Local volatility models and stochastic volatility models are two approaches used to model the behavior of stock prices in option pricing. The main difference between the two types of models is the way they handle volatility.
Local volatility models assume that the volatility of the underlying asset is a deterministic function of time and price. In other words, the volatility is assumed to vary smoothly with the price and time, and its value at any given point in time is known. These models are also known as deterministic volatility models or Dupire models, named after Bruno Dupire who introduced the concept.
Stochastic volatility models, on the other hand, assume that the volatility of the underlying asset is a stochastic process. This means that the volatility is assumed to vary randomly over time and is not known with certainty at any given point in time. Stochastic volatility models are also known as random volatility models.
In local volatility models, the volatility surface is calibrated to market data to fit the observed prices of vanilla options. However, these models have limitations when it comes to pricing exotic options, such as barrier options or options with early exercise features. This is because these models assume that the volatility is known with certainty, which may not be the case in reality.
Stochastic volatility models, on the other hand, allow for more complex modeling of the volatility surface. There are several different types of stochastic volatility models, such as the Heston model, the SABR model, and the GARCH model. These models are more flexible in their assumptions about the volatility process and can be used to generate more accurate prices for exotic options.
In summary, both local volatility models and stochastic volatility models have their own strengths and weaknesses. Local volatility models are simpler to implement and can provide accurate prices for vanilla options, while stochastic volatility models are more flexible and can be used to price more complex exotic options.