more than 4-6 iterations necessary to get this accuracy.
The following parameters are used for the simulation procedure:
b σ r T s0 K
0.06 0.2 0.1 1 100 100
This example was also examined by Gobet et al. [21] without variance reduction in the context of the backward Euler-type scheme. As reference value Lapeyre and Temam [32] give 7.04 for the linear BSDE.
We calculate the estimators for the initial price for a number of simulations from 500 up to 50.000 and repeat this procedure 100 times, thereby examining the convergence behavior with respect to the number of simulations. Figure 4.1 shows the empirical mean plus/minus two empirical standard deviations of the 100 independent repetitions for the estimated initial valueYbtn0stop,L in the Black-Scholes model using the crude least-squares Monte Carlo method of Bender and Denk [2] and the variance reduced algorithm respectively. We see that importance sampling reduces the empirical variance roughly by a factor of 10, which demonstrates the success of the variance reduction. The estimation of the price is at approximately 7.00 slightly below the reference value of 7.04. This error is mainly due to the crude time discretization of the average. It can be mended by adding more time steps or applying more sophisticated discretization schemes for the average as suggested by Lapeyre and Temam [32] or Jourdain and Sbai [30].
We get a similar result concerning the empirical variance, see Figure 4.2 (a), in the case of different interest rates for lending and borrowing. Here, we useR = 0.15,r =0.1 and all the other parameters stay the same. This method is even more effective in the out of the money case. We illustrate this feature with Figure 4.2 (b), where we again take the model with the nonlinear BSDE but now with a higher strike (K=120). In this case we receive an empirical variance reduction by a factor of 37 in average. The higher effectiveness in the out of the money case is not too surprising since this phenomenon also arises in the simulations of Glasserman [20].
90 4.2. Variance reduction methods from option pricing
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 104 7.5
8 8.5 9 9.5 10 10.5
number of paths Y0
Mean and std of Y as a function of the number of paths
Importance sampling for Asian call Crude least−squares Monte Carlo
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 104 0.8
1 1.2 1.4 1.6 1.8 2
number of paths Y0
Mean and std of Y as a function of the number of paths
Importance sampling for Asian call Crude least−squares Monte Carlo
(a) At the money option, nonlinear BSDE. (b) Out of the money option, nonlinear BSDE.
Figure 4.2: Convergence ofYbtn0stop,Lfor Asian call option.