Local volatility model ($d{S}_t=S_t(rdt+\sigma (t,S_t)d{W}_t)$)

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.

In the zero-interest local volatility model $d{S}_t=\sqrt{S_t}d{W}_t^Q$, the risk-neutral pricing value of the European contingent claim $e^{v{S}_T}$, for $v>0$ and $v(T-t)<2$, isThe compound Poisson transition law of a driftless square-root diffusion proves finiteness and this conditional-expectation formula. Alternatively, substituting $V(t,S)=e^{a(t)S+b(t)}$ into the pricing equation for a local volatility model gives $a^{\prime }+a^2/2=0$ and $b^{\prime }=0$, with terminal values $a(T)=v,b(T)=0$. Hence $a(t)=v/[1-v(T-t)/2]$ and $b(t)=0$. The conditional-expectation calculation verifies that this formal solution is the price of the unbounded claim. The delta hedge holds $a(t)V$ stocks and $(1-a(t)S_t)V/B_t$ bank units.

For a local volatility model $d{S}_t=S_t(rdt+\sigma (S_t)d{W}_t^Q)$ with bank account $B_t=B_0{e}^{rt}$, a nonnegative classical solution $V$ ofdefines a possible price of a European contingent claim. Indeed, Itô formula gives $d(V(t,S_t)/B_t)=B_t^{-1}\sigma (S_t)S_t{V}_S(t,S_t)d{W}_t^Q$, a local martingale. Thus the same equivalent local martingale measure works for the market augmented by this price. For an admissible trading strategy whose discounted wealth is bounded below, the discounted gains are a local martingale bounded below, hence a supermartingale by localization and the Fatou lemma. A zero-cost terminal gain that is nonnegative and strictly positive with positive probability would then contradict the supermartingale expectation inequality. This proves the absence of arbitrage for such strategies. If $V$ is bounded on the finite time interval, its discounted value is a bounded martingale and consequently $V(t,S_t)=B_t{\mathbb{E}}_Q[g(S_T)/B_T\mid {\mathcal{F}}_t]$. Nonnegativity alone guarantees the local martingale calculation, not this latter equality for every possible solution.

With $x=\log S$ and $u=e^{r(T-t)}V(e^x,t)$, a local volatility pricing equation becomes the displayed backward equation. Central finite differences and the explicit Euler method give $u_i^{j-1}=u_i^j+k[a_i^j{D}_{xx}{u}_i^j+b_i^j{D}_x{u}_i^j]$. Nonnegative weights require $k{\sigma }_i^2/h^2\le 1$ and $|b_i|h\le {\sigma }_i^2$. Under these conditions the update is a convex combination, yielding maximum-norm stability. With smooth coefficients, positive diffusion, fixed positive remaining maturity and suitable consistent boundary treatment, its ordinary error is $O(k+h^2)$. Volatility bounds alone do not imply all this regularity or a Greek error expansion.

The logarithmic coordinate requires the displayed chain-rule factors for option delta and option gamma. If only a value error bound $\Vert e_h{\Vert }_{\infty }=O(h^2)$ is known, central differentiation gives $D_x{e}_h=O(h)$ and $D_{xx}{e}_h=O(1)$. Thus first and second Greek convergence do not follow with the same order from a value bound alone. With a smooth spatial error expansion $e_h=h^{2}E+o(h^2)$ and corresponding derivative control, both Greeks can instead retain second-order convergence away from a nonsmooth payoff or boundary. Refinement and Greek-specific regularity are essential to distinguish these situations.

If $V$ satisfies the pricing equation for a local volatility model, hold $h_t^S=V_S(t,S_t)$ stocks and $h_t^B=[V(t,S_t)-S_t{V}_S(t,S_t)]/B_t$ units of the bank account. Their value is $V(t,S_t)$. The pricing equation for a local volatility model and Itô formula giveThus this is a self-financing strategy, with terminal value $g(S_T)$. It is a delta hedge, and nonnegative $V$ makes its wealth an admissible trading strategy. The stock and bank wealth amounts are $S_t{V}_S$ and $V-S_t{V}_S$, rather than the asset-unit holdings.

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.