Volatility Surfaces: Smiles, Skews, and Local Vol
You look at the market. Calls with the same expiry but different strikes have different implied volatilities. The Black-Scholes model says this should not happen. Constant volatility means one number for all strikes. But the market does not care what Black-Scholes says.
Chapter 50 takes this observation and builds an entire framework around it. The idea: if constant volatility does not fit market prices, make volatility a function of stock price and time. Calibrate it so all market prices match. This is the deterministic volatility surface, and despite some serious problems, it has become one of the most widely used tools in derivatives pricing.
Starting Simple: Time-Dependent Volatility
Before going full two-dimensional, let us start with the easy case. Suppose you see three European calls on the same stock, all with strike 105, but expiring in one, three, and seven months. Their implied volatilities are 21.2%, 20.5%, and 19.4% respectively.
Clearly volatility is not constant. But if we just let volatility depend on time, sigma(t), we can accommodate this. The key relationship is that implied volatility is the square root of the average variance:
sigma_implied^2 * (T - t) = integral of sigma(s)^2 ds from t to T
In plain language: variances add up over time, so implied volatility is like a time-weighted average of actual volatility. If short-term implied vol is higher than long-term, actual volatility must be declining. If it is lower, actual volatility must be increasing.
With a piecewise-constant assumption for sigma(t), the math is simple. For the first period, actual volatility equals the one-month implied volatility. For the second period, actual volatility comes from:
sigma_2^2 = (sigma_imp2^2 * T2 - sigma_imp1^2 * T1) / (T2 - T1)
This is just “backing out” the forward volatility from the term structure. Straightforward and useful.
The Problem: Different Strikes
Now here is where it gets complicated. Suppose you have two seven-month options, one struck at 100 and another at 105, with implied volatilities of 20.8% and 19.4%. You cannot reconcile both with time-dependent volatility alone. You need volatility to depend on the stock price as well.
Plot implied volatility against strike for options with the same expiry and you see the famous shapes:
The smile: Implied vol is higher for both low and high strikes, creating a U-shape. Common in FX markets.
The skew: Implied vol slopes downward from left to right. Low strikes (downside protection) are expensive, high strikes are cheap. Dominant in equity markets since the 1987 crash.
The frown: An upside-down smile. Rare but it exists.
Add another axis for expiry and you get a three-dimensional surface. This is the implied volatility surface, and it is the starting point for everything that follows.
From Implied Vol to Local Vol
The goal: find a local volatility function sigma(S, t) such that when you use it in the Black-Scholes PDE, the theoretical prices match all observed market prices.
We are going backwards. Normally you specify sigma, solve for option prices, and compute implied vols. Here we specify implied vols and ask: what local volatility function is consistent with these?
This is mathematically known as an inverse problem. And inverse problems are notoriously tricky.
The derivation is elegant. Wilmott uses the transition probability density function and the Fokker-Planck (forward Kolmogorov) equation. The beauty of using European call payoffs is that differentiating the call price twice with respect to strike gives you the risk-neutral probability density directly:
d^2 V / d E^2 = e^(-r(T-t)) * p(S*, t*; E, T)
This is a remarkable result. You can extract the market-implied probability distribution from option prices by taking the second derivative with respect to strike.
The final formula for the local volatility surface in terms of call prices is:
sigma^2(E, T) = 2 * (dV/dT + rE * dV/dE) / (E^2 * d^2V/dE^2)
Here V(E, T) represents the call price as a function of strike and expiry, evaluated at today’s spot price and time.
When there is a continuous dividend yield D, a small adjustment appears:
sigma^2(E, T) = 2 * (dV/dT + (r-D)E * dV/dE + DV) / (E^2 * d^2V/dE^2)
There is also an equivalent formula in terms of implied volatility rather than call prices. This is useful because it avoids dividing small numbers by small numbers far from the money, where both the numerator and denominator of the price-based formula become tiny.
The Practical Problems
The math is clean but the practice is messy.
Problem 1: You do not have a continuous surface of prices. You have a finite, discrete set of strikes and expiries. Some strikes are liquid, others barely trade. Some expiries are common (monthly), others are not. You need to interpolate and extrapolate, and the final answer depends heavily on how you do this.
Problem 2: The inverse problem is ill-posed. A small change in implied volatilities can lead to a huge change in the local volatility surface. If one implied vol shifts by 0.5%, the local surface can move by several percent. This is the mathematical opposite of smoothing: the forward problem (from local to implied) averages things out, the inverse problem amplifies noise.
This ill-posedness requires regularization. You need to impose some smoothness on the result, typically through the choice of interpolation method. Polynomial fits, splines, parametric forms, all have different properties. Wilmott shows a nice cautionary example: fitting a perfect polynomial through a set of data points produces a wiggly mess that looks nothing like the underlying relationship. A simple straight line, while imperfect, often captures the real signal better.
Problem 3: The surface is not stable over time. If the model were correct, you would calibrate today and the surface would stay the same forever. Come back next week, recalibrate, same surface. Does this happen? Wilmott’s answer: “It doesn’t.” The surface changes every time you look at it. This means the model is fundamentally wrong as a description of reality.
A Simple Parameterization
Given all these problems, Wilmott proposes a simple parameterization that captures the most important features without overfitting:
sigma_implied = a(T) + b(T) * (E - S*) / S*
This is just a straight line in moneyness. Two parameters per expiry: a(T) is the at-the-money level, b(T) is the skew. From market data, you get a(T) from the at-the-money straddle and b(T) from the risk reversal.
The corresponding local volatility surface can be computed analytically. It is fast, stable, and captures the two most important pieces of information: level and slope. More complex shapes (curvature, etc.) are there in the data but are harder to estimate reliably and add less value.
This follows the philosophy: a parsimonious model that captures the main features is better than a complex model that overfits the noise. There is evidence (Dumas, Fleming & Whaley, 1998) that overfitting the local volatility surface actually destroys the information content of market prices.
Market Conventions
Options traders often quote prices not by strike but by delta. For example, “the six-month 25-delta call has a volatility of 13%.” This means the option with a Black-Scholes delta of 0.25 (about 25% in-the-money equivalent) is priced using 13% volatility.
The strike is implicit. Given the delta, volatility, spot price, rate, and expiry, you can solve for the strike. This convention is especially common in FX markets where the concept of a “fixed strike” is less natural (since exchange rates can move a lot).
The simple volatility surface parameterization translates naturally into this delta-based convention.
How to Use the Local Volatility Surface
Wilmott gives two interpretations:
The naive interpretation: The local volatility surface is the market’s prediction of future volatility. Use sigma(S, t) to price exotic contracts. If the stock is at 100 today and the surface says sigma(110, 6 months) = 18%, then “the market believes” volatility will be 18% if the stock reaches 110 in six months.
The practical interpretation: The surface is a snapshot, not a prediction. Tomorrow it will be different. But you can use it to price non-traded contracts consistently with traded ones. If you price an exotic using the same volatility surface and simultaneously hedge with the vanillas that determined that surface, you reduce your model exposure.
The second interpretation is the right one. But it comes with a critical warning: you must actually hedge with the vanillas, not just delta hedge. If you price a barrier option using a calibrated surface but only delta hedge with the stock, you are exposed to the surface changing. When it does change (and it will), your delta-only hedge will not save you.
The right workflow: calibrate the surface, price the exotic, then statically hedge the exotic with vanilla options to reduce exposure to model error. Only the residual needs delta hedging.
Curve Fitting 101
The chapter ends with a beautiful little appendix on curve fitting that every quant should internalize.
Given a set of data points, you can always fit a polynomial that goes through every single point. Perfect fit. But the resulting curve wiggles wildly between the points and gives nonsensical extrapolations. The perfect fit is actually the worst model.
A simple straight line misses every point but captures the overall trend. It is “wrong” everywhere but “right” in the big picture.
The moral: think before you fit. More parameters is not always better. The best model is often the simplest one that captures the essential features without being fooled by noise.
This applies directly to volatility surfaces. Overfitting gives unstable, unreliable surfaces. A parsimonious fit (level plus skew) is often all you need.
Key Takeaways
The deterministic volatility surface is a widely used tool that, strictly speaking, fails its most basic scientific test (stability over time). But it remains useful as a consistency tool for relative pricing.
The forward problem (local vol to implied vol) is smooth and stable. The inverse problem (implied vol to local vol) is ill-posed and sensitive to noise. Always remember which direction you are working in.
Keep it simple. Two parameters per expiry (level and skew) capture most of the information. More complex parameterizations risk overfitting and actually reduce the usefulness of the model.
And always, always hedge with vanillas when pricing exotics off a calibrated surface. The surface is a tool for consistency, not a crystal ball for future volatility.
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