NOVEL IMPORTANCE SAMPLING FOR THE ...

NOVEL IMPORTANCE SAMPLING FOR THE VALUATION OF BASKET AND ASIAN OPTIONS J S DAGPUNAR, SCHOOL OF MATHEMATICS, UNIVERSITY OF EDINBURGH, UK Abstract. In this paper we discuss the use of a new method of importance sampling in the pricing of European options on a basket of assets and on average price (Asian) options. In standard Monte Carlo estimation the expectation is taken with respect to a vector which is distributed as n variate normal, where n is the number of assets in the basket, or the number of points in the averaging computation in the case of an Asian option. A change of measure which results in sampling the ratio of arithmetic to geometric average payo¤s is central to the importance sampling method employed. Since this ratio varies little, considerable variance reduction can be achieved, particularly when the call option is ’in the money’. Experimental results are given.

1. Introduction In recent years there has been signi…cant application of Monte Carlo methods to the pricing of …nancial derivatives; see for example Glasserman (2004) and Boyle, Broadie, and Glasserman (1997). In the present paper we are concerned with the pricing of European options on a basket of assets, and on the average asset price (Asian option) during a speci…ed time period. A good introduction to the terminology and analysis of derivatives is the book by Hull (2006). Suppose we have a basket of n assets. The price of the ith asset follows a geometric Brownian motion given by p 1 2 ( ) Xi i (t; Zi ) = xi exp t + i tZi i i 2 where xi ; i ; and i are the initial price, the growth rate, and the volatility for the ith asset. Let x0 = (x1 ; :::; xn ); 0 = ( 1 ; :::; n ), and 0 = ( 1 ; :::; n ): Let 0 Z = (Z1 ; :::; Zn ): Z is multivariate normal, M V Nn (0; V); where V is the matrix of correlations between the returns on the assets. We de…ne the arithmetic average price of the basket at time t by n X ( ) ( ) XA (t; Z) = wi Xi i (t; Zi ) (1) i=1

where the speci…ed weight vector, w0 = (w1 ; :::; wn ); has positive values with Pn i=1 wi = 1: Unfortunately, there is no closed form expression for the distrib( ) ution of XA (t; Z): Consider …rst a European call option on the basket with a strike price K and an exercise time t: Suppose that the ith asset earns interest continuously at rate qi : We de…ne the vectors q0 = (q1 ; :::; qn ) and r0 = (r; :::; r) where r is the risk free interest rate. The price of a call option CA on the basket is the Key words and phrases. Basket option, Asian option, Monte Carlo, Importance sampling, Multivariate analysis, Simulation, Finance. 1

2

J S DAGPUNAR, SCHOOL OF M ATHEM ATICS, UNIVERSITY OF EDINBURGH, UK

expected value of the payo¤ on the basket to a risk neutral investor, discounted by the risk free interest rate r: It follows that we set = r q in (1) and that !+ n X p 1 2 CA = e rt Ef wi xi exp (r qi K i )t + i tZi 2 i=1

where f is the M V Nn (0; V) density and x+ max(0; x): The value CP of the corresponding put option can be obtained by put-call parity. Of course, CA and CP can be estimated through conventional Monte Carlo methods, but with n large and with no e¤ective variance reduction procedures, it can take excessive computation time to obtain precise estimates. Joy et al. (1996) and Chiaia (1999) have used low discrepancy methods. Milevsky and Posner (1998a) use moment matching to approximate the distribution of the sum of (correlated) lognormal random variables by a reciprocal gamma density. Following Glasserman, Heidelberger, and Shahabuddin’s (1999a and b) example for Asian options, Dagpunar(2007, pp.123-126) used a combination of importance and strati…ed sampling for Basket options, although the importance sampling distribution selected there (change of drift in the underlying Brownian motion) is quite di¤erent from the one considered in the present paper. Gentle (1992) used approximations based on the price of the corresponding geometric basket. (There is a crucial error in the latter Q Q in that = exp( 0:5 2i T ) should be = exp( 0:5ci 2i T ): This a¤ects the numerical answers in the paper.) It is also possible to take the geometric average of the asset prices. In this case we de…ne the geometric average price of the basket at time t by n Y p 1 2 ( ) i xw XG (t; Z) = i )t + i tZi i exp wi ( i 2 i=1 or

( )

XG (t; Z) = x0 exp where

X x0 =

wi

n Y

i

1 2

2

t+

p

tZ

(2)

i xw i

i=1

is the geometric spot price, Z s N (0; 1); and where we de…ne n X 2 = wi 2i i=1

and

2

=

n X n X

wi wj

i j Vij :

(3)

j=1 i=1

( )

Since XG (t; Z) is lognormally distributed there is no di¢ culty in obtaining the price of geometric European put and call options. However, it is the arithmetic options that are used most frequently, and therefore e¢ cient computational methods are required here. Asian options are intimately connected with basket options. Consider a single stock with price X ( ) (t; Z) where p 1 2 t+ tZ X ( ) (t; Z) = x exp 2

NOVEL IM PORTANCE SAM PLING FOR THE VALUATIONOF BASKET AND ASIAN OPTIONS3

and where interest is earned continuously at rate Q: Divide the interval [0; t] into n sub-intervals each of length h with nh = t: Then the (discrete) arithmetic average price during [0; t] is given by ( )

n

X A (t; Z) =

1X x exp n i=1

1 2

2

ih +

p

ihZi

Pi where Zi = p1i j=1 Tj and where fTi ; i = 1; ::; ng are i.i.d. N (0; 1) random variables.. Therefore, the arithmetic average call and put prices are CA = e

rt

PA = e

rt

(r Q)

Ef X A

+

(t; Z)

K

and

respectively. Now Vim tion shows that

+

(r Q)

(t; Z) Ef K X A q = cov(Zi ; Zm ) = mi for 1 i m ( ) X A (t; Z)

=

n X

(

wi Xi

i)

n: A little manipula-

(t; Zi )

i=1

where we de…ne i

=

wi

=

i

=

xi

; 1 ; nr

ih ; t = x exp( [ih

t]):

Expressed in this way the arithmetic average price is the price of a basket of n assets, the ith asset price being the contribution of the stock price at time ih to the average. If we further set qi as Q for all i then the identi…cation is complete. There have been many approaches to the pricing of arithmetic Asian options. A lognormal density is used by Turnbull and Wakeman (1991) to approximate the density of the sum of lognormals, and by Curran (1994) to approximate the density of arithmetic minus geometric averages. In a paper which is a precursor to the one by Gentle (1993), Vorst (1992) used an approximation based on the analytical price of a geometric average price call. Milevsky and Posner (1998b) use the reciprocal gamma approximation mentioned previously. Examples of low discrepancy methods are those of Joy et al. (1996), Galanti and Jung (1997), and Jung (1998). Geman and Yor (1993) investigate the arithmetic continuous (as opposed to discrete) average. To date, the two most promising approaches for discrete averaging are those of Vázquez-Abad and Dufresne (1998) and Glasserman, Heidelberger, and Shahabuddin (1999a and b). The …rst employs a control variate (with estimated optimised constant of proportionality) together with a change of measure which makes the likelihood of a positive payo¤ greater. The second employs importance sampling (change of drift) together with strati…cation along preferred directions. Most of the variance reduction comes from the strati…cation. In this paper we employ a new approach. This obtains Monte Carlo estimates of call and put options on (arithmetic) basket and Asian options. The method employs an importance sampling technique which is quite di¤erent from those used

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in the papers mentioned previously. We will show that for call options in the money the approach leads to an algorithm which can be up to four orders of magnitude faster than the best of the existing methods. 2. Importance Sampling As we have seen, the price PA of a put option on a arithmetic weighted basket can be written as PA = e rt Ef (g (z)) (r q)

+

where the density f is M V Nn (0; V) and where g (z) = K XA (t; z) : A crude Monte Carlo procedure to estimate PA is to sample m random vectors z1 ; :::; zn from f and to evaluate the unbiased estimate m e rt X g (zi ) : PbA = m i=1

We now change the measure from f to h so that g (z) f (z) PA = e rt Eh h (z)

:

(4)

By choosing h in such a way that gf h varies little, the Monte Carlo estimate based upon (4) can be made quite precise. Recall that g (z) is the payo¤ for an arithmetic average. It follows that a natural choice for h is based upon a geometric weighted payo¤. We choose h (z) = Ae

rt

K

(r q)

XG

+

(t; z)

f (z)

(5)

where A is a normalising constant. Integrating (5) over z; we have 1 = AP where P is the price of a European call option on a single asset with volatility , as 2 given in (3), initial price x0 exp 0:5( 2 )t , strike price K; and exercise time t: Thus 0 1 + (r q) K X (t; z) A B C (6) PA = P Eh @ +A: (r q) K XG (t; z)

It is now clear why we have chosen to concentrate on the put option. Since a geometric mean never exceeds an arithmetic mean the ratio in (6) is always …nite. Had we chosen to estimate the call price, the ratio would be in…nite for some z; and the approach would fail. To obtain the call price we simply use put-call parity. The essential problem now is how to sample from h: We will need the following standard lemma concerning a multivariate normal distribution.

NOVEL IM PORTANCE SAM PLING FOR THE VALUATIONOF BASKET AND ASIAN OPTIONS5

Lemma 1 Let X0 be a random row vector of k elements which is M V Nk ( ; ) where X; ; are partitioned as X0 = (X01 ; X02 ) ;

and If

22

MV N

0

0 1;

= (

0 2) ;

=

11

12

21

22

:

is invertible then the conditional distribution of X1 given X2 = x2 is 1

+

12

1 22

(X2

2) ;

11

12

1 22

21

:

Proof. S ee for example Arnold (p214, 1990). The sampling method is implicit in the following theorem. Theorem 2 Pn Pn Let si = wi i ; where 2 = j=1 i=1 wi wj P n (r; :::; r), q0 = (q1 ; ::; qn ), q = i=1 wi qi : Let hZ (z) = Ae

rt

(r q)

K

XG

Let s0 = (s1 ; ::; sn ), r0 =

i j Vij :

+

(t; z)

f (z)

be an n-variate density function, where f is M V Nn (0; V): Let Y = s0 Z; be the standard normal density, P the price of a European put option on a single asset with 2 volatility ; initial asset price x0 exp 0:5( 2 )t ; strike price K; and exercise time t. Let r be the risk free interest rate. Then (a) the marginal density of Y is hY (y) =

rt

e

K

P

x0 exp

r

q

1 2

2

p

t+

+

ty

(y)

and (b) the conditional distribution of Z given that Y = y is 0

M V Nn yVs; V

Vs (Vs) :

Proof (a) Suppose for the moment that the density of Z is f (z) rather than hZ (z) : In this Pn case Z s M V Nn (0; V) and Y = 1 j=1 wj j Zj s N (0; 1): We would also have Pn that cov(Zi ; Y ) = 1 j=1 cov(Zi ; Zj )wj j = (Vs)i It would follow that the joint V Vs density of Z and Y would be M V Nn+1 (0; B) where B = . 0 (Vs) 1 (r q)

Now observe that hZ (z) = Ae rt K XG (t; z) Pn 1 Since Y = j=1 wj j Zj we have from (2) that K

(r q)

XG

+

(t; z)

=

K

x0 exp r

+

f (z) :

q

1 2

2

t+

2

t+

p

+

ty

:

Thus the joint density of Z and Y is hZ;Y (z;y) = Ae

rt

K

x0 exp

r

q

1 2

p

+

ty

(z;y)

(7)

6


where is the M V Nn+1 (0; B) density. Integrating this joint density over z we obtain the marginal density of Y as hY (y) = Ae

rt

K

x0 exp

r

q

1 2

2

p

t+

+

(y):

ty

(8)

Integrating h over all values of y we obtain 1 = AP which gives the required result. Proof (b) From (7) and (8) the conditional density of Z given that Y = y is (z;y) (y) : The required density is therefore the conditional density of Z given that (Z0 ; y) is M V Nn+1 (0; B) : The result follows directly from lemma 1, where we put k = n+1; 0 X0 = (Z0 ; Y ) ; 0 = ( 01 ; 02 ) = 0; = B; 11 = V, 12 = Vs; 21 = (Vs) ; 22 = 1: Theorem 2 indicates that we should …rst sample a value y from hY and then sample the vector z from the conditional M RV N distribution. We use inversion of y the distribution function of Y: Let HY (y) = 1 hY (u)du: Then HY (y) < 1 if and only if y