The concept of "local volatility" was introduced by Dupire (1994) as a way to generalize and extend the commonly used Black-Scholes model for option pricing. In the Black-Scholes model, the volatility of the underlying asset is assumed to be constant. However, in practice, it is observed that the options on the same underlying asset, but with different strikes and maturities, have different implied volatilities. This phenomenon is called "volatility smile" or "volatility skew". Local volatility is a more realistic approach to modeling the volatility of an asset, as it allows for a time-dependent and state-dependent (i.e., dependent on the value of the underlying asset) volatility structure.
The local volatility function, denoted by σ(S, t), is a deterministic function where S represents the underlying asset price, and t represents time. It is then used to model the asset price evolution under the risk-neutral probability measure (ℚ) according to the following stochastic differential equation (SDE):
dSt = rStdt + σ(St, t)StdWtℚ
where r is the risk-free interest rate, and dWtℚ is a ℚ-Brownian motion increment.
To extract the local volatility surface (i.e., the function σ(S, t)) from observed market prices of European options, one can use Dupire’s formula as follows:
$$\sigma^2 (S,K,T) = \frac{\frac{\partial C}{\partial T} + rK\frac{\partial C}{\partial K} + \frac{r}{2} K^2 \frac{\partial^2 C}{\partial K^2}}{\frac{1}{2}K^2\frac{\partial^2 C}{\partial K^2}},$$
where K is the option’s strike price, T is the time-to-maturity, and C(K, T) represents the market price of the European call option with strike K maturing at time T.
Local volatility models are used for several purposes in option pricing and risk management:
1. **Pricing exotic options**: Options with complex payoffs or early exercise features (e.g., barrier options or American options) can be more accurately priced using local volatility models compared to constant volatility models, as the latter fails to capture the true market dynamics.
2. **Hedging**: The local volatility model can be used to compute more realistic sensitivities (i.e., Greeks) of options to various factors, such as changes in underlying asset prices, volatility, interest rates, etc. This helps traders and portfolio managers to hedge their positions more accurately and reduce the risk associated with their options portfolios.
3. **Generating consistent scenarios**: In risk management, local volatility models can be used to generate future scenarios of asset returns that are consistent with the observed market option prices. This is useful for stress testing and value-at-risk (VaR) calculations, as well as for capital requirement and risk budgeting exercises.
Overall, local volatility models have become an essential tool for option pricing and risk management, providing a more realistic representation of the market’s behavior and allowing for better-informed decision-making.