Local volatility ($\sigma (t,s)$)

A local volatility model uses a deterministic function of current time and spot as its diffusion volatility, $\sigma (t,S_t)$. It can reproduce a surface of European call option prices through the Dupire equation.

The stock diffusion $d{S}_t=S_t(rdt+\sigma (t,S_t)d{W}_t)$ has spot-dependent local volatility. With positive volatility and the Brownian filtration, Brownian martingale representation theorem converts discounted payoff martingales into stock gains.

For a non-dividend-paying stock and constant interest rate $r$, discounted call prices satisfy $C_T=\frac{1}{2}{K}^2\sigma (T,K{)}^2{C}_{KK}-rK{C}_K$. Strike differentiation recovers the discounted terminal density as $C_{KK}$; differentiating the discounted payoff identity supplies the maturity derivative. Put-call parity implies the same equation for put prices.

Where $C_{KK}>0$, rearranging the Dupire equation gives $\sigma (T,K{)}^2=2(C_T+rK{C}_K)/(K^2{C}_{KK})$. The strike curvature represents discounted density, while the adjusted maturity derivative gives the local diffusion contribution.

For a square-integrable discounted claim martingale $dM=hdW$ and discounted stock $d\tilde{S}=\tilde{S}\sigma dW$, choose stock holdings $h/(\tilde{S}\sigma )$. The remaining wealth lies in the bond. A nonnegative claim makes this replication admissible; discounted admissible wealth is a supermartingale, establishing the minimal initial cost.