A B C D E F G H I J K __L__ M N O P Q R S T U V W X Y Z

# 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.

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.

Bruno Dupire. (1994). Pricing with a smile.

Questions: Magnus Wiktorsson

Last update: 2016 Sep 26 10:07:12. Validate: HTML CSS

Centre for Mathematical Sciences, Box 118, SE-22100, Lund. Telefon: +46 46-222 00 00 (vx)