Importance Sampling for Monte Carlo Simulation to Evaluate Collar Options under Stochastic Volatility Model
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Technická Univerzita v Liberci
Technical university of Liberec, Czech Republic
Technical university of Liberec, Czech Republic
Abstract
The collar option is one kind of exotic options which is useful when institutional investors wish to lock in the profi t they already have on the underlying asset. Under the constant volatility assumption, the pricing problem of collar options can be solved in the classical Black Scholes framework. However the smile-shaped pattern of the Black Scholes implied volatilities which extracted from options has provided evidence against the constant volatility assumption, so stochastic volatility model is introduced. Because of the complexity of the stochastic volatility model, a closed-form expression for the price of collar options may not be available. In such case, a suitable numerical method is needed for option pricing under stochastic volatility. Since the dimensions of state variable are usually more than two after the introduction of another volatility diffusion process, the classical fi nite difference method seems ineffi cient in the stochastic volatility scenario. For its easy and fl exible computation, Monte Carlo method is suitable for evaluating option under stochastic volatility. This paper presents a variance reduction method for Monte Carlo computation to estimate collar option under stochastic volatility model. The approximated price of the collar option under fast mean reverting stochastic volatility model is derived from the partial differential equation by singular perturbation technique. The importance sampling method based on the approximation price is used to reduce the variance of the Monte Carlo simulation. Numerical experiments are carried out under the context of different mean reverting rate. Numerical experiment results demonstrate that the importance sampling Monte Carlo simulation achieves better variance reduction effi ciency than the basic Monte Carlo simulation.
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importance sampling, Monte Carlo simulation, collar options, stochastic volatility
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1212-3609