1 Local Volatility Models
In mathematical finance Local Volatility Models are option pricing models that treat volatility as a function of both asset price \(S_{t}\) and time \(t\). The asset price process \((S_{t})\) is assumed to have dynamics given by the SDE \[ dS_{t}=(r_{t}-d_{t})S_{t}dt+\sigma(S_{t}, t)S_{t}dW_{t}. \] A simple derivation of the Black-Scholes PDE (BSPDE) tells us that if \(V_{t}=u(t,S_{t})\) denotes the value of the replicating portfolio at time \(t\), then \[ \partial _{t }u+rx\partial _{x}u+ \frac{1}{2}\sigma^2(t,x)x^2\partial_{x x}u-ru=0. \] We see that BS is quite inefficient in solving option prices since it requires the knowledge at all \((t,x)\) just to solve out a single number.
A special example of the local volatility model is the [[constant-elasticity-of-variance-cev-model]] which has dynamics given by \[ S_{t}= \mu S_{t}+\sigma S_{t}^\gamma dW_{t}. \]
1.1 Dupire’s PDE
However, in the local volatility model, there is a model dependent result called Dupire's Formula that efficiently solves out option price. The formula is \[ \frac{{\partial C}}{\partial T}(T,K)=-rK\frac{{\partial C}}{\partial K}(T,K)+ \frac{1}{2}\sigma^2(T,K)K^2 \frac{{\partial^2C}}{\partial K^2}(T,K) \] with natural initial condition \[ C(0,K)=(S_{0}-K)_{+}, \] since when \(T=0\) option price is just the immediate payoff.
Dupire’s PDE provides an explicit formula for implied volatility in the local volatility model.
1.2 Problems
The local volatility model is expected to produce implied volatility surface \(I(T,K)\) not flat in \(T\) or in \(K\) however it is still not perfect. The model suffers from the so-called \((t,T,K)\)-problem.
In the setting of local volatility \(\sigma=\sigma(t,x)\) so the volatility is always the same given that the time and stock price is the same. As a result, at time 0 if we calibrate and get the surface \(I(0,T,K)\) using options sold at time 0 and maturing at time \(T\), we expect to see the same surface \(I(t,T+t, K)\) based on options sold at \(t\) and maturing at time \(T+t\). However, empirical data implies that those two implied volatility surfaces are not the same under time translation which is a contradiction to the model.