How do you apply the Dupire local volatility model in the context of implied volatility and option pricing?

The Dupire Local Volatility model, also known as the Local Volatility Model, provides a method to find the local volatility function that generates a theoretical option price surface consistent with the observed market-implied volatility surface for options on the same underlying asset. This local volatility function, when applied in the context of option pricing, can be used to price exotic options or manage the risk of a portfolio of vanilla options.

We start with the Dupire equation, which describes the relationship between local volatility and implied volatility. Given a call option price, C(S, K, T), where S is the underlying stock price, K is the strike price, and T is the time to expiry, the Black-Scholes equation for European call options can be written as:

$$\frac{\partial C}{\partial T} + \frac{1}{2} \sigma^2_\text{loc}(S, T) S^2 \frac{\partial^2 C}{\partial S^2} + rS\frac{\partial C}{\partial S} - rC = 0$$

Here, σ_loc is the local volatility function, r is the risk-free interest rate, and $\frac{\partial C}{\partial T}$, $\frac{\partial^2 C}{\partial S^2}$, and $\frac{\partial C}{\partial S}$ are the first and second partial derivatives of the call option price with respect to time and price.

We know from the standard Black-Scholes model that the implied volatility, σ_imp, can be used to find the price of a European call option using the following formula:

C(S, K, T) = SN(d₁) − e^− rTKN(d₂),

where

$$d_{1} = \frac{\ln(S/K) + (r + \frac{1}{2}\sigma_\text{imp}^{2})T}{\sigma_\text{imp}\sqrt{T}},$$

$$d_{2} = d_{1} - \sigma_\text{imp}\sqrt{T},$$

and N( ⋅ ) is the cumulative distribution function of the standard normal distribution.

Now we can substitute the Black-Scholes formula for the call option price into the Dupire equation and then solve for the local volatility function σ_loc(S, T). To do this, we first need to find the derivatives of C with respect to T, S, and S². These derivatives can be calculated using Black-Scholes Greeks and their respective formulas. Once we have these derivatives, we can plug them into the Dupire equation and solve for the local volatility function.

After isolating σ_loc(S, T), we obtain the Dupire local volatility equation:

$$\sigma^2_\text{loc}(S, T) = \frac{\frac{\partial C}{\partial T} + rS\frac{\partial C}{\partial S} - rC}{\frac{1}{2} S^2 \frac{\partial^2 C}{\partial S^2}}$$

Now we can use the observed market prices of European call options with various strikes and maturities to calculate the corresponding implied volatilities, then deduce the local volatility function by solving the Dupire local volatility equation at each point (S, T).

In the context of option pricing, we can use the derived local volatility function in conjunction with a numerical method, such as a finite difference method, Monte Carlo simulation, or a binomial/lattice tree model, for exotic options. It helps us to price other derivatives that may not have liquid market quotes but have dependencies on the same underlying asset.

For example, if we want to price a European Barrier Option on the same underlying asset using the local volatility model, we can use the Monte Carlo simulation to simulate the underlying stock price paths, where the stock returns follow a Geometric Brownian Motion with the local volatility function as the instantaneous volatility. Finally, we can compute the discounted option payoffs over the simulations to estimate the option price.

In summary, the Dupire Local Volatility model provides a methodology for estimating local volatilities consistent with observed market-implied volatilities. This enables practitioners not only to price plain vanilla options more accurately but also to price exotic options and manage the risk of options portfolios more effectively.