Implementing Importance Sampling in the Least-Squares Monte Carlo Approach for American Options Manuel Morales ∗ Department of Mathematics and Statistics University of Montreal April 2006
Abstract We illustrate how importance sampling can be implemented in the Least-Squares Monte-Carlo approach (LSM) introduced by Longstaff and Schwartz (2001). The fact that the LSM estimates the optimal stopping rule, gives way to the application of a change of measure to accelerate the simulation. An Accelerated Least-Squares (ALSM) estimator is presented and compared with the straight forward LSM. The rate of convergence of this estimator is empirically studied as a function of the chosen change of measure. An algorithm to optimize the choice of the change of measure is discussed.
1
Introduction
In financial as in insurance mathematics, closed-form expressions are not always available for many quantities of interest. Simulation techniques are oftenly use to deal with these situations. Ruin probabilities and option ∗
Manuel Morales. Department of Mathematics and Statistics, University of Montreal. CP. 6128 succ. centre-ville. Montreal, Quebec. H3C 3J7. CANADA. Tel: 1-(514) 343 6697. Fax: 1-(514) 343 5700. Email:
[email protected]
1
prices are often the target of simulation techniques that allow for quantitative analysis. In finance, American-style options, i.e. options with early exercise opportunities are an example of derivatives for which closed-form solutions for their price are not available. Simulation methods have been successfully applied in these cases [Boyle (1977)], however, in the case of American-style securities there is a difficulty: the need to estimate optimal exercise policies as well. The American-style option pricing problem is to find A = max E e−rτ f (Sτ ) , (1) τ
under the risk neutral measure, where f is the payoff from the option at exercise, r is the riskless rate of interest, T the maturity and τ is any stopping time such that τ < T . Focusing only on a discrete time approximation of the problem, the exercise opportunities are restricted to be on the set 0 = t0 , t1 , . . . , tK = T . This optimal stopping time is [see Duffie (1996) or Karatzas (1988) for a further discussion on American options]
where
τ ∗ = min {tk 6 T |f (Stk ) > F (Stk )} ,
(2)
F (Stk ) = max E e−r(τ −tk ) f (Sτ )|Ftk .
(3)
>
τ tk
That is, the option is exercised as soon as the immediate exercise value f (Stk ) is at least as great as the value from continuation F (Stk ). Notice that this optimal rule is recursively defined, working backwards in time from (3) one can define a sequence of stopping times {τk }k=0,1,...,K representing the optimal stopping strategy in the interval [tk , T ]. If this optimal stopping rule were known, then it would be enough to simulate N paths for the stock price, then the estimate for each path would be erτ f (Sτ ) and one would just have to average over all simulated path estimates. But the optimal stopping rule is not known, it depends on F (Stk ) = E e−r(τk −tk ) f (Sτk )|Ftk , k = 0, 1, . . . , K . (4) This value also has to be estimated from the simulation. This brings some difficulties when implementing a simulation to estimate an American-style option. 2
There have been several attempts to solve this problem, we find in the literature examples such as Tilley (1993), Barraquand and Martineau (1995), Carri`ere (1996), Broadie and Glasserman (1997), Broadie et al. (1997) or Carr (1998). In some of these approaches they approximate the early exercise boundary or estimate the optimal stopping rule in some fashion in order to price an American-style security. The paper by Longstaff and Schwartz (2001) uses yet a fundamentally different approach. They focus directly on the conditional expectation function (4). This expectation can be estimated from simulated paths by using a simple least-squares regression. Recall that this expectation defines the optimal exercise strategy (2), the holder of an American-style option optimally compares the payoff from immediate exercise with the expected payoff from keeping the option alive, and then exercises whenever the immediate payoff is higher. Then, once they have an estimate Fb (Stk ) for (4) they obtain a complete description of the optimal stopping rule along each path by using (2) recursively. With this, American-style options can be estimated by simulation. Moreni (2003) discusses the theoretical aspects of implementing importance sampling within the Longstaff–Schwartz model for American options. A comparative study of variance reducion methods for American options can be found in Lemieux and La (2005). Arouna (2003) explores an adaptative application of importance sampling in option pricing. In this paper, we seek to provide yet another illustration of importance sampling as an efficient variance reduction technique for Monte Carlo option pricing. We also illustrate how to implement an adaptative importance sampling method in the Lonstaff–Schwartz algorithm. In Section 2 and 3 we describe the LSM approach of Longstaff and Schwartz (2001). In Section 4 we implement importance sampling as a mean to accelerate the convergence of the LSM estimator. We also present a way to find the optimal change of measure needed to speed up the simulations. Finally, Sections 5 and 6 give results, comparisons and conclusions.
2
The LSM Approach
Let (Ω, F , P) be a complete probability space where Ω is the set of all possible paths of the stock process {St }t∈[0,T ] , and let {Ft }t∈[0,T ] be the filtration generated by the stock process. Consistently with the no-arbitrage theory 3
there exists an equivalent martingale measure Q. Now, if the payoff function f (St ) of an American-style option is in L2 (Ω, F , Q) then the value A(St ) of such an option will be the maximized values of discounted cash flows from the options over all Ft -stopping times (5) A(St ) = max E e−rτ f (Sτ ) . τ
If we focus on the case where the option can be exercised at times 0 = t0 < t1 < . . . , < tK = T then the optimal stopping rule is given by (2). Basically this means that the option holder chooses, at every time tk , whether to exercise or to keep the option alive. This is done by comparing the value of immediate exercise to that of continuation. This algorithm aims at estimating the conditional expected value of continuation. In order to estimate (4) they approximate the conditional expectation of the value of continuation F (ω, t) with a linear combination of weighted Laguerre polynomials of the form X
Ln (X) = e− 2
eX dn (X n e−X ) , n dX n
n = 0, 1, . . . .
(6)
This approximation uses the fact that the Laguerre polynomials form a basis of the space where F (ω, t) is an element of. Then, F (ω, t) can be represented as ∞ X F (ω, tk ) = aj Lj (Stk ) . j=0
The estimation of the conditional expectation is done by performing a regression using the first M polynomials in (6). The regression is on N simulated paths, under the risk-neutral measure, of the stock price. The resulting FbM (ω, tk ) is then used to estimate the optimal strategy by substituting F (ω, tk ) by FbM (ω, tk ) in (2) [see Longstaff and Schwartz (2001) for details]. The choice of this basis is entirely arbitrary, there are many other choices for a basis of this space [see Stentoft (2004)]. Here, a simpler basis is used. Using only simple powers of X we have the following expression F (ω, tk ) =
∞ X
aj Stjk .
(7)
j=0
One of the nice features of this approach is that the method proved to be more or less robust with respect to the basis used. The number of terms of 4
the basis that are used in the regression is important in the accuracy of the estimator. Here we use the first four terms in (7). According to the study by Stentoft (2004), taking a cubic polynomial yields results within the same margin of error as those obtained using (6). Once we have a full description of the optimal rule τ ∗ we can apply it to a set of simulated paths to obtain the option value for that path ∗
LSM(ωi , M, K) = e−rτ f (Sτ ∗ ) , and then the estimate of the option price would be the average over all paths of this estimated values LSM(ωi , M, K).
3
The LSM Implementation
In the Black-Scholes framework St follows a geometric Brownian motion St = S0 e(r−
σ2 )t+σWt 2
,
t>0,
where Wt is a standard Brownian motion. We simulate N paths. Each path is composed of prices of S at times (n) k = 1, 2, . . . , K. Let St be the nth simulated path under the risk neutral measure and also let 0 = t0 , t1 , . . . , tK = T be the exercise times. If we have N paths we would have, at time T , a sequence of possible realization (1) (2) (N ) of ST . Namely ST , ST , . . . , ST . At this time the payoff from the option, conditioned on not exercising before time T , is simply f (ST ) as if it were an European-style option. Now, at time tK−1 , if the option is in the money, the option holder must decide whether to exercise or to continue the option’s life. Denote by X the vector composed by prices for which the option is in the money at time tK−1 , (n )
(n )
(n )
r 1 2 X = (StK−1 , StK−1 , . . . , StK−1 ).
Also denote by Y the vector of discounted payoffs received at time T , if the options is not exercised at time tK−1 , for those paths for which the option is in the money at time tK−1 , (n1 )
Y = (e−r(T −tK−1 ) f (ST
(n2 )
), e−r(T −tK−1 ) f (ST
5
(n )
), . . . , e−r(T −tK−1 ) f (StT r )) .
To estimate the expected cash flow from keeping the option alive, conditional on the stock price at time tK−1 , the vector Y is regressed on a linear combination of a certain type of functions Li of X Y =
M X
ai Li (X) ,
ai ∈ R .
(8)
i=1
In this paper we use Y = a1 +a2 X +a3 X 2 +a4 X 3 , which has been shown to be an adequate choice [Stentoft (2004)] since it produces similar results as those obtained by Longstaff and Schwartz (2001) but it is easier to implement. In all cases, the resulting function is an estimate of F (X) = E[Y |X] as defined in (3). With the estimate Fb (St ) we can compare the value of immediate exercise at time tK−1 to the value from continuation and define the optimal strategy n o ∗ τK−1 = min t ∈ [tK−1 , T ]|f (St ) > Fb (St ) , (9) where Fb is the function obtained with the regression. This defines an optimal ∗ at time tK−1 . For those paths where the option is out of the strategy τK−1 money, the holder should not exercise it and let the option continue to time T . For those paths where the option is in the money the holder should use (9) to decide whether or not to exercise, i.e., compare f (StK−1 ) with Fb (StK−1 ) and exercise if the former value is larger, otherwise keep the option alive and exercise at maturity tK = T . This defines cash flows from the option at times tK−1 and T . Then, in a similar way, we can build an optimal strategy for time tK−2 Redefine the vector (n )
(n )
(n )
1 2 r0 , StK−2 , . . . , StK−2 ) X = (StK−2
to be the set of paths for which the option is in the money at time tK−2 . There are nr0 of such paths. Also redefine the vector −r(τ −t ) (nr 0 ) (n1 ) (n2 ) K−1 K−2 f (SτK−1 ), f (SτK−1 ), . . . , f (StτK−1 ) Y = e to be the set of discounted payoffs at time tK−2 obtained using the strategy τK−1 . Notice that the discounting factor is different for each entry of the 6
vector. It depends on whether the option was exercised at time τK−1 = tK−1 or τK−1 = T . A new regression of the form Y = a1 + a2 X + a3 X 2 + a4 X 3 is performed. This yields new estimated values for ai and consequently a new expression for the conditional expectation of the cash flows given the stock price at tK−2 . With this, the optimal strategy is updated to τK−2 . In a recursive way one can work backwards to obtain a complete estimate of the optimal strategy τ0 at time 0. Once we have identified an optimal strategy τ0 , the payoff from each path can be computed by ). e−rτ0 f (Sτ(n) 0 The simulation estimate is then the average over all the sample payoffs LSM(ω, K) =
N X
e−rτ0 f (Sτ(n) ). 0
n=1
4
An Accelerated LSM
Longstaff and Schwartz (2001) and Stentoft (2004) use antithetic simulation as a variance reduction technique for the LSM approach. Here, we implement an accelerated LSM based on a change of probability measure. Theoretical aspects of this implementation in the Longstaff–Schwartz algorithm have been discussed in Moreni (2003). Importance sampling has also been used by Broadie and Glasserman (1997) and Dufresne and Vazquez-Abad (1998) to improve the accuracy of Monte Carlo simulations. Notice that in our estimation of the price of the option there might exist several simulated paths that stay out-of-the-money, and therefore do not contribute to the estimation. The number of these zero-paths increases for options with high volatility or with a high (or low) strike price that makes the option stay in an out-of-the-money region. Under this change of measure the objective is to simulate fewer of these zero-paths, and then get back the estimate under the risk-neutral measure using the Radon-Nikodym theorem. The American option price is ∗ (10) A = EP e−rτ f (Sτ ∗ ) , where P is the risk-neutral measure and τ ∗ = min {tk 6 T |f (Stk ) > F (Stk )} . 7
(11)
In the LSM approach, F (Stk ) is approximated by Fb (Stk ) = atk + btk Stk + ctk St2k + dtk St3k ,
k = 1, 2, . . . , K ,
where atk , btk , ctk and dtk are constants for k = 1, 2, . . . , K. This defines a stopping rule n o τLSM = min tk 6 T |f (Stk ) > Fb (Stk ) ,
(12)
(13)
and then the LSM estimator is LSM(K, N ) = EP e−rτLSM f (SτLSM ) .
(14)
Notice that the payoff f depends on the previously simulated values of S up to time τLSM where the option is finally exercised. This is because at each time k we evaluate the stopping rule, until we stop at time τLSM . In turn, each one of the Stk is a function of a normal random variable Zk . This is because in the Black-Scholes framework, S follows a geometric Brownian motion, i.e. σ2 Stk = Stk−1 e(r− 2 )tk +σZk , k = 1, 2, . . . , K . Then, we can write (14) as LSM(K, N ) = EP [h(Z1 , Z2 , . . . , ZτLSM )] ,
(15)
where Zi are i.i.d. normal random variables with mean r(T /K) and variance σ 2 (T /K) (T is the maturity of the option and K is the number of discrete times in which it is divided). The function h is described through (14) and (13), and merely indicates this dependance on the Zi ’s. If we simulate the price process under a new measure, where Zi are i.i.d. normal variates with mean µ(T /K) and variance σ 2 (T /K), then by the Radon-Nikodym theorem we can rewrite this last equation as dP , LSM(K, N ) = EQ h(Z1 , Z2 , . . . , ZτLSM ) dQ where the Radon-Nikodym derivative is the likelihood ratio between two normal densities with common variance (Zi −r(T /K))2
τY − LSM 2σ 2 T /K dP e (Z1 , Z2 , . . . , ZτLSM ) = (Z −µT /K)2 dQ − i 2 2σ T /K i=1 e
8
,
which takes the simple form " #2 p p τX LSM (µ − r) (T /K) dP τLSM (µ − r) (T /K) = exp − − Zi . dQ 2 σ σ i=1
This holds since any normal variate is absolutely continuous with respect to another normal variate. In order to estimate (14) we simulate N paths, each one composed of prices at times k = 1, 2, . . . , K as in the straight forward LSM. At the same time, we compute the Radon-Nikodym dP (i) (i) (Z , Z2 , . . . , Zτ(i) ), LSM dQ 1 derivative for each path i = 1, 2, . . . , N . Then our estimate for (14) is N 1 X −rτLSM dP (i) (i) \ LSM (K, M) = (Z1 , . . . , Zτ(i) e f (SτLSM (Z1 , . . . , Zτ(i) ) ), LSM LSM N i=1 dQ
where the {Zj |j = 1, 2, . . . , τLSM } are simulated under the measure Q. Now, if we can find a change of measure Q (parameterized by µ) such dP (Z1 , . . . , ZτLSM ) < 1 [see L’Ecuyer (1994)] and such that the option that dQ remains in-the-money then the reduction of the variance of the estimator of (14) is assured. A variance reduction implies a more accurate estimate. The problem of finding the optimal µ under which the estimator has the minimum variance is pointed out in the following example.
4.1
Pricing an American Put Option
We revisit the example in Longstaff and Schwartz (2001). Consider an American put option on a share of a non-dividend paying stock, where the riskneutral price process is driven by the dynamics specified in Black and Scholes (1973) t>0, dSt = rSt dt + σSt dWt , where r = 0.06 is the risk-free rate, σ = 0.2 the volatility, and W is a standard Brownian motion. The option is exercisable 50 times per year at a strike price of 40 and for maturity of T = 1. For this option the payoff function is f (Stk ) = (40 − SτLSM )+ , 9
Figure 1: Variances of the ALSM for Different Values of S0 .
14 12
Variance
10 S=36 S=38 S=40 S=44
8 6 4 2
0.08
0.04
0.00
-0.04
-0.08
-0.12
-0.16
-0.20
-0.24
-0.28
-0.32
-0.36
-0.40
-0.44
0
mu
where τLSM is the LSM stopping rule. First the stopping rule is estimated with a first simulation. Then this rule is used on the simulated paths generated under different new measures Q where the option is in-the-money. The desired estimator and its variance are obtained by switching back to the risk-neutral measure P. In Figure 1 the estimated variance of different American put are presented for the accelerated LSM algorithm. We ran only 1200 simulations. Notice the concavity of the variance as a function of µ and how the optimal change of measure is attained at values to the left of the risk neutral measure r (0.06 in this case). We can see how for an S0 = 36 the optimal µ is somewhere around −0.12 and for S0 = 44 the optimal is around −0.29. These µ∗ are negative because the transformed paths under the new measure Q have to stay in-the-money and that implies staying below the strike price of 40. We can also see that as the option starts deeper out-of-the-money the flatter the variance curve it becomes. We then choose the µ∗ that shows the minimum
10
variance according to the graphs and perform simulations of another 100,000 paths to estimate the price under the optimal change of measure µ∗ . Figure 2: Results for LSM and ALSM Simulations.
S
Sigma
T
FD
LSM
S.E.
36 36 36 36
0.2 0.2 0.4 0.4
1 2 1 2
4.478 4.840 7.101 8.508
4.458 4.799 7.085 8.470
0.010 0.012 0.019 0.023
38 38 38 38
0.2 0.2 0.4 0.4
1 2 1 2
3.250 3.745 6.148 7.670
3.234 3.714 6.121 7.625
40 40 40 40
0.2 0.2 0.4 0.4
1 2 1 2
2.314 2.885 5.312 6.920
44 44 44 44
0.2 0.2 0.4 0.4
1 2 1 2
1.110 1.690 3.948 5.647
0.003 0.004 0.008 0.009
Time (sec) 29 28 29 28
-0.19 -0.18 -0.37 -0.3
3.242 3.725 6.146 7.652
0.003 0.004 0.007 0.009
28 27 29 28
-0.2 -0.16 -0.42 -0.3
15 15 16 16
2.308 2.869 5.311 6.905
0.003 0.004 0.007 0.009
26 26 27 27
-0.28 -0.21 -0.49 -0.3
11 12 14 15
1.106 1.678 3.947 5.635
0.002 0.002 0.006 0.008
22 23 25 25
-0.33 -0.18 -0.55 -0.29
Time
ALSM
S.E.
19 18 19 18
4.472 4.820 7.098 8.491
0.010 0.011 0.019 0.023
18 17 19 18
2.294 2.851 5.289 6.875
0.009 0.011 0.018 0.022
1.108 1.676 3.930 5.601
0.007 0.009 0.016 0.021
(sec)
mu*
In Figure 2 LSM (straight forward simulation) and ALSM (accelerated simulation) estimates for different American puts are presented for an optimal choice of µ∗ (100,000 paths). We also show an estimate obtained using a finite difference method as a benchmark. Recall that the LSM is a biased estimator of the true price, its accuracy depends on the number of polynomial terms in the regression. Longstaff and Schwartz (2001) use the first four Laguerre polynomials in their estimation and argue that their estimates lie within an acceptable distance from the true value. In this context, the ALSM is an estimate of the biased LSM estimator and not of the true price, however it lies within the same acceptable distance from the true price as the LSM of Longstaff and Schwartz (2001). 11
Notice in Figure 2 that the variance improvement of ALSM with respect to the straight forward LSM is important. This reduction is slightly better than the one obtained when using antithetic simulation as in Stentoft (2004). As for the computational cost of implementing importance sampling, we can see that the increase in time is not significant An arising problem is to automatize the choice of µ so that the proposed ALSM is really an improvement with respect to the LSM. This problem is dealt with in the following section.
4.2
Sensitivity Analysis
We have seen that for different choices of parameters of the American put the optimal µ varies. We need an optimization algorithm that searches for µ at the same time that estimates the option price. In order to do so we implement stochastic approximation procedures commonly use in optimization theory [see Kushner and Vazquez-Abad (1996)]. More recently, Arouna (2003) discussed such procedure in option pricing. The recursive method is given by µn+1 = µn − n Jb0 (µn ) ,
n = 0, 1, 2, . . . ,
where Jb0 is an estimator for the derivative of the objective function J, in this case J is the estimated variance of the ALSM. Under certain conditions [Kushner and Yin (1997)] the sequence {µn } converges to the optimum µ∗ that minimizes J. There exist many methods to estimate derivatives in the literature. We use a finite differences approach. The estimator for the derivative of the variance is given by J(µ + h) − J(µ − h) , Jb0 (µ) = 2h
h>0.
This implies estimating two extra variances, one under µ + h and the other under µ − h, with M replications; then the value of µ is updated and another M replications are ran. Remark that at the same time that we are searching for the optimal µ the ALSM is also estimated. This automatizes the original algorithm. Notice that the recursion still needs an initial value for µ0 . For our results we use µ0 = 0, since we know that the optimal µ has to be negative, we also use n = 0.1/n and h = 0.01. Figure 3 shows the behavior of the sequence {µn } as n increases, for several choices of µ0 , fixed T = 1 and σ = 0.04. In total 10, 000 simulations 12
Figure 3: Convergence of the sequence {µn }.
9700
9100
8500
7900
7300
6700
6100
5500
4900
3700
3100
2500
1900
1300
700
100
9900
9200
8500
7800
7100
6400
5700
5000
4300
3600
2900
2200
1500
800
100
0.1
0.100 0.000 -0.100 -0.200 -0.300 -0.400 -0.500 -0.600 -0.700
0 -0.1 MU
MU
4300
S=38
S=36
-0.2 -0.3 -0.4 -0.5
n
n
9700
9100
8500
7900
7300
6700
6100
5500
4900
4300
3700
3100
2500
1900
1300
700
100
S=44
0.1 0
MU
-0.1 -0.2 -0.3 -0.4 -0.5 -0.6 n
were used for the plots and the value of µn was updated in batches of size 100. The algorithm seems to convergence nicely, independently of the initial value. The time of computation is not greatly affected by the implementation of the finite differences. However as we will see, this automatization is achieved at the expense of an increase in the variance. This due to the fact that at onset (small values of n), when µ is still away from the optimum, the observations used to estimate the ALSM have a greater variability than the observations obtained when n is large, therefore µ is closer to the optimum. This problem can be overcome by merely performing more simulations. The choice of the estimator of the derivative Jb0 is also important. The better the approximation, the faster µ will converge to the minimum and fewer observations fall under µ’s far from the optimum. There exist other methods, such as Infinitesimal Perturbation Analysis, that was successfully
13
applied in Dufresne and Vazquez-Abad (1998) to price Asian options.
5
Results and Comparisons
Figure 4 compares the estimates for the same American put as in Section 4 using the Modified ALSM to the ALSM estimations. The LSM* estimates of Longstaff and Schwartz (2001) and the estimation made using a finite difference method are also presented. Figure 4: Results for the LSM and Accelerated LSM Simulations.
S.E.
mu*
Time (sec)
S
FD
LSM*
LSM
S.E.
ALSM
S.E.
MALSM
36
7.101
7.091
7.085
0.019
7.098
0.008
7.096
0.008 -0.390
36
38
6.148
6.139
6.121
0.019
6.146
0.007
6.143
0.007 -0.397
38
40
5.312
5.308
5.289
0.018
5.311
0.007
5.311
0.007 -0.409
40
44
3.948
3.957
3.930
0.016
3.947
0.006
3.948
0.006 -0.410
40
The results are for puts with T = 1 and σ = 0.16. When using the automatized MALSM an increase of variance is observed in most of the cases, this reflects in the estimate for the option price. However, the option estimates stay within the same safe distance from the benchmark as the estimates from Longstaff and Schwartz (2001). In Figure 4 we present only those estimates that still show an improvement, despite the increase of the variance, with respect to the estimates from Longstaff and Schwartz (2001). Note that these options are those with a large volatility. Notice the large improvement in terms of variance from LSM to MALSM, specially in those cases where the option starts out-of-the-money. As it was expected, the change of measure 14
estimator seems to perform better in those cases where the option is deeply out-of-the-money and where the volatility is large. The optimal µ is also reported, this µ was estimated at the same time as the MALSM, using the stochastic approximation method. The estimation made by a finite difference approach is also included in Figure 4. Notice in Figure 4 that the variance of the MALSM is the same as that of ALSM which was estimated directly under an optimal µ∗ . This is not the case for the other options illustrated in Figure 2 and that do not appear in Figure 4, for which a small increase of the variance is observed due to the automatization. Despite this increase, these estimates are as good as the estimates in Longstaff and Schwartz (2001), so that accuracy is not lost if importance sampling is chosen over antithetic simulation. Based on the variance improvement in our results, it seems that the accelerated ALSM and the automatized MALSM would perform slightly better than the LSM as presented in Longstaff and Schwartz (2001) and Stentoft (2004), specially in those cases with large volatility. The increased computational time was of only 10 seconds.
6
Conclusions
Dufresne and Vazquez-Abad (1998) used methods from optimization theory to improve straight forward simulations in the case of Asian options. In the same spirit, we use similar techniques to improve estimates of the price of American options. This illustration relies on the new LSM approach of Longstaff and Schwartz (2001). Our implementation seeks to offer another example of the use of standard variance reduction techniques in option pricing. The presented ALSM and MALSM estimators give an alternative variance reduction technique to pricing American-style options using the LSM approach. Implementing importance sampling is not computationally costly and yields estimators with smaller variance than those obtained using antithetic simulation. Although this difference is not overwhelming, the low computational cost of implementing importance sampling is an incentive to prefer the latter over the antithetic simulation method used in Stentoft (2004), specially for options with large volatilities. As a consequence of the reduction in the variance of the estimator, fewer simulations are needed to attain more accurate estimations. This proves to be particularly useful in those cases where 15
the option is deep out-of-the-money and a lot of paths yield non-exercisable options. Importance sampling does not represent a great improvement over antithetic simulation except for those options with high volatility. However, if combined with other variance reduction techniques, such as control variates [Dufresne and Vazquez-Abad (1998)], it should yield estimators showing a significantly smaller variance which would reflect in its accuracy.
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[11] Karatzas, I. (1988). On the Pricing of American Options. Applied Mathematics and Optimization. 17. 37-60. [12] Kushner, H. and Vazquez-Abad, F. (1996). Stochastic Approximation Methods for Systems over an Infinite Horizon. SIAM. Journal of Control and Optimization. 34 (2). 712-756. [13] Kushner, H. and Yin, G. (1997). Stochastic Approximation and Applications. New York. Springer Verlag. [14] L’Ecuyer, P. (1994). Efficiency Improvement and Variance Reduction. Proceedings of the SIAM Winter Simulation Conference. [15] Lemieux, C. and La, J. (2005). A Study of Variance Reduction Techniques for American Option Pricing. Proceedings of the 2005 SIAM Winter Simulation Conference. [16] Longstaff, F.A. and Schwartz, E. (2001). Valuing American Options By Simulation: A Simple Least-Squares Approach. Review of Financial Studies. 14. 113-147. [17] Moreni, N. (2003). Pricing American Options: A Variance Reduction Technique for the Longstaff-Schwartz Algorithm. Technical Report. 2003-256. CERMICS-ENPC, Champs-sur-Marne. [18] Stentoft, L. (2004). Assessing the Least Squares Monte-Carlo Approach to American Option Valuation. Review of Derivatives Research. 7. 129-168. [19] Tilley, J.A. (1993). Valuing American Options in a Path Simulation Model. Transaction of the Society of Actuaries. 45. 499-520.
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