Pricing multi-asset American-style options by memory reduction Monte Carlo methods
Introduction
Monte Carlo method is one of the main methods for computing American-style options, see for instance [12], [2], [3], [9]. These algorithms are computationally inefficient because they require the storage of all asset prices at all simulation times for all simulated paths. Thus the total storage requirement grows like O(dMN) where d is the number of underlying assets, M is the number of simulated paths and N is the number of time steps. The accuracy of the simulation is hence severely limited by the storage requirement.
The apparent difficulties in using Monte Carlo methods to price American-style options come from the backward nature of the early exercise feature. There is no way of knowing whether early exercise is optimal when a particular asset price is reached at a given time. One can look at this problem from another point of view. In Monte Carlo method, the simulated paths are all generated in the time-increasing direction, i.e. they start from the initial asset prices x0 and march to the expiry date according to a given geometric Brownian motion. But since the pricing of American options is a backward process starting from the expiry date back to x0, the usual approach is to save all the intermediate asset prices along all the paths.
In this paper, we use our simulation method in [4] for computing multi-asset American-style options that does not require storing of all the intermediate asset prices. The storage is reduced from O(dMN) to (d + 1)M + N only. Our main idea is to generate the paths twice: one in a forward sweep to establish the asset prices at the expiry date, and one in a backward sweep that computes the intermediate asset prices only when they are needed. The only additional cost in our method is that we have to generate each random number twice instead of once. The resulting computational cost is less than twice of that of the methods where all the intermediate asset prices are stored.
The remainder of this paper is organized as follows. Section 2 recalls the usual full-storage approach for computing multi-asset American-style options. Section 3 gives some background about random number generators in computers. In Section 4, we introduce our memory reduction method. In Section 5, we show how to use our method to compute multi-asset American options by adopting it to the least-squares method proposed by Longstaff and Schwartz [9]. Section 6 gives some numerical results to illustrate the effectiveness of our method.
We will use the matlab language [11] to explain how the codes are to be written as the language is easier to comprehend. The corresponding commands in fortran 90 [5] and mathematica [13] are given in Appendix A.
Section snippets
The full-storage method
As usual, we let the prices of d non-dividend paying assets x = (x1, x2, … , xd)T follow the geometric Brownian motionwhere r is the risk-free interest rate, σk is the volatility of asset k, and dWk is the Wiener process for asset k. By Ito’s Lemma, we havewhere xk(t) is the price of asset k at time t, z is a d-vector of standard normal random variables, and is the volatility matrix. The
The random number generator
According to (2), we need to generate one standard normal random number for each asset at each time step on each path. Most computer languages already have built-in functions to generate these random numbers. In matlab, it is “randn”. By using the concept of a seed, one has the flexibility to change or fix the sequence of random numbers each time they are generated. For example, the matlab commands
randn(ˈseedˈ,s);
e = randn;
The forward-path method
In this section, we apply our method which does not need to store the intermediate asset price vectors when computing the multi-asset option prices. In this method, each vector of random numbers is generated twice, but the intermediate asset price vectors are generated only once as in the full-storage method.
From (2), the intermediate asset price vectors are given bywhere · is the Hadamard product, and σ · σ = ((σ1)2, (σ2)2, … , (σd)2)T.
The least-squares method
Our path generating technique can reduce the memory requirement of Monte Carlo methods for pricing multi-asset American-style options. In this section, we illustrate this by pricing an American put option on the maximum of multi-assets using the least-squares approach developed by Longstaff and Schwartz [9].
At the final exercise date, the optimal exercise strategy for an American option is to exercise the option if it is in the money. Prior to the final date, however, the optimal strategy is to
Numerical examples
In this section, we test our method on an example given in [1, p. 400]. It is an American put option on the maximum of three assets (i.e. d = 3) with strike price E. The current prices of the three assets S0 are ($40, $40, $40)T, the riskless rate r is 0.05, the volatilities of the assets σ are 0.2, 0.3 and 0.5 respectively, and the expiry date is T months. Their correlation coefficients are given by ρ. We emphasize that the results obtained by the full-storage method and the forward-path method
Acknowledgment
The research was partially supported by the Hong Kong Research Grant Council Grant CUHK4243/01P and CUHK DAG 2060220.
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