Local Volatility Surface Construction¶

Constructing the local volatility surface from market data is the computational realization of Dupire's formula. While the formula itself is elegant — a ratio of partial derivatives of the call price surface — its numerical implementation faces substantial challenges. The call surface must be differentiated twice in strike and once in time, operations that amplify noise and can produce negative or unbounded values if the input data is not carefully prepared. This section develops the complete pipeline from smoothed market data (the output of interpolation and smoothing) to a validated local volatility surface $\sigma_{\text{loc}}(K, T)$, covering numerical differentiation, regularization, and diagnostic checks.

Learning Objectives

After completing this section, you should be able to:

Implement Dupire's formula numerically using finite differences on a smoothed call price surface
Apply Dupire's formula in implied volatility space using the Gatheral-Jacquier formulation
Identify and mitigate sources of numerical instability (small denominator, wing extrapolation, sparse maturity grid)
Apply Tikhonov regularization and positivity constraints to produce a well-behaved surface
Validate the constructed surface by repricing vanilla options

Numerical Dupire Formula¶

The Formula in Price Space¶

Dupire's formula expresses local volatility in terms of call price derivatives:

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

Given a smooth call price surface $\hat{C}(K, T)$ (from the interpolation step), the derivatives are approximated by finite differences.

Finite Difference Approximations¶

Time derivative (centered difference):

\[ \frac{\partial C}{\partial T}\bigg|_{K_i, T_j} \approx \frac{\hat{C}(K_i, T_{j+1}) - \hat{C}(K_i, T_{j-1})}{T_{j+1} - T_{j-1}} \]

At the boundary maturities, use one-sided differences:

\[ \frac{\partial C}{\partial T}\bigg|_{K_i, T_1} \approx \frac{\hat{C}(K_i, T_2) - \hat{C}(K_i, T_1)}{T_2 - T_1} \]

Strike derivatives (centered differences on a non-uniform grid):

\[ \frac{\partial C}{\partial K}\bigg|_{K_i, T_j} \approx \frac{\hat{C}(K_{i+1}, T_j) - \hat{C}(K_{i-1}, T_j)}{K_{i+1} - K_{i-1}} \]

\[ \frac{\partial^2 C}{\partial K^2}\bigg|_{K_i, T_j} \approx \frac{2}{K_{i+1} - K_{i-1}}\left(\frac{\hat{C}(K_{i+1}, T_j) - \hat{C}(K_i, T_j)}{K_{i+1} - K_i} - \frac{\hat{C}(K_i, T_j) - \hat{C}(K_{i-1}, T_j)}{K_i - K_{i-1}}\right) \]

For a uniform grid with spacing $\Delta K$, this simplifies to the standard centered second difference:

\[ \frac{\partial^2 C}{\partial K^2}\bigg|_{K_i, T_j} \approx \frac{\hat{C}(K_{i+1}, T_j) - 2\hat{C}(K_i, T_j) + \hat{C}(K_{i-1}, T_j)}{(\Delta K)^2} \]

Assembly¶

Combining these approximations yields $\sigma_{\text{loc}}^2(K_i, T_j)$ at each grid point. Points where the denominator is too small (below a threshold $\epsilon_{\min}$) or the numerator is negative are flagged and handled by the regularization step described below.

Dupire Formula in Implied Volatility Space¶

Motivation¶

An alternative to differentiating call prices is to express Dupire's formula directly in terms of implied volatility derivatives. This is often more stable because the implied volatility surface is smoother than the price surface.

The Gatheral-Jacquier Formula¶

Given the implied volatility surface $\sigma_{\text{IV}}(K, T)$, the local volatility is:

\[ \sigma_{\text{loc}}^2(K, T) = \frac{\sigma_{\text{IV}}^2 + 2\sigma_{\text{IV}}T\frac{\partial \sigma_{\text{IV}}}{\partial T} + 2(r - q)K\sigma_{\text{IV}}T\frac{\partial \sigma_{\text{IV}}}{\partial K}}{\left(1 + Kd_1\sqrt{T}\frac{\partial \sigma_{\text{IV}}}{\partial K}\right)^2 + \sigma_{\text{IV}}K^2 T\left(\frac{\partial^2 \sigma_{\text{IV}}}{\partial K^2} - d_1\sqrt{T}\left(\frac{\partial \sigma_{\text{IV}}}{\partial K}\right)^2\right)} \]

where $d_1 = \frac{\ln(S_0/K) + (r - q + \sigma_{\text{IV}}^2/2)T}{\sigma_{\text{IV}}\sqrt{T}}$.

This formula requires only the first and second derivatives of $\sigma_{\text{IV}}$ with respect to strike, and the first derivative with respect to maturity. When the IV surface is represented by an analytic parametrization (SVI, SSVI), these derivatives can be computed analytically, avoiding finite differences entirely.

Total Variance Formulation¶

In terms of total variance $w(y, T) = \sigma_{\text{IV}}^2(y, T) \cdot T$ and log-moneyness $y = \ln(K/F)$, the formula simplifies to:

\[ \sigma_{\text{loc}}^2 = \frac{\frac{\partial w}{\partial T}}{\left(1 - \frac{y}{w}\frac{\partial w}{\partial y}\right)^2 - \frac{1}{4}\left(\frac{\partial w}{\partial y}\right)^2\left(\frac{1}{w} + \frac{1}{4}\right) + \frac{1}{2}\frac{\partial^2 w}{\partial y^2}} \]

The denominator is precisely Durrleman's condition $g(y, T) \geq 0$, so positivity of the denominator is equivalent to absence of butterfly arbitrage.

Analytic Derivatives from SVI

For the SVI parametrization $w(y) = a + b(\rho(y-m) + \sqrt{(y-m)^2 + \sigma^2})$, the derivatives are:

\[ \frac{\partial w}{\partial y} = b\left(\rho + \frac{y - m}{\sqrt{(y-m)^2 + \sigma^2}}\right) \]

\[ \frac{\partial^2 w}{\partial y^2} = \frac{b\sigma^2}{((y-m)^2 + \sigma^2)^{3/2}} \]

These are smooth, bounded, and free of numerical differentiation noise, making SVI-based construction particularly robust.

Regularization¶

The Ill-Posedness Problem¶

Even after smoothing, the local volatility surface can exhibit problematic behavior:

Spikes: Where $C_{KK}$ is close to zero (deep OTM/ITM), $\sigma_{\text{loc}}$ diverges
Oscillations: Residual noise in $C$ produces high-frequency oscillations in $\sigma_{\text{loc}}$
Negative values: Numerical errors can produce $\sigma_{\text{loc}}^2 < 0$ at isolated points

These issues stem from the ill-posedness of the inverse problem: small perturbations in $C$ produce large changes in $\sigma_{\text{loc}}$.

Tikhonov Regularization¶

Instead of applying Dupire's formula pointwise, solve the inverse problem with a regularization penalty:

\[ \min_{\sigma_{\text{loc}}} \left\{\sum_{i,j} w_{ij}\left(C_{\text{model}}(K_i, T_j; \sigma_{\text{loc}}) - \hat{C}(K_i, T_j)\right)^2 + \lambda_1 \int\!\!\int \left(\frac{\partial \sigma_{\text{loc}}}{\partial K}\right)^2 dK \, dT + \lambda_2 \int\!\!\int \left(\frac{\partial \sigma_{\text{loc}}}{\partial T}\right)^2 dK \, dT\right\} \]

where $C_{\text{model}}(K_i, T_j; \sigma_{\text{loc}})$ is the price computed by solving the forward PDE with the candidate $\sigma_{\text{loc}}$, and $\lambda_1, \lambda_2 > 0$ are regularization parameters penalizing roughness.

This formulation:

Ensures $\sigma_{\text{loc}}$ is smooth (via the penalty terms)
Matches market prices approximately (via the data-fidelity term)
Requires solving a PDE at each optimization step (computationally expensive)

Positivity Constraints¶

Enforce $\sigma_{\text{loc}}^2(K, T) > 0$ by either:

Parametrization: Write $\sigma_{\text{loc}} = e^{f(K, T)}$ where $f$ is unconstrained
Projection: After each Dupire computation, set $\sigma_{\text{loc}}^2 = \max(\sigma_{\text{loc}}^2, \sigma_{\min}^2)$ with $\sigma_{\min} = 1\%$
Constrained optimization: Include $\sigma_{\text{loc}}^2 \geq \sigma_{\min}^2$ as an explicit constraint

Bounding the Surface¶

In addition to positivity, practitioners typically enforce upper bounds:

\[ \sigma_{\min}^2 \leq \sigma_{\text{loc}}^2(K, T) \leq \sigma_{\max}^2 \]

with $\sigma_{\min} \in [0.5\%, 2\%]$ and $\sigma_{\max} \in [100\%, 300\%]$. Values outside this range almost always indicate data or numerical problems rather than genuine market dynamics.

Handling Sparse Data¶

Maturity Gaps¶

Market data often has large gaps between available maturities (e.g., 1M, 2M, 3M, 6M, 1Y, 2Y). Between maturities, the local volatility must be interpolated from the total variance interpolation performed in the smoothing step. The time derivative $\partial C / \partial T$ is particularly sensitive to the maturity gap:

\[ \text{error in } \partial_T C \sim O\left(\frac{(\Delta T)^2}{12}\frac{\partial^3 C}{\partial T^3}\right) \]

Larger $\Delta T$ increases the error, leading to inaccurate local volatility between maturities.

Mitigation: Use the total variance formulation $w_T = \partial w / \partial T$, which is directly available from the interpolation step without additional finite differencing.

Strike Gaps¶

In the wings, strikes become sparse and option prices are small, making $C_{KK}$ both noisy and near zero. Strategies include:

SVI extrapolation: Use the SVI parametrization to extend the smile to extreme strikes with controlled wing behavior
Flat local vol extrapolation: Beyond the last reliable data point, set $\sigma_{\text{loc}}(K, T) = \sigma_{\text{loc}}(K_{\text{last}}, T)$
Density tail modeling: Fit parametric tails (e.g., generalized Pareto) to the risk-neutral density and back out the corresponding local volatility

Short-Maturity Challenges¶

Near expiry ($T < 1$ month), the option surface is steep in time and the forward variance can change rapidly. The local volatility surface near short maturities requires:

Dense maturity interpolation points
Careful handling of the $T \to 0$ limit, where $\sigma_{\text{loc}}(K, 0) = \sigma_{\text{IV}}(K, 0)$ (Berestycki-Busca-Florent)
Explicit incorporation of known events (earnings, dividends) that create term structure jumps

Validation¶

Repricing Test¶

The most important diagnostic: if the constructed $\sigma_{\text{loc}}(K, T)$ is correct, solving the local volatility PDE (or running Monte Carlo) should reproduce the market call prices used as input.

Procedure:

Solve the forward PDE $\partial_T p = \frac{1}{2}\partial_{KK}[\sigma_{\text{loc}}^2 K^2 p]$ with the constructed surface
Compute call prices: $C_{\text{reprice}}(K, T) = e^{-rT}\int_K^\infty (S - K) p(S, T) \, dS$
Compare: $|C_{\text{reprice}} - C_{\text{market}}| < \epsilon_{\text{tol}}$

Typical tolerance: within 0.5 vol points in implied volatility terms.

Visual Diagnostics¶

Useful plots for quality assessment:

Local vol surface: 3D plot of $\sigma_{\text{loc}}(K, T)$ — look for spikes, negative regions, and unrealistic values
Term structure slices: $\sigma_{\text{loc}}(K, T)$ at fixed $K$ as a function of $T$ — should be smooth and bounded
Smile slices: $\sigma_{\text{loc}}(K, T)$ at fixed $T$ as a function of $K$ — typically decreasing for equity indices
Implied vol comparison: Overlay the original market IV and the IV implied by the repriced surface
Risk-neutral density: $q(K) = e^{rT} C_{KK}$ — should be positive and integrate to 1

Stability Test¶

Perturb the input data by adding noise within bid-ask spreads and reconstruct the surface. A stable construction should produce local volatility surfaces that differ by at most a few percentage points. Large sensitivity indicates insufficient regularization.

Construction Diagnostic

After constructing the local volatility surface for SPX options, a typical diagnostic summary might show:

Metric	Value	Acceptable Range
$\sigma_{\text{loc}}$ range	8%--65%	1%--200%
Repricing RMSE (IV)	0.12 vol pts	< 0.5 vol pts
Negative $\sigma_{\text{loc}}^2$ points	0	0
Max $	\Delta\sigma_{\text{loc}}	$ under noise

All metrics are within acceptable bounds, indicating a reliable construction.

From $(K, T)$ to $(S, t)$¶

Coordinate Transformation¶

Dupire's formula produces $\sigma_{\text{loc}}(K, T)$ — the local volatility as a function of strike and maturity. For PDE pricing and Monte Carlo simulation, we need $\sigma_{\text{loc}}(S, t)$ — the local volatility as a function of spot and time.

For a European option evaluated at $t = 0$ with current spot $S_0$, the two are related by a simple relabeling: the strike $K$ in Dupire's formula plays the role of the future spot $S$, and the maturity $T$ plays the role of the future time $t$:

\[ \sigma_{\text{loc}}(S, t) = \sigma_{\text{loc}}(K = S, T = t) \]

This identification holds because Dupire's formula gives the instantaneous volatility the diffusion has when it reaches level $K$ at time $T$.

Grid Mapping¶

When the Dupire surface is computed on a $(K, T)$ grid and the FDM or Monte Carlo scheme uses an $(S, t)$ grid, interpolation is required. The same bilinear or bicubic interpolation used within each framework applies.

Summary¶

Local volatility surface construction transforms market data into a usable function $\sigma_{\text{loc}}(S, t)$ through a multi-stage pipeline:

Input: Smoothed, arbitrage-free call price or implied volatility surface (from interpolation and smoothing)
Numerical differentiation: Compute $\partial_T C$, $\partial_K C$, $\partial_{KK} C$ via finite differences or analytic derivatives of parametric models
Dupire's formula: Apply in price space or implied volatility space to obtain $\sigma_{\text{loc}}^2(K, T)$
Regularization: Enforce smoothness, positivity, and boundedness to handle ill-conditioning
Validation: Reprice vanilla options to verify consistency, check for negative values and spikes
Coordinate mapping: Convert from $(K, T)$ to $(S, t)$ for use in pricing engines

The quality of the local volatility surface depends critically on the quality of the input data and the smoothing step. Garbage in, garbage out -- no amount of regularization can compensate for fundamentally flawed input data.

Exercises¶

Exercise 1. Given a smooth call price surface with $C(100, 1) = 10.45$, $C(105, 1) = 7.80$, $C(95, 1) = 13.60$, $C(100, 0.9) = 9.90$, $C(100, 1.1) = 10.95$, and parameters $r = 3\%$, $q = 1\%$, compute the local volatility $\sigma_{\text{loc}}(100, 1)$ using Dupire's formula with centered finite differences. Use $\Delta K = 5$ and $\Delta T = 0.1$.

Solution to Exercise 1

Using Dupire's formula with centered finite differences:

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

Time derivative (centered, $\Delta T = 0.1$):

\[ \frac{\partial C}{\partial T}\bigg|_{100,1} \approx \frac{C(100, 1.1) - C(100, 0.9)}{1.1 - 0.9} = \frac{10.95 - 9.90}{0.2} = 5.25 \]

First strike derivative (centered, $\Delta K = 5$):

\[ \frac{\partial C}{\partial K}\bigg|_{100,1} \approx \frac{C(105, 1) - C(95, 1)}{105 - 95} = \frac{7.80 - 13.60}{10} = -0.58 \]

Second strike derivative (centered, $\Delta K = 5$):

\[ \frac{\partial^2 C}{\partial K^2}\bigg|_{100,1} \approx \frac{C(105, 1) - 2C(100, 1) + C(95, 1)}{(\Delta K)^2} = \frac{7.80 - 2(10.45) + 13.60}{25} = \frac{0.50}{25} = 0.02 \]

Numerator:

\[ \frac{\partial C}{\partial T} + qC + (r-q)K\frac{\partial C}{\partial K} = 5.25 + 0.01(10.45) + (0.03 - 0.01)(100)(-0.58) \]

\[ = 5.25 + 0.1045 - 1.16 = 4.1945 \]

Denominator:

\[ \frac{1}{2}K^2 \frac{\partial^2 C}{\partial K^2} = \frac{1}{2}(100)^2(0.02) = 100 \]

Local variance:

\[ \sigma_{\text{loc}}^2(100, 1) = \frac{4.1945}{100} = 0.041945 \]

Local volatility:

\[ \sigma_{\text{loc}}(100, 1) = \sqrt{0.041945} \approx 0.2048 = 20.48\% \]

Exercise 2. The Gatheral-Jacquier formula expresses local volatility in terms of implied volatility derivatives. Explain why this formulation is often more numerically stable than differentiating call prices directly. Under what conditions does it avoid finite differences entirely?

Solution to Exercise 2

The Gatheral-Jacquier formula expresses $\sigma_{\text{loc}}^2$ directly in terms of implied volatility and its derivatives, rather than call price derivatives. This is more numerically stable for several reasons:

Smoother input surface. The implied volatility surface is much smoother than the call price surface. Call prices range from zero (deep OTM) to $S - Ke^{-rT}$ (deep ITM) with a steep gradient near ATM, whereas implied volatility typically varies by only a few percentage points across strikes. Differentiating a smoother function produces more stable derivatives.

Better-conditioned denominator. In the price-space Dupire formula, the denominator $\frac{1}{2}K^2 C_{KK}$ involves the second derivative of call prices, which can be extremely small deep OTM or ITM (where the risk-neutral density is near zero). In IV space, the denominator involves quantities related to the curvature of the smile, which are typically better behaved.

Avoids differentiating near-zero quantities. Deep OTM call prices are very small, making finite difference ratios of prices highly sensitive to noise. Implied volatilities remain well-defined and of moderate magnitude even for deep OTM options.

The formulation avoids finite differences entirely when the implied volatility surface is represented by an analytic parametrization such as SVI or SSVI. In that case, the derivatives $\partial \sigma_{\text{IV}}/\partial K$, $\partial^2 \sigma_{\text{IV}}/\partial K^2$, and $\partial \sigma_{\text{IV}}/\partial T$ can all be computed analytically from the parametric formulas, eliminating discretization error in the differentiation step completely.

Exercise 3. Consider the total variance formulation of Dupire's formula:

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

Verify that when the total variance surface is flat ($w(y, T) = \sigma_0^2 T$, independent of $y$), the formula correctly gives $\sigma_{\text{loc}} = \sigma_0$.

Solution to Exercise 3

When $w(y, T) = \sigma_0^2 T$ (flat total variance, independent of $y$), all $y$-derivatives vanish:

\[ \frac{\partial w}{\partial y} = 0, \quad \frac{\partial^2 w}{\partial y^2} = 0 \]

and the time derivative is:

\[ \frac{\partial w}{\partial T} = \sigma_0^2 \]

Substituting into the total variance formulation of Dupire's formula:

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

The numerator becomes $\sigma_0^2$.

The denominator simplifies term by term:

First term: $\left(1 - \frac{y}{w} \cdot 0\right)^2 = 1$
Second term: $-\frac{1}{4}(0)^2(\cdots) = 0$
Third term: $\frac{1}{2}(0) = 0$

So the denominator equals 1, and:

\[ \sigma_{\text{loc}}^2 = \frac{\sigma_0^2}{1} = \sigma_0^2 \]

Therefore $\sigma_{\text{loc}} = \sigma_0$, confirming that a flat implied volatility surface correctly yields a constant local volatility equal to the implied volatility. This is the expected result: when the market is consistent with Black-Scholes (flat smile, constant implied vol), the local volatility model recovers constant volatility.

Exercise 4. Tikhonov regularization adds penalty terms $\lambda_1 \int\!\!\int (\partial_K \sigma_{\text{loc}})^2 \, dK\,dT + \lambda_2 \int\!\!\int (\partial_T \sigma_{\text{loc}})^2 \, dK\,dT$ to the objective function. Explain the effect of increasing $\lambda_1$ relative to $\lambda_2$. In what market scenario (e.g., steep skew vs flat skew, sparse vs dense maturities) would you choose a larger $\lambda_2$?

Solution to Exercise 4

The regularization penalty consists of two terms:

\[ \lambda_1 \int\!\!\int \left(\frac{\partial \sigma_{\text{loc}}}{\partial K}\right)^2 dK\,dT + \lambda_2 \int\!\!\int \left(\frac{\partial \sigma_{\text{loc}}}{\partial T}\right)^2 dK\,dT \]

Effect of $\lambda_1$: Penalizes variation of $\sigma_{\text{loc}}$ across strikes. Increasing $\lambda_1$ forces the local volatility surface to be flatter in the strike direction, smoothing out the skew structure. A large $\lambda_1$ produces a surface that is nearly constant in $K$ at each time — essentially removing the smile from the local volatility.

Effect of $\lambda_2$: Penalizes variation of $\sigma_{\text{loc}}$ across maturities. Increasing $\lambda_2$ forces the local volatility to be flatter in the time direction, smoothing out the term structure. A large $\lambda_2$ produces a surface that changes slowly over time.

Increasing $\lambda_1$ relative to $\lambda_2$ means penalizing strike variation more than time variation. This flattens the smile more aggressively while still allowing the term structure to vary. This is appropriate when the smile structure in the data is noisy but the term structure is reliable.

When to choose a larger $\lambda_2$: A larger $\lambda_2$ is appropriate when maturities are sparse (e.g., large gaps between available expiries such as 3M, 6M, 1Y). With sparse maturities, the time derivative $\partial C / \partial T$ is poorly estimated, causing oscillations in $\sigma_{\text{loc}}$ along the time axis. A larger $\lambda_2$ smooths these oscillations. In contrast, if the skew is steep (e.g., equity index options with pronounced put skew) but well-observed across many strikes, one would keep $\lambda_1$ small to preserve the genuine smile structure and increase $\lambda_2$ to stabilize the less reliable time direction.

Exercise 5. After constructing a local volatility surface, the repricing test yields an RMSE of 2.5 implied volatility points. List three possible causes of this large repricing error and for each cause describe a corrective action.

Solution to Exercise 5

An RMSE of 2.5 implied volatility points is large (typical acceptable tolerance is below 0.5 vol points). Three possible causes and corrective actions:

Cause 1: Insufficient smoothing of the input call price surface. If the smoothing step left too much noise in the call prices, the differentiation step produces a noisy local volatility surface. When this surface is used to reprice, the noise accumulates and the repriced prices deviate from market prices. Corrective action: Increase the smoothing parameter $\lambda$ in the smoothing spline or tighten the SVI/SSVI fit tolerances. Use cross-validation to find a better balance between fit and smoothness.

Cause 2: Arbitrage violations in the smoothed surface. If the interpolated call surface violates no-arbitrage conditions (negative $C_{KK}$, negative $C_T$, or monotonicity failure), Dupire's formula produces negative or infinite local volatility values at those points. Clamping or projecting these values introduces systematic error. Corrective action: Enforce all three no-arbitrage constraints (strike monotonicity, strike convexity, calendar monotonicity) explicitly during the interpolation step. Use constrained optimization or arbitrage-free parametrizations (SSVI).

Cause 3: Poor wing extrapolation. If the implied volatility or call price surface is extrapolated poorly beyond the range of observed strikes, the local volatility in the wings can be wildly inaccurate. Since option prices (especially for OTM puts on equity indices) have non-trivial wing contributions, errors in the wing local volatility affect repricing across a range of strikes. Corrective action: Use a principled wing extrapolation method (e.g., SVI linear wings consistent with Lee's moment formula). Verify that the extrapolated local volatility is bounded and smooth, and test sensitivity of repriced prices to extrapolation assumptions.

Exercise 6. Explain the coordinate transformation from $(K, T)$ to $(S, t)$ in the context of Dupire's formula. Why is it valid to set $\sigma_{\text{loc}}(S, t) = \sigma_{\text{loc}}(K = S, T = t)$? Under what circumstances does this simple relabeling fail, and what additional steps are needed?

Solution to Exercise 6

Dupire's formula produces $\sigma_{\text{loc}}(K, T)$, where $K$ is the strike and $T$ is the maturity of the European option used in the calibration. The PDE or Monte Carlo simulation requires $\sigma_{\text{loc}}(S, t)$, where $S$ is the spot price and $t$ is the calendar time.

Why the relabeling $\sigma_{\text{loc}}(S, t) = \sigma_{\text{loc}}(K = S, T = t)$ is valid: Dupire's formula gives the local volatility at the point where the risk-neutral diffusion process reaches level $K$ at time $T$. By construction, $\sigma_{\text{loc}}(K, T)$ is the instantaneous volatility that the spot process $S_t$ has when $S_T = K$ at time $T$. Therefore, when we simulate the SDE forward and the process reaches level $S$ at time $t$, the correct volatility to use is $\sigma_{\text{loc}}(K = S, T = t)$. The strike variable in Dupire's output plays exactly the role of the future spot level.

When this simple relabeling fails:

When the risk-free rate $r$ or dividend yield $q$ is not constant, the forward price $F(t, T)$ introduces a time-dependent drift that shifts the relationship between strike and spot. One must account for the forward mapping: $K$ in Dupire's formula corresponds to a forward-adjusted spot, not the raw spot.
When discrete dividends are present, the spot process has jumps at dividend dates, and the local volatility surface in $(K, T)$ space must be adjusted to account for the known drop in spot at each ex-dividend date.
For stochastic interest rate models, the discounting and drift depend on the rate process, complicating the correspondence.

In these cases, the additional step is to map through the forward price: first express Dupire's output in terms of the forward moneyness $K/F(0,T)$ and then invert using the time-dependent forward curve to recover $\sigma_{\text{loc}}(S, t)$ accounting for the deterministic drift components.

Exercise 7. Using the SVI parametrization $w(y) = a + b(\rho(y-m) + \sqrt{(y-m)^2 + \sigma^2})$, compute the analytic first and second derivatives $\partial_y w$ and $\partial_{yy} w$. Verify that $\partial_{yy} w > 0$ for all $y$, which guarantees the butterfly no-arbitrage condition is not violated by the convexity term alone.

Solution to Exercise 7

Starting from the SVI parametrization:

\[ w(y) = a + b\left(\rho(y - m) + \sqrt{(y-m)^2 + \sigma^2}\right) \]

First derivative. Let $u = y - m$. Then:

\[ \frac{\partial w}{\partial y} = b\left(\rho + \frac{u}{\sqrt{u^2 + \sigma^2}}\right) = b\left(\rho + \frac{y - m}{\sqrt{(y-m)^2 + \sigma^2}}\right) \]

Second derivative. Differentiating the second term using the quotient/chain rule:

\[ \frac{\partial}{\partial y}\left(\frac{u}{\sqrt{u^2 + \sigma^2}}\right) = \frac{\sqrt{u^2 + \sigma^2} - u \cdot \frac{u}{\sqrt{u^2 + \sigma^2}}}{u^2 + \sigma^2} = \frac{u^2 + \sigma^2 - u^2}{(u^2 + \sigma^2)^{3/2}} = \frac{\sigma^2}{(u^2 + \sigma^2)^{3/2}} \]

Therefore:

\[ \frac{\partial^2 w}{\partial y^2} = \frac{b\sigma^2}{\left((y-m)^2 + \sigma^2\right)^{3/2}} \]

Verification that $\partial_{yy} w > 0$ for all $y$:

$b > 0$ by assumption (SVI parameter constraint)
$\sigma^2 > 0$ since $\sigma > 0$
$((y-m)^2 + \sigma^2)^{3/2} > 0$ for all $y$, since the base is strictly positive

Therefore $\partial_{yy} w > 0$ for all $y$, unconditionally. This means the SVI parametrization always produces a strictly convex total variance slice in log-moneyness, which ensures that the contribution of the $\frac{1}{2}\partial_{yy}w$ term in the Durrleman condition is always positive. While this alone does not guarantee full absence of butterfly arbitrage (the other terms in Durrleman's condition must also be checked), it ensures the convexity term never causes a violation.

Metric	Value	Acceptable Range
\(\sigma_{\text{loc}}\) range	8%--65%	1%--200%
Repricing RMSE (IV)	0.12 vol pts	< 0.5 vol pts
Negative \(\sigma_{\text{loc}}^2\) points	0	0
Max $	\Delta\sigma_{\text{loc}}	$ under noise