Local Volatility Dupire: Calibrate Exotics to the Smile

Key Takeaways

• Local volatility dupire recovers a unique σ(S,t) function from vanilla prices using Dupire's formula, then runs Monte Carlo under that surface for exotic pricing.
• For a downward-sloping SPX skew, local vol at the 90-strike is roughly twice the implied vol there, the rule-of-thumb factor-of-two relationship.
• A common mistake: feeding a non-arbitrage-free vanilla surface into Dupire, calendar or butterfly violations produce negative local variances and unusable functions.
• Local volatility reproduces today's vanilla surface perfectly but implies forward smiles that flatten and shift rigidly with spot, mispricing cliquets and forward-start trades.

What It Is

Bruno Dupire published "Pricing with a Smile" in Risk magazine in 1994. Independently, Emanuel Derman and Iraj Kani published "Riding on a Smile" in the same magazine that year, reaching the same result via implied trees. The insight: if you know vanilla call prices for every strike K and every maturity T, there is a unique deterministic volatility function sigma(S, t) such that the underlying's diffusion

dS_t = r * S_t * dt + sigma(S_t, t) * S_t * dW_t

reproduces every observed vanilla price. Dupire's formula inverts the vanilla surface to recover that function.

The Intuition

Black-Scholes uses one constant volatility. Heston adds a second stochastic process for variance. Local volatility takes a middle path: volatility still depends on one source of randomness (the spot itself), but the function sigma(S, t) varies by level and time. Because the function has as many degrees of freedom as the vanilla surface, it can always be calibrated to fit the smile exactly.

The practical value is consistency. Once you have sigma(S, t) calibrated to the vanilla surface, every exotic option priced by Monte Carlo under that surface is automatically consistent with the market prices of vanilla hedges. You can price a barrier, cliquet, or lookback and know the price will be arbitrage-free relative to vanilla options at every maturity and strike.

How It Works

Dupire's formula expresses the local variance in terms of observable option prices:

sigma_L(K, T)^2 = 2 * (dC/dT + (r - d) * K * dC/dK + d * C) / (K^2 * d^2C/dK^2)

Where C = C(K, T) is the call price as a function of strike and maturity, r is the risk-free rate, and d is the dividend yield. The partial derivatives are taken across the vanilla price surface.

Equivalently, one can rewrite Dupire in terms of implied volatility sigma_IV(K, T), which is often more numerically stable because implied vol surfaces are smoother than price surfaces.

Implementation steps:

1. Observe call prices C(K_i, T_j) across a grid of strikes and maturities
2. Interpolate/extrapolate to a smooth, arbitrage-free surface
3. Compute partial derivatives dC/dT, dC/dK, d^2C/dK^2
4. Apply Dupire's formula to get sigma_L(K, T) at each grid point
5. Use sigma_L(S, t) in Monte Carlo or PDE pricing for exotics

The tricky step is (2). Raw market quotes are noisy and sparse. Badly interpolated surfaces produce negative local variances and unusable local vols. Production systems use SVI, SSVI, or arbitrage-free splines.

Worked Example

Take an S&P 500 vanilla surface on a given day. Observed implied vols for one-month expiries might run from 24 at the 85 percent strike (deep OTM put) down to 15 at the 115 percent strike (deep OTM call). Apply Dupire's formula at the 90 percent strike and you might recover a local volatility of 28, higher than the implied vol at that strike.

This is a general result known as the rule of thumb: for a downward-sloping IV skew, local volatility is steeper than implied volatility, roughly by a factor of two at short maturities. The intuition is that implied vol averages the path through spot values, while local vol is the instantaneous function at a specific spot level.

Now price an exotic like a one-year down-and-in barrier put struck at 90 percent with a barrier at 70 percent. Run a Monte Carlo simulation under the calibrated sigma_L(S, t) surface. The barrier is hit on paths where spot falls through 70 percent, and on those paths local volatility is high, so the terminal put payoff is inflated. The price reflects that consistently with the skew the vanilla surface is already showing.

Common Mistakes

1. Feeding a non-arbitrage-free surface into Dupire. If the vanilla surface violates butterfly or calendar no-arbitrage conditions, Dupire's formula produces negative local variances. The fix is to enforce arbitrage-free interpolation up-front, not to patch the local vol after the fact.

2. Assuming local volatility predicts future smiles correctly. Local vol reproduces today's vanilla surface but implies that the future smile will flatten and move rigidly with spot. Empirical smile dynamics do not behave that way, which is why local vol prices for forward-starting exotics (cliquets, forward-start options) can be systematically wrong.

3. Using local vol for stochastic-vol sensitive trades. Products whose value depends heavily on the random evolution of volatility (variance swaps, VIX options, some cliquets) are better priced under Heston or a hybrid stochastic-local vol model. Pure local vol suppresses vega dynamics.

4. Ignoring the short-maturity blow-up. Near T = 0, the denominator K^2 * d^2C/dK^2 is tiny and the formula becomes numerically unstable. Practitioner implementations apply a floor to the shortest maturity bucket and extrapolate carefully below it.

5. Re-calibrating daily without tracking drift. The local volatility surface shifts from day to day. Traders who re-calibrate every morning and price exotics off fresh surfaces can produce noisy P&L swings that look like risk when they are calibration artifacts.

Frequently Asked Questions

Q: What is the local volatility Dupire model in simple terms? Dupire's model finds the one volatility function σ(S,t), varying by stock price and time, that makes a diffusion process reproduce every observed vanilla option price simultaneously. It is the pricing surface analogue of finding a curve that passes through every data point.

Q: How does the local volatility model affect investment decisions? It lets traders price exotic options, barriers, cliquets, lookbacks, consistently with the vanilla market. Without calibrating to the surface, an exotic price could be arbitraged against vanilla hedges. Local vol eliminates that inconsistency.

Q: What is a real-world example of the Dupire model in action? SPX with ATM one-month IV at 14 and 90-strike IV at 20. Dupire's formula recovers a local vol of roughly 28 at the 90-strike, steeper than the implied vol because local vol is the instantaneous function, not the path-averaged expectation.

Q: How can practitioners implement local volatility correctly? Start with an arbitrage-free vanilla surface interpolated using SVI or SSVI. Only then apply Dupire's formula. Check for negative local variances, which signal arbitrage violations in the input surface. Apply a maturity floor for the near-zero-time instability.

Q: How is local volatility different from stochastic volatility (Heston)? Local vol fits the vanilla surface exactly but implies the smile flattens and moves with spot. Stochastic vol (Heston) fits less perfectly but implies more realistic forward smile dynamics, which matters for cliquets and variance swaps where forward vol determines the value.

Sources

1. Dupire, B. (1994). "Pricing with a Smile." Risk magazine, 7(1), 18-20. https://www.risk.net/derivatives/1500261/pricing-with-a-smile
2. World Scientific. "Local Volatility and Dupire's Equation." https://www.worldscientific.com/doi/pdf/10.1142/9789811212772_0001
3. Columbia University. "Local Volatility, Stochastic Volatility and Jump-Diffusion Models." http://www.columbia.edu/~mh2078/ContinuousFE/LocalStochasticJumps.pdf
4. Derman, E. and Kani, I. (1994). "Riding on a Smile." Risk magazine. https://emanuelderman.com/wp-content/uploads/1994/02/gs-riding_on_a_smile.pdf

Disclaimer

This article is educational content only and is not financial advice. Nothing here is a recommendation to buy, sell, or hold any security. Consult a licensed advisor before making investment decisions.