Stochastic Volatility
نویسندگان
چکیده
In the original Black-Scholes model, the risk is quantified by a constant volatility parameter. It has been proposed by many authors that the volatilities should be modeled by a stochastic process to obtain a more realistic model. The volatility that corresponds to actual market data for option prices in Black-Scholes model is called the implied volatility. This volatility is in general dependent on the strike price, in contrast to the underlying assumption of Black-Scholes model. As a function of strike it forms a curve called ”volatility smile”. To explain this smile it has been proposed to study models allowing for a volatility driven by a stochastic process. In the present paper a review of stochastic volatility is presented and three stochastic volatility models are studied in some detail. We study the volatility smile of these models and show that in some cases we can reproduce a smile similar to the curves occuring in reality. We also study a corrected Black-Scholes pricing formula.
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