Higher Moments: Implied Skew & Kurtosis
Objectives
- State the Bakshi-Kapadia-Madan result: risk-neutral variance, skewness, and kurtosis as price-weighted integrals of the out-of-the-money option strip.
- Read implied skew and implied kurtosis as forward-looking measures of crash-fear and tail-fatness, and distinguish a level shift from a shape shift in the option surface.
- Judge when a move in the SVQ implied moment series is genuine evidence the market is repricing tail risk versus when it is volatility-of-volatility noise.
In the last chapter you recovered an entire probability distribution from option prices — the risk-neutral density, the market's own forecast of where the underlying might land. A whole density is a lot to hold in your head. So the natural next question is the one a statistician always asks: how do you summarize it? You summarize a distribution with its moments. The first tells you where it is centered, the second how wide it is, the third how lopsided, the fourth how fat its tails are. This chapter is about the third and fourth — the higher moments — because they are where the real information about crashes lives.
Let us build the intuition before any integrals.
Volatility, the second moment, is symmetric by construction. It cannot tell a market that fears a sudden collapse apart from one that dreams of a sudden melt-up. Both show up as "wide." But you and I both know those are not the same market. The information that separates them is the third moment, skewness, and in equity index options it is almost always negative. A negative risk-neutral skew means the density has a long left tail: the market is paying up for downside protection, pricing a small chance of a large drop more richly than the mirror-image rally. That price is the famous volatility smile tilting into a smirk. Implied skew is, quite literally, the market's crash-fear gauge — and it is forward-looking, because it is extracted from options that have not yet expired.
The fourth moment, kurtosis, asks a different question: forget which side, how fat are the tails on both ends? High implied kurtosis says the market is pricing a meaningful chance of a large move in either direction — outcomes far from the center are more likely than a normal distribution would ever allow. Excess kurtosis is the option market admitting, in price, that returns are not Gaussian.
Now the beautiful part, and the reason this is a Tier 300 result rather than a hand-wave. You do not need to fit a model, assume a dynamic, or guess a parametric density to get these moments. In 2003 Bakshi, Kapadia, and Madan showed that the risk-neutral variance, skewness, and kurtosis of the return over any horizon can be computed directly from the prices of the options expiring at that horizon. No diffusion assumed. The recipe is purely the price strip.
Here is the engine. Any smooth payoff that depends on the terminal price can be replicated by a static portfolio of options — a continuum of calls above the forward and puts below it. Bakshi, Kapadia, and Madan apply this to three specific payoffs: the squared log-return, the cubed log-return, and the fourth power of the log-return. Each becomes its own price, recovered as an integral over strikes. They name these the volatility contract, the cubic contract, and the quartic contract, with theoretical prices usually written V, W, and X. Each is a strike-weighted integral of out-of-the-money option prices — out-of-the-money calls for strikes above the forward, out-of-the-money puts below — with the weighting kernel growing steeper for the higher contracts. The cubic kernel carries a sign that splits puts from calls, which is exactly why W can be negative and why it is what encodes asymmetry.
From those three contract prices, the moments fall out by the standard definitions. Risk-neutral variance is essentially V itself, discounted, minus a small mean-correction term. Risk-neutral skewness is the third central moment over variance-to-the-three-halves — built from W, V, and the mean. Kurtosis is the fourth central moment over variance-squared — built from X, W, V, and the mean. The mean of the log-return, often called mu, is itself pinned down by no-arbitrage from these same contract prices, so the whole calculation closes on the option surface and the risk-free rate alone.
What you should take away is the shape of the dependence. Skewness leans on the cubic weighting, which punishes deep out-of-the-money puts hardest — so a richening of crash insurance drives skew more negative. Kurtosis leans on the quartic weighting, fatter still in the wings — so it is dominated by the very tails of the strike range, the cheap far-out options that most people ignore.
The SVQ implied moment series puts these formulas to work. SVQ runs the Bakshi-Kapadia-Madan machinery on each expiry's live option chain: it takes the out-of-the-money strip, builds the V, W, and X contracts as discrete strike-weighted sums, and publishes a time series of model-free implied volatility, implied skewness, and implied kurtosis. Around a feared event, implied volatility may barely twitch, but the skewness line plunges more negative as traders bid up puts — the market is repricing the left tail, not the width. That divergence is the payoff of separating the moments: a falling skew with flat vol is genuine evidence of crash repricing that a single volatility number would have hidden completely.
Tie this back to the institute's throughline. The whole game is judging when a number is real evidence an asset will move. A jump in implied volatility is ambiguous — it tells you something is coming but not which way. A sharp drop in implied skew, with kurtosis fattening alongside it, is far more specific: the market is paying, in hard dollars, for downside. That is evidence with a direction.