Abstract
Monte-Carlo simulations have been utilized greatly in the pricing of derivative securities. Over the years, several variance reduction techniques have been developed to curb the instability, as well as, increase the simulation e?ciencies of the Monte-Carlo methods. Our approach in this research work will consider the use of antithetic variate techniques to estimate the fair prices of barrier options. Next, we use the quasi-Monte Carlo method, together with Sobol sequence to estimate the values of the same option. An extended version of the Black-Scholes model will serve as basis for the exact prices of these exotic options. The resulting simulated prices will be compared to the exact prices. The research concludes by showing some results which proves that when random numbers are generated via low discrepancy sequences in contrast to the normal pseudo-random numbers, a more efficient simulation method is ensued. This is further applicable in pricing complex derivatives without closed formsolutions.
Highlights
The path-dependent nature of barrier options classified them as exotic options and this becomes difficult during delta-hedging
Research works have been done in the class of barrier options, but relatively no such work has been carried out on the comparative study focusing on the use of antithetic and quasi Monte-Carlo simulation method to price zero-rebate barrier call option with European knock-out features
The parameters K = 50, r = 0.1, σ = 0.2, T = 1 were used to value the DO put options. We compare their results on Antithetic Monte-Carlo Simulation (AMCS) and the normal MCS using M = 10000, with our quasi Monte Carlo simulation methods (QMCS) value using 213 = 8192 paths
Summary
The path-dependent nature of barrier options classified them as exotic options and this becomes difficult during delta-hedging. The significance of this research is to conduct a comparative study on zero rebate knock-out barrier option pricing using the MCS, AMCS and QMCS methods. This section gives an overview of the simulation techniques employed by the standard MCS, AMCS and the QMCS methods to barrier option pricing.
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