Stochastic volatility refers to the idea that the volatility of an asset price can vary randomly over time. This is in contrast to the traditional Black-Scholes model, which assumes that volatility is constant and known. Stochastic volatility models attempt to capture this randomness and uncertainty by introducing a stochastic process for volatility.
One popular model for stochastic volatility is the Heston model, which was introduced by Steven Heston in 1993. In the Heston model, the volatility of an asset is modeled as a mean-reverting process that is driven by a Brownian motion. Specifically, the model assumes that the volatility of the asset follows the following stochastic differential equation:
$$d\sigma_t = \kappa(\theta - \sigma_t)dt + \xi\sqrt{\sigma_t}dW_t^{\sigma}$$
where σt is the volatility at time t, κ is the rate of mean reversion, θ is the long-term mean of the volatility, ξ is the volatility of the volatility, and Wtσ is a Brownian motion that is correlated with the Brownian motion of the asset price.
When it comes to option pricing, stochastic volatility models can be used to price options by incorporating the randomness of volatility. One popular approach is to use Monte Carlo simulation, where the model is simulated many times to generate possible paths for the asset price and volatility. These paths are then used to calculate the expected value of the option payoff at expiration.
Another approach is to use partial differential equations (PDEs) to derive an analytical solution for the option price. This is typically done by applying the Feynman-Kac formula, which relates the option price to the solution of a PDE. The resulting PDE will depend on the particular stochastic volatility model being used, and may involve additional variables beyond the asset price and time.
Overall, stochastic volatility models provide a more realistic framework for option pricing by accounting for the randomness of volatility. While these models can be more complex than the traditional Black-Scholes model, they are widely used in quantitative trading and investment due to their ability to generate more accurate pricing and risk measures.