Local Volatility

Dupire's Formula

Bruno Dupire famously derived the formula that extracts the local volatility surface from the market prices of European calls. The local volatility $sigma_{loc}(K, T)$ is given by: $$ sigma_{loc}^2(K, T) = rac{rac{partial C}{partial T} + rK rac{partial C}{partial K}}{ rac{1}{2} K^2 rac{partial^2 C}{partial K^2} } $$ Where: * $C(K, T)$ is the market price of a Call option with strike $K$ and maturity $T$. * $rac{partial C}{partial T}$ is the change in call price with respect to time (Theta). * $rac{partial C}{partial K}$ is the change in call price with respect to strike (Dual Delta). * $rac{partial^2 C}{partial K^2}$ is the curvature of the call price with respect to strike (Dual Gamma). * $r$ is the risk-free rate. **Interpretation:** The numerator represents the time-decay of the option, and the denominator represents the convexity (density) of the probability distribution. The ratio gives the instantaneous variance required to satisfy the forward Fokker-Planck equation.

The Local Volatility Surface

The result of the model is a **Local Volatility Surface**, a 3D plot mapping Volatility vs. Spot Price vs. Time. ### Key Characteristics 1. **Inverse Relation:** In equity markets, the local volatility function typically slopes downward with respect to price. As the stock price falls, local volatility increases. This mimics the "leverage effect" and panic selling. 2. **Sticky Local Volatility:** The model assumes that the volatility function $sigma(S, t)$ itself is fixed. As the market moves, the asset "slides" along this pre-determined surface. * *Example:* If the spot is 100 and vol is 20%, and the spot drops to 90 where the surface says vol is 25%, the model assumes vol becomes 25%. * *Reality Check:* In reality, the *entire surface* often shifts when the spot moves (Sticky Strike vs. Sticky Delta dynamics), which is a limitation of the LV model.

Pricing Exotics

The primary use case for Local Volatility is pricing **Exotic Options**, particularly those with path dependency (Barriers, Asians, Cliquets). Why? Because vanilla options (calls/puts) are inputs to the model, so the model matches them by definition. The value add is taking that calibrated surface and using it to price something that *isn't* traded. **Example: Barrier Option (Down-and-Out Call)** * A Down-and-Out Call becomes worthless if the price touches a barrier $B$. * To price this, you need to know the volatility *at the barrier*. * Black-Scholes uses a constant average volatility. * Local Volatility uses the specific volatility $sigma(B, t)$ at the barrier level. Since LV typically shows higher volatility at lower prices (downside skew), LV often prices Down-and-Out calls lower than Black-Scholes (higher probability of hitting the barrier).

Local Volatility vs. Stochastic Volatility

Two approaches to "Solving the Smile".

Local-Stochastic Volatility (LSV)

Because Local Volatility implies rigid dynamics (if Price is X, Vol *must* be Y) and Stochastic Volatility struggles to fit the short-term smile, banks often use **Local-Stochastic Volatility (LSV)** models. LSV combines both: $$ dS_t = sigma_{loc}(S_t, t) cdot alpha_t cdot S_t dW_t^1 $$ Where $sigma_{loc}$ handles the static smile calibration, and $alpha_t$ (the stochastic component) handles the dynamic behavior of volatility (vol-of-vol). This is the state-of-the-art for pricing complex FX and Equity exotics.