Dupire's Local Volatility Inversion Formula

Scenario

In continuous-time quantitative finance, the local volatility model specifies the underlying asset dynamics as $dS_t = r S_t dt + \sigma_{\text{loc}}(S_t, t) S_t dW_t^{\mathbb{Q}}$. The local volatility function must be calibrated to match the entire market volatility surface.

Your Task

1. Using the Breeden-Litzenberger result for the risk-neutral transition probability density $p(K, T) = e^{r T} \frac{\partial^2 C}{\partial K^2}$, differentiate the call pricing integral with respect to maturity $T$.
2. Apply the forward Fokker-Planck (Kolmogorov forward) PDE to express $\frac{\partial C}{\partial T}$ in terms of strike derivatives.
3. Derive Dupire's local volatility equation: $\sigma_{\text{loc}}^2(K, T) = \frac{\frac{\partial C}{\partial T} + r K \frac{\partial C}{\partial K}}{\frac{1}{2} K^2 \frac{\partial^2 C}{\partial K^2}}$
4. Construct a call pricing surface under constant Black-Scholes volatility $\sigma_0 = 0.25$ and numerically confirm that the finite-difference Dupire formula recovers $\sigma_{\text{loc}}(K, T) = 0.25$ identically.

Constraints

• Continuous dividend yield $q = 0$.
• Strictly positive convex call price profile in strike $K$.

Examples Under flat Black-Scholes surfaces, $\frac{\partial C}{\partial T} + r K \frac{\partial C}{\partial K} = \frac{1}{2}\sigma_0^2 K^2 \frac{\partial^2 C}{\partial K^2}$, confirming ratio equals $\sigma_0^2$.