An approximate method to option pricing in the Heston model

The Heston model is one of the most popular stochastic volatility models for derivativespricing, and it is a mathematical model describing the evolution of the volatility of an un-derlying asset. The model proposed by Heston(1993), takes into account non-lognormaldistribution of the assets returns, leverage e ect and the important mean-reverting prop-erty of volatility. In addition, it has a semi-closed form solution for European options.In this paper by means of classical It^o calculus, we decompose option prices as the sumof the classical Black-Scholes formula.This decomposition allows us to develop rst- andsecond-order approximation formulas for option prices and implied volatilities in the Hes-ton volatility framework, as well as to study their accuracy for short maturities. Moreover,we show that the corresponding approximations for the implied volatility are linear( rst-order approximation) and quadratic(second-order approximation) in the log stock price.

Keywords:

Stochastic volatility , Heston model , It^o calculus , Black-Scholes formula