Dupire local vol¶

Dupire local volatility replaces the constant-volatility assumption with a deterministic surface \(\sigma_{\mathrm{loc}}(S,t)\) chosen so that the model reproduces a continuum of European vanilla prices.

Context¶

Under the risk-neutral measure, the local-vol model writes

\[ dS_t = \bigl(r(t)-q(t)\bigr) S_t\,dt + \sigma_{\mathrm{loc}}(S_t,t)\,S_t\,dW_t^{\mathbb{Q}}, \qquad S_0 > 0. \]

Here \(r(t)\) is the risk-free rate and \(q(t)\) is the dividend yield. The objective is not to forecast future realized volatility. It is to find a deterministic diffusion that matches the market's vanilla option prices across strike and maturity.

Core Dupire math¶

From call prices to density¶

Let \(C(K,T)\) be the discounted price of a European call with strike \(K\) and maturity \(T\). In the absence of static arbitrage:

\(C(K,T)\) is decreasing in \(K\),
\(C(K,T)\) is convex in \(K\),
and the curvature \(C_{KK}(K,T)\) is proportional to the risk-neutral density.

That convexity matters because Dupire uses second strike derivatives. A surface that is visually smooth but not convex enough can produce negative or unstable local variance.

Dupire formula in strike space¶

For discounted call prices, the codebase uses

\[ \sigma_{\mathrm{loc}}^2(K,T) = \frac{2\left(C_T + \bigl(r(T)-q(T)\bigr)K C_K + q(T) C\right)} {K^2 C_{KK}}. \]

If you work with forward or undiscounted call prices \(c(K,T)\), the numerator changes to

\[ \sigma_{\mathrm{loc}}^2(K,T) = \frac{2\left(c_T + \bigl(r(T)-q(T)\bigr)\bigl(K c_K - c\bigr)\right)} {K^2 c_{KK}}. \]

Why the code often works in log strike¶

With \(x = \log K\), the derivatives satisfy

\[ K C_K = C_x, \qquad K^2 C_{KK} = C_{xx} - C_x. \]

So the discounted-call formula becomes

\[ \sigma_{\mathrm{loc}}^2(x,T) = \frac{2\left(C_T + \bigl(r(T)-q(T)\bigr) C_x + q(T) C\right)} {C_{xx} - C_x}. \]

This is often numerically preferable because it removes the explicit \(K^2\) factor from the denominator. In noisy wing data, that can reduce avoidable amplification.

Why matching marginals is not the whole story¶

Breeden-Litzenberger tells you about the marginal distribution at one maturity. Dupire is stronger: it specifies a full time-inhomogeneous diffusion. Two models can agree on a single maturity distribution and still produce very different dynamics between maturities. That difference matters for hedging, path-dependent pricing, and PDE behavior.

Engineering notes¶

Local vol is derivative-hungry. You need a surface that is stable in maturity and sufficiently smooth in strike.
Piecewise-linear behavior in total variance can make \(w_T\) piecewise constant, which shows up as visible banding in local vol.
That is why the docs recommend a smooth eSSVI projection for Dupire-oriented workflows rather than a raw stack of repaired slices.
Boundary points are the least trustworthy part of the surface. Trimming and masking are a feature, not a failure.

Assumptions behind local-vol pricing¶

The market is modeled as complete enough for a deterministic local volatility surface to be meaningful.
Vanilla option prices are arbitrage-consistent across strike and maturity.
The required derivatives exist in a stable numerical sense after interpolation or smoothing.

Diagnostics and failure modes¶

The main warning signs are:

noisy or sign-flipping curvature \(C_{KK}\),
denominator values that are too small or non-positive,
negative local variance after differentiation,
boundary artifacts from sparse wings or short maturity spacing,
and seam or banding artifacts when \(w_T\) is not time-smooth.

When a local-vol report masks points as invalid, that usually means the differentiation problem is ill-conditioned at those coordinates, not that the diagnostics are being overly conservative.

References¶

Bruno Dupire, Pricing and Hedging with Smiles
Jim Gatheral, The Volatility Surface