Dupire Local Volatility Model & Implied Surface Lab

Extract Bruno Dupire's (1994) instantaneous continuous diffusion surface $\sigma_L(K, T)$ from Black-Scholes implied variance smiles. Verify the 2x slope rule and calendar/butterfly no-arbitrage bounds.

Market Surface Presets:
Market Surface Controls
Dupire Local Vol ($\sigma_L$)

23.84%

Instantaneous Diffusion
Implied Vol ($\sigma_{\text{BS}}$)

20.48%

Black-Scholes Quote
ATM Slope Ratio

1.98x

Theoretical Benchmark: 2.0x
Forward Price $F(T)$

$101.51

Cost of Carry Horizon
Calendar Slope ($\frac{\partial w}{\partial T}$)

4.182%

Total Variance Rate
No-Arbitrage Status

Arbitrage-Free

Calendar & Butterfly
Strike Spectrum: Implied vs. Local Volatility Comparison
Horizon: T = 0.50Y
Strike ($K$) Moneyness Implied $\sigma_{\text{BS}}$ Local $\sigma_L$ Ratio (Local/Impl) $\frac{\partial w}{\partial T}$ Density Denom Status

Compares Dupire's instantaneous local volatility curve $\sigma_L(K)$ against the Black-Scholes implied smile $\sigma_{\text{BS}}(K)$. Notice that the local volatility smile is notably steeper and exhibits wider wings.

Displays local volatility smile slices across multiple maturities ($T = 0.15Y, 0.35Y, 0.75Y, 1.50Y$), demonstrating how term mean-reversion flattens the smile across longer horizons.

Validates the classical institutional rule of thumb: at at-the-money ($K = F$), the slope of local volatility is almost exactly twice the slope of implied volatility ($2.0\times$).

5x5 Sensitivity: Strike ($K$) vs. Implied Skew on Local Volatility (%)
5x5 Sensitivity: Tenor ($T$) vs. ATM Vol on ATM Local Volatility (%)

Quantitative Foundations of Dupire Local Volatility

1. Continuous Diffusion & Fokker-Planck Forward PDE

In 1994, Bruno Dupire published "Pricing with a Smile", answering a foundational question in quantitative derivatives: does there exist a continuous, risk-neutral diffusion process that exactly replicates all vanilla European option prices across all strikes and maturities?

$dS_t = (r - q) S_t dt + \sigma_L(S_t, t) S_t dW_t$

By applying the Fokker-Planck (forward Kolmogorov) equation to the risk-neutral transition probability density $p(S_0, 0; S_T, T)$ and differentiating the European call integral $C(K, T) = e^{-rT} \int_K^\infty (S_T - K) p(S_T, T) dS_T$, Dupire derived the celebrated formula:

$\sigma_L^2(K, T) = \frac{\frac{\partial C}{\partial T} + (r - q) K \frac{\partial C}{\partial K} + q C}{\frac{1}{2} K^2 \frac{\partial^2 C}{\partial K^2}}$

2. The Total Variance Formulation $w(k, T)$

Direct differentiation of option prices $C(K, T)$ in market practice is highly susceptible to numerical instability. In modern quant libraries (Gatheral 2006), Dupire's formula is expressed in terms of dimensionless log-moneyness $k = \ln(K/F_T)$ and total implied variance $w(k, T) = \sigma_{\text{BS}}^2(k, T) \cdot T$:

$\sigma_L^2(k, T) = \frac{\frac{\partial w}{\partial T}}{1 - \frac{k}{w} \frac{\partial w}{\partial k} + \frac{1}{4}\left(-\frac{1}{4} - \frac{1}{w} + \frac{k^2}{w^2}\right)\left(\frac{\partial w}{\partial k}\right)^2 + \frac{1}{2}\frac{\partial^2 w}{\partial k^2}}$

This formulation directly connects Dupire local variance to the shape of the implied volatility surface. The numerator represents calendar time decay rate, while the denominator contains the strike skew and curvature terms.

3. The Institutional "Factor of 2" Rule

A celebrated stylized fact on derivatives trading desks is the factor-of-two rule connecting the slope of the local volatility smile to the implied volatility smile at the money ($k = 0$):

$\left.\frac{\partial \sigma_L}{\partial K}\right|_{K=F} \approx 2 \times \left.\frac{\partial \sigma_{\text{BS}}}{\partial K}\right|_{K=F}$

Why does this occur? Implied volatility $\sigma_{\text{BS}}(K, T)$ acts as a harmonic/path average of local volatilities from the initial spot price $S_0$ to the terminal strike $K$. To change the cumulative average by a slope of $m$, the marginal instantaneous rate of change at the destination must be approximately $2m$.

4. No-Arbitrage Constraints & Exotic Forward Smile Bias

Dupire's formula guarantees a well-defined positive local variance $\sigma_L^2 > 0$ if and only if two static no-arbitrage conditions hold:

  • Calendar Monotonicity: $\frac{\partial w}{\partial T} \ge 0$. If total variance decreases with maturity, calendar spread arbitrage exists.
  • Butterfly Convexity: The denominator must be strictly positive, which is equivalent to Breeden-Litzenberger density non-negativity ($\frac{\partial^2 C}{\partial K^2} \ge 0$).

While Dupire's model prices vanilla European options perfectly by construction, it assumes zero volatility-of-volatility. As a consequence, it predicts that future implied smiles will flatten, creating systemic underpricing of forward smile risk in barrier options and cliquets compared to stochastic volatility models like Heston or SABR.

Dupire Local Volatility Mastery Quiz

1. How does Black-Scholes implied volatility relate to Dupire local volatility?

2. What does the "Factor of 2" rule of thumb state regarding at-the-money volatility slopes?

3. What static market arbitrage condition is indicated if $\frac{\partial w}{\partial T} < 0$ in Dupire's formula?

4. Why do pure local volatility models struggle to price cliquets and barrier options reliably?

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Frequently Asked Questions

Black-Scholes implied volatility is a single constant parameter quoted in the market that equates the theoretical model price to the observed option price. In contrast, Bruno Dupire's (1994) local volatility $\sigma_L(S, t)$ is an instantaneous, deterministic function of asset price and time in the continuous diffusion process $dS_t = (r-q)S_t dt + \sigma_L(S_t, t)S_t dW_t$. Implied volatility represents a non-linear path-integral average of local volatilities across the entire lifetime of the option.

Near the money ($K \approx F$), the slope of the local volatility smile with respect to strike is approximately twice the slope of the implied volatility smile: $\frac{\partial \sigma_L}{\partial K} \approx 2 \times \frac{\partial \sigma_{\text{BS}}}{\partial K}$. Intuitively, because implied volatility is an average of local volatility from spot to strike, the local volatility at the margin must change twice as aggressively to move the average by that amount.

Dupire's formula can produce negative or undefined local variance when market prices contain static arbitrage. In the numerator, $\frac{\partial w}{\partial T} < 0$ violates calendar spread monotonicity (longer-dated options having lower total variance than shorter-dated options). In the denominator, a non-positive value corresponds to negative second derivative of call price with respect to strike, violating Breeden-Litzenberger risk-neutral density non-negativity (butterfly arbitrage).

Although Dupire's model calibrates perfectly to all vanilla option prices across strikes and maturities today, it assumes volatility is purely deterministic in price and time. As time progresses forward, the model predicts that future implied volatility smiles will flatten out toward a flat line. This forward smile flattening leads to severe mispricing of path-dependent exotic options like cliquets, reverse convertibles, and Bermudans, requiring stochastic volatility models (Heston or SABR) or local-stochastic volatility (LSV) hybrids.

Instead of taking noisy derivatives of option prices with respect to strike and maturity, institutional quants express Dupire's equation in terms of total implied variance $w(k, T) = \sigma_{\text{BS}}^2 \cdot T$ across log-moneyness $k = \ln(K/F_T)$. This algebraic formulation avoids numerical instability and yields the local variance directly from partial derivatives $\frac{\partial w}{\partial T}$, $\frac{\partial w}{\partial k}$, and $\frac{\partial^2 w}{\partial k^2}$.