7.5 AMERICAN OPTION
7.5.1 Simulation: Least Squares Approach
128 PATH-DEPENDENT OPTIONS
This simulation gives the arithmetic Asian call (AAC) price to be 0.1698.
The analytical price for the geometric Asian call (GAC) is computed as 0.1318.
The AAC is a bit more expensive than the GAC because the arithmetic mean always dominates the geometric mean. The computational time is about 10
seconds. U
AMERICAN OPTION 129
~~
Path 1 2 3 4 5 6 7 8
t = 113 8.3826 11.9899 13.1381 6.8064 7.0508 11.2214 8.9672 11.5336
~~ ~
t = 213 9.9528 13.8988 17.4061 7.8115 9.1293 8.3600 8.7787 10.9398
t = l 6.7581 14.5060 13.4123 10.6520 7.4551 9.2896 9.0822 8.6958
~
Y3 = max(K - S(1),0) 5.2419
0 0 1.3480 4.5449 2.7104 2.9178 3.3042 Table 7.1 Sample paths.
S(2/3) for in-the-money paths in Table 7.2, where At = 1/3. We model the expected payoff from continuation at time t = 2/3 as a quadratic polynomials,
f2(St), of asset values at time t = 2/3. Coefficients of the polynomials are estimated from the data in Table 7.2 by the least squares method. Therefore, we estimate 6g,61 and 62 from the regression line:
~ 3 e - ~ ~ ~ = 60
+
61 [ ~ ( 2 / 3 ) ]+
62[s(2/3)I2+
6 .The resulting formula is
E[Y3e-'At]S(2/3)] = -82.5347
+
17.7788[S(2/3)] - O.9063[S(2/3)l2 := f i ( S ) .Path Y3e-TAt S(2/3) Exercise in-the-money?
5.1898 9.9528
~ 13.8988 __ 17.4061 1.3346 7.8115 4.4997 9.1293 2.6834 8.3600 2.8888 8.7787 3.2714 10.9398
Yes No No Yes Yes Yes YeS Yes Table 7.2 Regression at t = 213.
With this conditional expectation function, f2(S), we are able to compare the value of immediate exercise, K - S(2/3), and compute payoffs, Y2, for each path at t = 2/3. The value of Y2 is obtained by the formula,
K - s(2/3), if K - W / 3 )
2
h ( s ( 2 / 3 ) ) ,y2 =
{
e-Aty3, otherwise.130 PATH-DEPENDENT OPTIONS
This formula asserts that the payoff at time t = 2/3 is K - S if exercising the option is worth more than the expected payoff from holding it; otherwise, the payoff a t time 2/3 becomes the discounted cash flow in the next exercise time. The last column of Table 7.3 gives the expected payoffs, Y2, for each sample path.
Path Exercise Continuation eWrAt Y3 YZ K - W / 3 ) fdS(2/3))
2.0472
~
4.1885 2.8707 3.6400 3.2213 1.0602
4.6380 5.1898
~ 0
0
____
1.0428 1.3346 4.2388 4.4997 2.7554 2.6834 3.6959 2.8888 3.4968 3.2714
5.1898 0 0 4.1885 4.4997 3.6400 2.8888 3.2714
Table 7.3 Optimal decision at t = 2/3.
Next, we repeat the procedure for t = 1/3. In Table 7.4, all sample paths are in-the-money except path three. Then, the least squares estimation cor- responding to in-the-money paths gives
E[Y2e-'At[S(1/3)] = -8.9488
+
3.31049(1/3) - 0.2036[S(1/3)]2 := fi(S).This regression function determines the exercising policy a t t = 1/3.
Path YZe-rAt S( 1/3) Exercise in-the-money?
5.1381 0 0 4.1468 4.4549 3.6038 2.8600 3.2388
8.3826 11.9899 13.1381 6.8064 7.0508 11.2214 8.9672 11.5336
Yes Yes No Yes Yes Yes Yes Yes Table 7.4 Regression at t = 1/3.
AMERICAN OPTION 131
Once again, the Y1 in Table 7.5 is computed according t o the optimal decision by the rule,
K - S(1/3), if K - S(1/3)
2
f l ( W / 3 ) ) ,' 1 =
{
e-Aty2, otherwise.Finally, the current price of the American option is estimated by the average of e-rAtYl, i.e., $3.0919, which is higher than the European option price
$2.4343. 0
3.6174 0.0101 5.1936 4.9492 0.7786 3.0328 0.4664 __
4.4921 1.4689 4.1494 4.2688 2.5572 4.3620 2.1440 __
5.1381 0 0 4.1468 4.4549 3.6038 2.8600 3.2388
5.1381 0 0 5.1936 4.9492 3.6038 2.8600 3.2388 Table 7.5 Optimal decision at t = 1/3.
7.5.2
Consider an American put option with exercise rights a t tl
<
. . .<
t, = T . To simplify matters, we assume t,+l- t j = At for j = 1 , 2 , .. . ,
n - 1. Given a sample path of the underlying asset price, { S ( t , ) , S(t2),. . . ,
S(t,)}, we study possible payoffs received by the option holder at each of the exercise time points. Clearly, if the option is not exercised prematurely, then the holder receives the terminal payoff, denoted as Y, = max(K - S(t,),O). At time t = t n - l , the corresponding payoff, Yn-l, depends on the holder's decision of exercising the option. Therefore,Analyzing the Least Squares Approach
K - S(tn-l), exercise,
yn-l =
{
e-rAtyn, continue.This formula indicates that the option holder receives K - S(tn-l) if the optimal decision is to exercise the option. Otherwise, the holder will receive a cash flow of Y, at the next time step. The present value of this cash flow is obtained through multiplying a discounted factor Inductively, the payoff y3 at time tj can be described as
K - S ( t j ) , exercise,
%
={
e-rAty 3+1, continue. (7.5)132 PATH-DEPENDENT OPTIONS
This iterative process stops until Y1 is obtained. Since the option holder has no exercise right in the time period [ O , t l ) , the American put option can be viewed as a European option that expires at tl with payoff Y1. Risk-neutral valuation allows us t o value the American put, PA(O, S), as
PA(o, S ) =
E
[ e P * ~1 = S ]Therefore, a typical simulation algorithm generates N sample paths, each mated by
follows the algorithm t o obtain {Y:'),
. . . ,
Yl ") }. The American put is esti-The above simulation is incomplete, however. To simulate the American put, the payoff, Y1, a t time tl should be obtained via simulation. This re- quires the simulation algorithm t o detect optimal exercise at each time point successively. In other words, we have t o clarify the condition of exercising the option in (7.5). It is crucial that the optimal decision should not be made by simply comparing the values of K - S(t,) and e-'Atq+l in (7.5). The reason is that the decision at time t, should be based on the information up to t,. However, the value depends on the asset value a t t,+l. The correct approach is t o compare the immediate exercise cash flow K - S(t,) with the expectation on the discounted cash flow conditional on the asset price S(t,).
This leads (7.5) to
(7.7) K - S(t,), if K - S(t,)
2
f,(S(t,)),e-'Atq+l, if K - S(t,)
<
f,(S(t,)),where f j ( S ( t j ) ) is the conditional expectation function at t j , that is,
The key to the Longstaff and Schwartz (2001) approach is the use of least squares to estimate the function, f j ( S ) . Under certain technical conditions, it can be shown that the function f j ( S ( t j ) ) can be approximated by a polynomial of S ( t j ) . In other words,
M
k=O
where {@} converges t o zero rapidly. Therefore, one way to approximate f j ( S ) is by truncating the polynomial of infinite order t o a finite order poly- nomial. Coefficients of the finite order polynomials are estimated through the least squares method.
AMERICAN OPTION 133
In Example 7.3, we use a polynomials of degree 2 to approximate f j ( S ) . The simulation starts by generating N asset price paths, { S z ( t l ) , .
. . ,
Si(tn)}for i = 1 , 2 , .
. .
, N . When t = t,, it is clear that Yiz) = max[K - S,(t,), 01 for the path i. We go one step back t o the time point t = tn-l, where N possible asset prices have been generated. Then, the coefficients ao, a l , anda2 are obtained by taking least squares estimation to the regression line:
fn-l(s)
=E
[ e - r A t ~ n l ~ ] = a0+
al[s(t,-i)]+
a 2 [ ~ ( t , - i ) l ~ . (7.9) The estimation is based on the sample ((Si(t,-~),Yn(~))lK>
Si(t,-l),Z = 1,.. . ,
N } , i.e., in-the-money paths. Then, payoffs a t t,-l are calculated via the rule (7.7). Having a sample of payoffs{YJ!lli
= 1,2,. ..
, N } at t n - 1 , we go one step back to the time point tn-2 and repeat the process. Eventually, we obtain N possible payoffs,{Y{li = 1,2,..
.,
N } , a t tl. Monte Carlo simulation estimates the current option price by the average in (7.6).Remarks:
1. In the regression equation (7.9), only in-the-money paths are used in the least squares estimation as these paths are sensitive to immediate exercise. Remember that the option holder will exercise the option only when it is in-the-money.
2. An obvious way t o improve the accuracy is to increase the number of terms in (7.9). However, one has to strike a balance between increasing the number of terms and the quality of estimates. Numerical experi- ments show that polynomials of degree 3 is a reasonable choice.
3. Instead of using ordinary monomials as basis functions in (7.9), one may consider other basis functions, like Hermite, Laguerre, Legendre, Chebyshev, Gegenbauer, and Jacobi polynomials. Numerical tests of Moreno and Navas (2003) show that the least squares approach is quite robust t o the choice of basis functions. For more complex derivatives, this choice can slightly affect option prices.
4. The recent analysis of Stentoft (2004) indicates that a modified spec- ification using ordinary monomials is preferred over the specification based on Laguerre polynomials used in Longstaff and Schwartz (2001).
Furthermore, the least squares method is computationally more efficient than other numerical methods, such as finite difference, especially when high dimensional problems are concerned.
5 . The paper by Longstaff and Schwartz (2001) points out that the R2 values of the regressions are often low. This means that the volatility of unexpected cash flows is large relative to the expected cash flows.
However, since the least squares simulation is based on conditional first
134 PATH- DEPENDENT 0 PTIO NS
moments rather than higher moments, the R2's of the regression should have little impact on estimated American option price.
6. If the user is really concerned about the R2, it may be more efficient to use other techniques such as weight least squares and GMM in estimat- ing the conditional expectation function.
Example 7.4 Using the parameters in the preceding example, simulate the American put price with continuous exercise rights and hence determine the optimal exercise policy. The simulation is based on 10,000 sample paths with At = 1/100.
The SPLUS code is as follows:
N <- 10000
n <- I00
dt<- l/n
r <- 0.03
sigma <- 0.4
so
<- 10K <- 12
nu <- r-sigma-2/2
stock <- matrix(O,N,n+l)
y <- c(rep(0,N))
put <- c(rep(0,N))
boundary <- c(rep(0,n)) stock[,l] <- SO
check1 <- proc.time() #the first check point
# generate asset price paths for (i in l:n>(
stock[,i+ll <- stock[,i]
*
exp(nu*dt+sigma*sqrt(dt)*rnorm(N,O,l))3
y <- pmax( (K-stockC,n+ll), 0for (j in n:2)(
a <- which( stock[,j]<K ) # identify in-the-money paths
if ( length(a) >= 3 ) ( # ensure there's a solution for the regression
# Compute the conditional expectation function
S <- stockCa,jl
A <- coef ( lm( y[a] S + S-2, singular.ok="T" ) )
if ( is.na(AF31) 1 ( A[31<-0 3
X <- matrix( c(rep(1,N) ,stock[,j] ,stock[,j]-2), ncol=3 )
put <- X %*% A
put <- exp(-r*dt)*pmax(put ,0>
# determinate Y & find boundary of K-S(t) <f(S(t))
AMERlCAN OPTlON 135
b <- which( (K-stock[,jl) > put )
y <- exp(-r*dt)*y
y[b] <- K - stock[b, j] # assign y as K-S when K-S > P
if ( length(b)==O ) {boundary[j]<-NA) # boundary cannot be estimated else { boundary[j] <- max(stock[b,jl)
1
3
else1
y <- exp(-r*dt)*y
boundary[j] <- NA 1
1
check2 <- proc.time() boundary[n+l] <- K
boundary[1:20] <- NA #give up the estimate for t(0.2 time<-c(O:n)/n
plot(tirne,boundary,type="h" ,ylim=c(O,K) ,xlab="time",ylab="asset price")
#the second check point
price <- exp(-r*dt)* mean(y)
check2-check1 #Check the CPU time for the valuation price #price of American Put Option
"j
m
I I 1
0.2 0.4 0.6 0.8 1 .o
time
Fig. 7.1 The exercising region of the American put option.
By using quadratic conditional expectation functions, our simulation esti- mates the American put price as 2.739 within 15 seconds, which is consistent
136 PATH-DEPENDENT OPTIONS
with the binomial model of Hull (2006). For the early exercise policy, we collect the maximum asset value that belongs to the exercising region at each time. For t 2 0.2, Fig. 7.1 plots the exercise policy against time. The option is optimal to exercise if the stock price falls into the shaded region. It is seen that the early exercise boundary looks like an increasing function of calendar time and hence a decreasing function of option maturity. For t < 0.2, our simulation has no path in the exercising region so that we are unable to graph
the exercising boundary. 0