Option Pricing And Monte Carlo Simulations

The advantage of Monte Carlo simulations is attributed to the flexibility of their implementation. In spite of their prevalence in finance, we address their efficiency and accuracy in option pricing from the perspective of variance reduction and price convergence. We demonstrate that increasing the number of paths in simulations will increase computational efficiency. Moreover, using a t-test, we examine the significance of price convergence, measured as the difference between sample means of option prices. Overall, our illustrative results show that the Monte Carlo simulation prices are not statistically different from the Black-Scholes type closed-form solution prices. INTRODUCTION onte Carlo simulations (MCS) have recently been an important technique for option pricing in finance. MCS avoid complicated mathematics and have a straightforward implementation conceptually and practically. For example, to price a European down-and-out call barrier option 1 by MCS, just treat it as a normal option unless the underlying asset price reaches the pre-determined level, as opposed to setting boundary conditions and solve a partial differential equation. In practice, MCS are procedures of sampling random outcomes for a particular process. However, while many academics and practitioners acknowledge the merits of MCS, some studies discuss their weaknesses in option pricing. Clewlow and Strickland (1998) and Hull (2000) point out that MCS generate high variances that lead to computational inefficiency. This problem can not be overlooked because such inefficiency may produce a biased estimator of the option price. In this paper, our focus is on the efficiency and accuracy of MCS in option pricing. We demonstrate that the estimated standard errors of MCS option prices can be reduced by increasing the number of paths in the simulations. Additionally, we use a t-test to examine whether MCS prices converge to Black-Scholes type of closed-form solution prices. The empirical evidence does not suggest any significant difference between those prices. Moreover, the results show that these two types of prices converge as the number of paths in simulations increases. The layout of this paper is as follows: section 2 provides a quick literature review. Section 3 examines variance reduction and price convergence of MCS. Section 4 provides the conclusions. LITERATURE REVIEW Originated from studies in physics, MCS have been very successfully applied in finance 2 . Hull and White (1987) use MCS to price options with stochastic volatilities. Schwartz and Torous (1989) apply MCS to the valuation of mortgaged-backed securities. Boyle et al. (1997) use MCS to price American options. On the other hand, the disadvantages of MCS are also discussed in some studies. Clewlow and Strickland (1998) and Hull (2000) argue that MCS are computationally inefficient due to the generated high variances. THE EFFICIENCY AND ACCURACY OF MONTE CARLO SIMULATIONS Variance Reduction The efficiency of MCS increases with the number of paths used in the simulations. Since MCS are sampling random variables, option prices are random as well. The estimated standard error (ESE) is calculated as the sample 1 A barrier option is a contingent claim whose payoff depends on whether the underlying asset has reached a certain pre-determined level for a specific path. 2 See Jackel (2002) for a thorough summary of the applications of MCS in finance. Journal Of Business & Economics Research – September 2005 Volume 3, Number 9 2 standard deviation of MCS option prices (SD) divided by the square-root of the number of paths (m):