Dupire local volatility

Definition A local volatility model assumes a constant rate $r$, a constant dividend yield $q$ and, under a pricing measure $\mathbb{Q}$ with Brownian motion $W^{\mathbb{Q}}$,

with $S_0 > 0$, where the local volatility $\sigma_{\mathrm{loc}}$ is a deterministic positive function of time and the current price. The model needs conditions that make this equation well posed, for example $\sigma_{\mathrm{loc}}$ bounded and $s \mapsto \sigma_{\mathrm{loc}}(t, s)\,s$ Lipschitz with a constant that does not depend on $t$. Then there is a unique solution, it stays positive, and the drift makes $e^{-(r - q)t} S_t$ a $\mathbb{Q}$-martingale. A constant $\sigma_{\mathrm{loc}} = \sigma$ gives the Black-Scholes model. The volatility depends on the path only through $(t, S_t)$, so $S$ is a Markov process driven by one Brownian motion. Calibration asks for the $\sigma_{\mathrm{loc}}$ whose European prices match a given call price surface, which fixes the law of $S_T$ under $\mathbb{Q}$ for each $T$, the one-time marginals. It does not fix the joint law of the path. Stochastic volatility and jump models fitted to the same European prices have the same marginals but can give different prices for barrier and forward-start options and different hedges. A local volatility that matches the surface is a model choice, not evidence about the physical dynamics under $\mathbb{P}$.

Definition The call price surface is the function $(K, T) \mapsto C(K, T)$ of time-$0$ prices of European calls on one underlying, for $K \ge 0$ and $T \ge 0$, with $C(K, 0) = (S_0 - K)^{+}$. With a constant rate and a pricing measure,

Its coordinates are the strike $K$ and the maturity $T$. They are the variables of a forward equation, in which the contract varies and the starting point $(0, S_0)$ is fixed. This differs from the backward Black-Scholes equation, whose variables are the current time $t$ and price $S$ for one fixed contract. Strike derivatives recover the marginal laws. The right derivative is $C_K(K+, T) = -e^{-rT}\,\mathbb{Q}(S_T > K)$, and when $S_T$ has a density $p(T, \cdot)$ under $\mathbb{Q}$,

the Breeden-Litzenberger relation. An arbitrage-free surface is nonincreasing and convex in $K$, with slopes in $[-e^{-rT}, 0]$, and its normalized prices are nondecreasing in $T$ at fixed normalized strike, as derived in Static arbitrage and volatility surfaces.

Definition Assume the local volatility model of the first card with $\sigma_{\mathrm{loc}}$ continuous, and assume that for every $T > 0$ the law of $S_T$ under $\mathbb{Q}$ has a density $p(T, s)$ on $(0, \infty)$, once continuously differentiable in $T$ and twice in $s$, that solves the forward Kolmogorov equation

where $a(T, s) = \sigma_{\mathrm{loc}}^2(T, s)\,s^2\,p(T, s)$. Assume also that differentiation under the integral sign is allowed and that the boundary terms below vanish as $s \to \infty$, namely $(s - K)\,s\,p$, $a$ and $(s - K)\,\partial_s a$. Then for $K > 0$ and $T > 0$ the call price surface satisfies the Dupire equation

with the initial condition $C(K, 0) = (S_0 - K)^{+}$. To derive it, write $D = e^{-rT}$ and $C = D\int_K^\infty (s - K)\,p\,ds$. Differentiating in $T$ gives $-rC$ plus $D\int_K^\infty (s - K)\,\partial_T p\,ds$, and the forward equation splits this integral into two parts. In the drift part one integration by parts moves the derivative onto $s - K$, whose derivative is $1$, and the identities $\int_K^\infty (s - K)\,p\,ds = C/D$ and $\int_K^\infty p\,ds = -C_K/D$ give

In the diffusion part two integrations by parts leave only the value at the strike, and $D\,p(T, K) = C_{KK}$ gives

Adding $-rC$, $(r - q)C$, $-(r - q)KC_K$ and the diffusion term gives the equation, because $-rC + (r - q)C = -qC$. With deterministic time-dependent $r$ and $q$ the same equation holds with their values at the maturity $T$. Stochastic rates need a different numeraire and are not covered here.

Definition Under the assumptions of the Dupire equation, at a point with $C_{KK}(K, T) > 0$, the local variance is

and the local volatility is its positive square root. The units are per year, since $A$ is in currency units per year and $K^2 C_{KK}$ in currency units. With $F(0, T) = S_0 e^{(r - q)T}$, $k = K/F(0, T)$ and the normalized price $c(k, T) = C(K, T)/(e^{-rT}F(0, T))$, the chain rule gives

the maturity derivative at fixed normalized strike. The numerator is therefore nonnegative exactly when the surface is calendar consistent at that point, and the denominator is nonnegative exactly when it is convex in strike. A given surface with $A < 0$ or $C_{KK} < 0$ at some point is not the price surface of any such model, contains a static arbitrage, and has no real local volatility there. At a point with $C_{KK} = 0$ the formula is undefined, and where $C_{KK}$ is small it divides by a small number. A positive value at every point still does not prove that the resulting coefficient defines a diffusion, which needs regularity and growth conditions on the whole function.

Definition A problem is well-posed in the sense of Hadamard when a solution exists, is unique, and depends continuously on the data. Recovering derivatives from prices fails the third condition. A perturbation $\eta(K) = \varepsilon \sin(\omega K)$ changes every price by at most $\varepsilon$ but changes the second derivative by up to $\varepsilon\omega^2$, which is as large as we like for a fast enough oscillation. On a grid the same effect appears in the stencils. Suppose the quotes $\widetilde{C}$ differ from the true prices by at most $\varepsilon$, and $C$ is four times continuously differentiable in $K$ with $\lvert C_{KKKK} \rvert \le M$ on $[K - h, K + h]$. The second difference of the quotes, $\widehat{C}_{KK} = (\widetilde{C}(K + h) - 2\widetilde{C}(K) + \widetilde{C}(K - h))/h^2$, then satisfies

where the first term is amplified quote noise and the second is the truncation error from Taylor’s theorem. The centered maturity difference with step $\Delta T$ amplifies quote errors by $1/\Delta T$. Refining the grid lowers the truncation error and raises the noise, so a finer grid does not by itself give a better local volatility. Local variance divides by the noisy curvature, which makes it most sensitive where $C_{KK}$ is small. That happens in the wings, and at short maturities, where the payoff kink concentrates nearly all the curvature close to $S_0$ and leaves very little elsewhere. Stable recovery fits a smooth surface that satisfies the static arbitrage constraints and controls noise before differentiating. How to do that for market data is outside this lesson.