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Local volatility

A local volatilty model is an extension of the the Black-Scholes model where the volatility $\sigma$ is replaced by a deterministic function $\sigma(t,S(t))$ depending on the time $t$ and the stock price $S(t)$. We thus have the following risk neutral stock price model: \[ \text{d}S(t)=rS(t)\text{d}t+\sigma(t,S(t))S(t)\text{d}W(t),\] where $W$ is standard Brownian motion.This model can perfectly fit any set of arbitage free option prices. Assuming that we have European call option prices for all strikes $K$ and times to maturity $T$ we can find the local volatility function $\sigma(t,S)$ through the Dupire formula \[ \sigma(T,S)^2=\frac{\frac{\partial}{\partial T} C(T,S_0,K)+rK\frac{\partial}{\partial K} C(T,S_0,K)}{\frac{K^2}{2}\frac{\partial^2}{\partial K} C(T,S_0,K)}\mid_ {K=S}, \] where $C(T,K,S_0)$ is the observed price of a European call option with strike $K$ and time to maturity $T$ with current stock price at $S_0$.
In practice we only have option prices on some grid in strike and time to maturity. This makes this model hard to fit to data in a stable way. It is often more numerically stable to fit the model using implied volatilities instead of option prices.
References
Bruno Dupire. (1994). Pricing with a smile. Risk , 7(1),18-20.

 

Questions: Magnus Wiktorsson
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