Estimating option greeks under the stochastic volatility using simulation

As the Black-Scholes (BS) equation being widely used to price options, which is based on a hypothesis that the underlying (bonds & stocks) volatility is constant. Many scholars proposed the extended version of this formula to predict the behavior of the volatility. So stochastic volatility model is the improved version in which fixed volatility is replaced. The purpose of this study is to adopt one of the famous stochastic volatility models, Heston Model (1993), to price European call options. Put option values can easily obtained by call-put parity if it is needed. Simulation has proved to be a valuable tool for estimating options price derivatives i.e. “Greeks”. This paper proposes the method for the simulation of stock prices and variance under the Heston stochastic volatility model. We consider three different models based on the Heston model. We present two direct methods a Path-wise method and Likelihood ratio method for estimating the derivatives of Options. Then we compare it with Black-Scholes equation, and make a sensitivity analysis for its parameters by using estimator's approaches.