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Dupire Equation as an Inverse Problem

Local volatility models provide an exact fit (in principle) to a continuum of vanilla option prices. This exactness comes at a cost: constructing the local volatility surface is a highly ill-posed inverse problem because it requires differentiating noisy market data.

The local volatility model

In the (risk-neutral) local volatility model, the underlying \(S_t\) follows

\[ dS_t = (r-q)S_t\,dt + \sigma_{\text{loc}}(t,S_t)\,S_t\,dW_t, \]

where \(r\) is the risk-free rate, \(q\) is the dividend yield (or convenience yield), and \(\sigma_{\text{loc}}(t,S)\) is the local volatility function to be calibrated.

Given \(\sigma_{\text{loc}}\), the model implies a unique surface of European option prices \(C(K,T)\).

Dupire’s forward equation

Let \(C(K,T)\) denote the time-0 price of a European call with strike \(K\) and maturity \(T\). Under standard smoothness assumptions, Dupire derived a forward PDE relating \(C\) and \(\sigma_{\text{loc}}\).

A common (simplified) form is:

\[ \partial_T C(K,T) = \frac12\,\sigma_{\text{loc}}^2(T,K)\,K^2\,\partial_{KK} C(K,T) - (r-q)K\,\partial_K C(K,T) + q\,C(K,T), \]

where derivatives are with respect to strike.

Solving this equation forward is the pricing problem.

Inverting Dupire: extracting local volatility

Rearranging the forward equation yields an expression for local variance

\[ \sigma_{\text{loc}}^2(T,K) = \frac{2\left(\partial_T C + (r-q)K\partial_K C - q C\right)} {K^2\,\partial_{KK} C}. \]

This reveals why calibration is an inverse problem:

  • it requires \(\partial_T C\) and \(\partial_{KK} C\),
  • these are derivatives of market prices, which are observed only at discrete points and contaminated by noise.

Thus, the mapping

\[ C(\cdot,\cdot) \longmapsto \sigma_{\text{loc}}(\cdot,\cdot) \]

is an unstable inversion involving differentiation and division by curvature.

Relationship to implied volatility

Market data are often given as implied vol \(\sigma_{\text{impl}}(K,T)\), not prices. One may:

  1. convert implied vol to prices via Black–Scholes,
  2. interpolate to a smooth surface,
  3. apply Dupire to the smoothed price surface.

Alternatively, Dupire can be written directly in terms of implied vol or total variance, but the same inverse-problem issues remain: derivatives are needed.

Key takeaways

  • Dupire provides a theoretical route from a smooth call price surface to \(\sigma_{\text{loc}}(T,K)\).
  • Calibration is an inverse problem because it requires differentiating noisy, discrete data.
  • Practical implementation requires smoothing, arbitrage filtering, and careful numerics.

Further reading

  • Dupire (1994), “Pricing with a Smile”.
  • Gatheral, The Volatility Surface (local vol and implied vol geometry).
  • Fengler, Semiparametric Modeling of Implied Volatility (surface construction).

Exercises

Exercise 1. Starting from Dupire's forward PDE

\[ \partial_T C = \frac{1}{2}\sigma_{\text{loc}}^2(T,K)\,K^2\,\partial_{KK}C - (r-q)K\,\partial_K C + qC \]

derive the inversion formula for \(\sigma_{\text{loc}}^2(T,K)\) by solving algebraically for the local variance. State explicitly which quantity appears in the denominator and explain why its sign is guaranteed under no-arbitrage conditions.

Solution to Exercise 1

Starting from the Dupire forward PDE

\[ \partial_T C = \frac{1}{2}\sigma_{\text{loc}}^2(T,K)\,K^2\,\partial_{KK}C - (r-q)K\,\partial_K C + qC \]

we isolate the term containing \(\sigma_{\text{loc}}^2\) by moving all other terms to the left-hand side:

\[ \partial_T C + (r-q)K\,\partial_K C - qC = \frac{1}{2}\sigma_{\text{loc}}^2(T,K)\,K^2\,\partial_{KK}C \]

Dividing both sides by \(\frac{1}{2}K^2\,\partial_{KK}C\) yields the inversion formula:

\[ \sigma_{\text{loc}}^2(T,K) = \frac{2\bigl(\partial_T C + (r-q)K\,\partial_K C - qC\bigr)}{K^2\,\partial_{KK}C} \]

The denominator is \(K^2\,\partial_{KK}C\). Since \(K^2 > 0\) always, the critical factor is \(\partial_{KK}C\).

Under no-arbitrage conditions, \(\partial_{KK}C \ge 0\) is guaranteed by the butterfly spread condition. To see this, consider the butterfly payoff at strikes \(K - h\), \(K\), \(K + h\):

\[ \text{Butterfly} = C(K-h) - 2C(K) + C(K+h) \ge 0 \]

This non-negativity holds because the butterfly payoff \((S_T - (K-h))^+ - 2(S_T - K)^+ + (S_T - (K+h))^+\) is non-negative for all \(S_T\), and a non-negative payoff must have a non-negative price under no-arbitrage. Dividing by \(h^2\) and taking \(h \to 0\) gives \(\partial_{KK}C \ge 0\).

Moreover, strict convexity (\(\partial_{KK}C > 0\)) holds whenever the risk-neutral density is strictly positive, since \(\partial_{KK}C\) equals the discounted risk-neutral density \(e^{-rT}q(K)\) at that strike. This ensures the denominator is strictly positive, making the Dupire inversion well-defined.

Exercise 2. Consider a flat implied volatility surface \(\sigma_{\text{impl}}(K,T) = \sigma_0\) for all \(K, T\). Compute \(\partial_T C\) and \(\partial_{KK} C\) from the Black--Scholes formula, and show that the Dupire formula recovers \(\sigma_{\text{loc}}(T,K) = \sigma_0\) everywhere.

Solution to Exercise 2

Under a flat implied volatility surface \(\sigma_{\text{impl}}(K,T) = \sigma_0\), the call price is given by the Black--Scholes formula:

\[ C(K,T) = S_0 e^{-qT}N(d_1) - K e^{-rT}N(d_2) \]

where \(d_1 = \frac{\ln(S_0/K) + (r - q + \frac{1}{2}\sigma_0^2)T}{\sigma_0\sqrt{T}}\) and \(d_2 = d_1 - \sigma_0\sqrt{T}\).

Computing \(\partial_T C\): By the Black--Scholes PDE in forward variables, we have

\[ \partial_T C = \frac{S_0 e^{-qT}\sigma_0 \,\phi(d_1)}{2\sqrt{T}} + qS_0 e^{-qT}N(d_1) - rKe^{-rT}N(d_2) \]

where \(\phi(\cdot)\) is the standard normal density.

Computing \(\partial_{KK} C\): Taking the first derivative with respect to \(K\):

\[ \partial_K C = -e^{-rT}N(d_2) \]

Taking the second derivative:

\[ \partial_{KK} C = e^{-rT}\frac{\phi(d_2)}{K\sigma_0\sqrt{T}} \]

Substituting into Dupire's formula:

\[ \sigma_{\text{loc}}^2 = \frac{2\bigl(\partial_T C + (r-q)K\partial_K C - qC\bigr)}{K^2\,\partial_{KK}C} \]

For the numerator, using the relation \(S_0 e^{-qT}\phi(d_1) = K e^{-rT}\phi(d_2)\) (which follows from the log-normal structure), we substitute:

\[ \partial_T C + (r-q)K\partial_K C - qC \]
\[ = \frac{Ke^{-rT}\phi(d_2)\sigma_0}{2\sqrt{T}} + qS_0 e^{-qT}N(d_1) - rKe^{-rT}N(d_2) \]
\[ \quad - (r-q)Ke^{-rT}N(d_2) - q\bigl(S_0 e^{-qT}N(d_1) - Ke^{-rT}N(d_2)\bigr) \]

The terms involving \(N(d_1)\) cancel: \(qS_0 e^{-qT}N(d_1) - qS_0 e^{-qT}N(d_1) = 0\). The terms involving \(N(d_2)\) also cancel: \(-rKe^{-rT}N(d_2) - (r-q)Ke^{-rT}N(d_2) + qKe^{-rT}N(d_2) = Ke^{-rT}N(d_2)(-r - r + q + q) = -2(r-q)Ke^{-rT}N(d_2)\)... Let us proceed more carefully by collecting all \(N(d_2)\) terms:

\[ -rKe^{-rT}N(d_2) - (r-q)Ke^{-rT}N(d_2) + qKe^{-rT}N(d_2) = Ke^{-rT}N(d_2)(-r - r + q + q) = 0 \]

since \(-r -(r-q)+q = -2r+2q\)... Actually, let us recount: \(-r -(r-q)+q = -r -r +q +q = -2r+2q\). This is zero only when \(r=q\).

A cleaner approach uses the Black--Scholes PDE directly. The call price satisfies

\[ \partial_T C = \frac{1}{2}\sigma_0^2 K^2 \partial_{KK}C - (r-q)K\partial_K C + qC \]

This is precisely Dupire's forward equation with \(\sigma_{\text{loc}}^2(T,K)\) replaced by \(\sigma_0^2\). Substituting into the Dupire inversion formula:

\[ \sigma_{\text{loc}}^2(T,K) = \frac{2\bigl(\partial_T C + (r-q)K\partial_K C - qC\bigr)}{K^2\,\partial_{KK}C} \]
\[ = \frac{2 \cdot \frac{1}{2}\sigma_0^2 K^2 \partial_{KK}C}{K^2\,\partial_{KK}C} = \sigma_0^2 \]

Therefore \(\sigma_{\text{loc}}(T,K) = \sigma_0\) everywhere, confirming that the Dupire formula correctly recovers a constant local volatility from a flat implied volatility surface.

Exercise 3. Suppose observed call prices are contaminated by additive noise: \(\hat{C}(K_i, T_j) = C(K_i, T_j) + \varepsilon_{ij}\) where \(\varepsilon_{ij} \sim N(0, \delta^2)\). Using a finite-difference approximation for \(\partial_{KK}C\) with strike spacing \(\Delta K\), show that the variance of the estimated second derivative scales as

\[ \operatorname{Var}\!\left(\widehat{\partial_{KK}C}\right) \sim \frac{\delta^2}{(\Delta K)^4} \]

Explain why this makes the Dupire inversion ill-posed and discuss the trade-off between bias and variance as \(\Delta K\) varies.

Solution to Exercise 3

Let the observed prices be \(\hat{C}(K_i, T_j) = C(K_i, T_j) + \varepsilon_{ij}\) with \(\varepsilon_{ij} \sim N(0, \delta^2)\) independent. The central finite-difference approximation for the second derivative in strike is

\[ \widehat{\partial_{KK}C}(K,T) = \frac{\hat{C}(K+\Delta K,T) - 2\hat{C}(K,T) + \hat{C}(K-\Delta K,T)}{(\Delta K)^2} \]

Substituting the noisy observations:

\[ \widehat{\partial_{KK}C} = \frac{(C_{+} + \varepsilon_{+}) - 2(C_0 + \varepsilon_0) + (C_{-} + \varepsilon_{-})}{(\Delta K)^2} \]
\[ = \underbrace{\frac{C_{+} - 2C_0 + C_{-}}{(\Delta K)^2}}_{\text{true value}+O((\Delta K)^2)} + \frac{\varepsilon_{+} - 2\varepsilon_0 + \varepsilon_{-}}{(\Delta K)^2} \]

The noise term is \(Z = (\varepsilon_{+} - 2\varepsilon_0 + \varepsilon_{-})/(\Delta K)^2\). Since the \(\varepsilon\) terms are independent with variance \(\delta^2\):

\[ \operatorname{Var}(Z) = \frac{1^2 + (-2)^2 + 1^2}{(\Delta K)^4}\,\delta^2 = \frac{6\delta^2}{(\Delta K)^4} \]

This confirms that \(\operatorname{Var}(\widehat{\partial_{KK}C}) \sim \delta^2/(\Delta K)^4\) (with proportionality constant 6).

Why this makes Dupire ill-posed: The Dupire formula divides by \(\partial_{KK}C\), so the estimated local variance inherits the noise from this estimate. As \(\Delta K \to 0\), the variance of the estimated second derivative grows as \(1/(\Delta K)^4\), which diverges rapidly. This means that refining the strike grid does not improve accuracy---it makes it worse.

Bias-variance trade-off: The total error in estimating \(\partial_{KK}C\) has two components:

  • Bias (truncation error): \(\sim C^{(4)}(\Delta K)^2/12\), which decreases as \(\Delta K \to 0\).
  • Variance (noise amplification): \(\sim 6\delta^2/(\Delta K)^4\), which increases as \(\Delta K \to 0\).

There is an optimal \(\Delta K\) that balances these. Setting the derivative of the total mean squared error to zero gives

\[ (\Delta K)^{\star} \sim \left(\frac{\delta}{|C^{(4)}|}\right)^{1/3} \]

Below this optimal spacing, noise dominates; above it, truncation error dominates. This fundamental trade-off is the hallmark of ill-posed inverse problems.

Exercise 4. A trader observes European call prices at strikes \(K \in \{80, 90, 100, 110, 120\}\) and maturities \(T \in \{0.25, 0.50\}\) with \(S_0 = 100\), \(r = 0.03\), \(q = 0\). Using finite differences, write down explicit formulas for approximating \(\partial_T C(K, T)\) and \(\partial_{KK} C(K, T)\) at the grid point \((K=100, T=0.25)\). Identify which market quotes are needed.

Solution to Exercise 4

The trader needs to approximate two derivatives at the grid point \((K=100, T=0.25)\).

Approximation of \(\partial_T C(100, 0.25)\): Using a forward difference in maturity with \(\Delta T = 0.50 - 0.25 = 0.25\):

\[ \partial_T C(100, 0.25) \approx \frac{C(100, 0.50) - C(100, 0.25)}{\Delta T} = \frac{C(100, 0.50) - C(100, 0.25)}{0.25} \]

This requires the market quotes \(C(100, 0.25)\) and \(C(100, 0.50)\).

Approximation of \(\partial_{KK}C(100, 0.25)\): Using the central difference in strike with \(h = 10\):

\[ \partial_{KK}C(100, 0.25) \approx \frac{C(110, 0.25) - 2C(100, 0.25) + C(90, 0.25)}{h^2} = \frac{C(110, 0.25) - 2C(100, 0.25) + C(90, 0.25)}{100} \]

This requires the market quotes \(C(90, 0.25)\), \(C(100, 0.25)\), and \(C(110, 0.25)\).

Approximation of \(\partial_K C(100, 0.25)\) (needed for the full Dupire formula): Using the central difference:

\[ \partial_K C(100, 0.25) \approx \frac{C(110, 0.25) - C(90, 0.25)}{2h} = \frac{C(110, 0.25) - C(90, 0.25)}{20} \]

Summary of required quotes: The five market prices needed are:

  • \(C(90, 0.25)\), \(C(100, 0.25)\), \(C(110, 0.25)\) (for strike derivatives)
  • \(C(100, 0.50)\) (for the time derivative)

Note that \(C(100, 0.25)\) is used in all three approximations. The quotes at \(K \in \{80, 120\}\) are not directly needed for these first-order finite-difference formulas, although they could be used for higher-order stencils or for computing \(\partial_K C\) and \(\partial_{KK}C\) at neighboring strikes.

Exercise 5. The butterfly spread condition requires \(\partial_{KK}C \ge 0\) for absence of arbitrage. Show that if this condition is violated at some point \((K_0, T_0)\) in the interpolated surface, the Dupire formula yields a negative local variance \(\sigma_{\text{loc}}^2(T_0, K_0) < 0\). Propose a practical remedy using arbitrage filtering before applying the Dupire inversion.

Solution to Exercise 5

The Dupire inversion formula (with \(r = q = 0\) for simplicity) is

\[ \sigma_{\text{loc}}^2(T_0, K_0) = \frac{2\,\partial_T C(K_0, T_0)}{K_0^2\,\partial_{KK}C(K_0, T_0)} \]

The numerator \(\partial_T C\) represents the time value of the option. Under no-arbitrage, call prices are non-decreasing in maturity (for \(q = 0\)), so \(\partial_T C \ge 0\). Thus the numerator is non-negative.

The denominator involves \(K_0^2 > 0\) and \(\partial_{KK}C\). If the butterfly spread condition is violated at \((K_0, T_0)\), then \(\partial_{KK}C(K_0, T_0) < 0\).

Combining a non-negative numerator with a negative denominator:

\[ \sigma_{\text{loc}}^2(T_0, K_0) = \frac{2\,\partial_T C}{K_0^2\,\partial_{KK}C} = \frac{(\ge 0)}{(< 0)} \le 0 \]

Since variance must be non-negative, a negative \(\sigma_{\text{loc}}^2\) is unphysical and signals an arbitrage violation in the input data. More precisely, \(\partial_{KK}C < 0\) means the risk-neutral density implied by the interpolated surface is negative at that point, which is impossible for a probability density.

Practical remedy using arbitrage filtering:

  1. Detection: After constructing the interpolated call surface, compute \(\partial_{KK}C\) at all grid points. Flag any point where \(\partial_{KK}C \le 0\) (or below a small positive threshold \(\epsilon\)).

  2. Local adjustment: For isolated violations, adjust the call prices at and around the offending strikes to restore convexity. This can be done by solving a constrained optimization:

    \[ \min_{\tilde{C}} \sum_i (\tilde{C}_i - C_i^{\text{obs}})^2 \quad \text{subject to} \quad \tilde{C}(K_{i-1}) - 2\tilde{C}(K_i) + \tilde{C}(K_{i+1}) \ge 0 \;\;\forall i \]

  3. Global constrained fitting: Fit a parametric model (such as SVI) that enforces convexity by construction. Since SVI total variance has \(\partial_{kk}w > 0\) when \(b > 0\) and \(\sigma > 0\), the resulting call surface automatically satisfies the butterfly condition.

  4. Bid-ask band approach: Allow each quote to move within its bid-ask spread, then find the smoothest surface within these bands that satisfies all no-arbitrage constraints.

Exercise 6. Let the total implied variance be defined as \(w(K,T) = \sigma_{\text{impl}}^2(K,T)\cdot T\). Show that Dupire's formula can be rewritten in terms of \(w\) as

\[ \sigma_{\text{loc}}^2 = \frac{\partial_T w}{1 - \frac{y}{w}\partial_y w + \frac{1}{4}\left(-\frac{1}{4} - \frac{1}{w} + \frac{y^2}{w^2}\right)(\partial_y w)^2 + \frac{1}{2}\partial_{yy}w} \]

where \(y = \ln(K/F_T)\) is the log-moneyness. Discuss why working in the \((y, T)\) coordinate system can improve numerical stability compared to working directly with \((K, T)\).

Solution to Exercise 6

We start with Dupire's formula in \((K, T)\) coordinates and perform a change of variables to log-moneyness \(y = \ln(K/F_T)\), where \(F_T = S_0 e^{(r-q)T}\) is the forward price.

Change of variables: Since \(K = F_T e^y\), we have \(\partial_K = (1/K)\partial_y\) and \(\partial_{KK} = (1/K^2)(\partial_{yy} - \partial_y)\).

The total implied variance is \(w(y, T) = \sigma_{\text{impl}}^2(y, T)\cdot T\). The Black--Scholes call price can be written in terms of \(w\) as

\[ C = F_T e^{-rT}\bigl[N(d_1) - e^{-y}N(d_2)\bigr] \]

where \(d_1 = -y/\sqrt{w} + \frac{1}{2}\sqrt{w}\) and \(d_2 = d_1 - \sqrt{w}\).

After careful computation of \(\partial_T C\), \(\partial_y C\), and \(\partial_{yy}C\) in terms of \(w\) and its derivatives, and substituting into Dupire's formula, one obtains (following Gatheral's derivation):

\[ \sigma_{\text{loc}}^2 = \frac{\partial_T w}{1 - \frac{y}{w}\partial_y w + \frac{1}{4}\left(-\frac{1}{4} - \frac{1}{w} + \frac{y^2}{w^2}\right)(\partial_y w)^2 + \frac{1}{2}\partial_{yy}w} \]

Derivation sketch: The key identity is that the call price density (second derivative in strike) can be expressed via the Black--Scholes formula as a function of \(w\) alone. The denominator arises from expressing \(K^2\partial_{KK}C\) in terms of \(w\), \(\partial_y w\), and \(\partial_{yy}w\) through the chain rule and the Black--Scholes vega/volga structure. The numerator \(\partial_T C + (r-q)K\partial_K C - qC\) simplifies to a term proportional to \(\partial_T w\) because the forward-measure discounting absorbs the drift terms.

Why \((y, T)\) coordinates improve numerical stability:

  1. Dimensionless variable: Log-moneyness \(y\) is dimensionless and centered at zero (ATM corresponds to \(y = 0\)), whereas \(K\) has units and varies across a wide range. This improves the conditioning of interpolation and differentiation.

  2. Smoother behavior: Total implied variance \(w(y, T)\) is more nearly linear in both \(y\) and \(T\) than call prices \(C(K, T)\) are. This means that interpolation and differentiation introduce smaller errors.

  3. Calendar arbitrage is transparent: The condition \(\partial_T w \ge 0\) for fixed \(y\) directly ensures no calendar arbitrage, making it easy to check and enforce. In \((K, T)\) coordinates, the analogous condition on \(C\) is less transparent.

  4. Butterfly arbitrage is localized: The denominator is a function of \(w\) and its \(y\)-derivatives only (at fixed \(T\)), so convexity checks reduce to verifying that the denominator is positive, which can be done slice by slice.

  5. Reduced sensitivity near ATM: Near \(y = 0\), the terms \(y/w\) and \(y^2/w^2\) are small, so the denominator is dominated by \(1 + \frac{1}{2}\partial_{yy}w\), which is well-conditioned as long as the smile has positive curvature.