Stochastic volatility models

Since the market crash of 1987, it has been observed that market implied volatility for options of lower strike prices are typically higher than for higher strike prices, suggesting that volatility is stochastic, varying both for time and for the price level of the underlying security. Stochastic volatility models have been developed including one developed by S.L. Heston.^[11] One principal advantage of the Heston model is that it can be solved in closed-form, while other stochastic volatility models require complex numerical methods.^[11]