Split-Surface Analytics: Call vs Put

By now you can read a volatility smile, and you know that the downside puts trade richer than the upside calls. That is skew, and in the last chapter you measured its slope. But you have been treating the surface as a single object — one smile, one curve through strike. In this chapter we split it in two, and a great deal of hidden information falls out.

Here is the intuition first. For any strike below the forward, you can observe an out-of-the-money put. For any strike above the forward, you can observe an out-of-the-money call. The standard practice is to stitch these together into one continuous implied-volatility curve and call it the smile. That stitching quietly assumes the two wings are telling the same story. They usually are not. The put wing is priced by people buying crash protection; the call wing is priced by people selling covered upside or chasing a melt-up. These are different crowds, different flows, different fears. When you fuse them into one curve, you average away the disagreement — and the disagreement is exactly the signal.

So instead, build two surfaces. Take every put across strike and maturity and fit a put-only implied-volatility surface. Then take every call and fit a call-only surface. SVQ calls these the split-surface, and the pair of them, indexed together, is the IVS — the implied-volatility-split object. Where put-call parity held perfectly and markets were frictionless, these two surfaces would be identical, because parity ties a call and a put at the same strike to the same implied volatility. In the real world they come apart, and how they come apart is the lesson.

Now let us formalize, because a number you cannot reproduce is a number you cannot trust.

Start with the single-wing skew. For a fixed maturity, the skew is the slope of implied volatility with respect to strike, conventionally measured in log-moneyness. If you write implied volatility as a function of log-moneyness k, equal to the log of strike over forward, then the local skew is the partial derivative of implied vol with respect to k. A useful summary is the twenty-five-delta risk reversal: the implied vol of the twenty-five-delta call minus the implied vol of the twenty-five-delta put. That single difference is the crudest skew decomposition there is — it literally subtracts the put wing from the call wing, and in the normal equity regime, where the downside puts trade richer, it comes out negative.

The PSI — the Put–Skew Index — generalizes that. Rather than sampling two delta points, PSI integrates the call-wing slope across the upside and subtracts the integrated put-wing slope across the downside, preserving the same call-minus-put orientation as the risk reversal. The result is a signed quantity: negative when the put surface is steeper than the call surface, which is the normal equity regime, and shrinking toward zero or flipping positive when the call wing starts to steepen — a tell that the crowd is now paying up for upside, not downside. Watching PSI move is watching the asymmetry of fear change in real time.

But skew is a property of the volatility curve, and the volatility curve is one step removed from what you actually care about, which is the probability the market assigns to where the asset lands. That is where the dual RND comes in. There is a classical result — Breeden and Litzenberger — that says the risk-neutral density of the terminal price is recovered from the second derivative of the option price with respect to strike. The density at strike K is the discount factor times the second partial derivative of the call price with respect to K, evaluated at that strike. Differentiate the call surface twice in strike and you get a probability density; do the same to the put surface and, by parity, you should get the same density.

Here is the move. Do not assume parity. Compute the density from the call surface alone, and separately compute the density from the put surface alone. Now you have two risk-neutral densities — the dual RND. The call-implied density is the market's distribution as priced by the upside instruments; the put-implied density is the distribution as priced by the downside instruments. Overlay them. In a calm, well-arbitraged market they sit almost on top of each other. When they separate — when the put-implied left tail is fatter than the call-implied left tail, or the two disagree about the location of the mode — you are looking directly at a market that has not agreed with itself about the shape of risk.

To make that separation a single tradeable number, SVQ computes the Split-Divergence Index, the SDI. Conceptually the SDI is a distance between the two densities — you can think of it as the integrated absolute difference between the call-implied and put-implied RNDs, or, more sharply, as a Kullback–Leibler-style divergence of one from the other. A near-zero SDI says the wings agree and there is no information in the split. A large, persistent SDI says the call market and the put market are pricing different futures, and the gap between them is your evidence.

And that word — evidence — is the throughline of this entire institute. A split-surface disagreement is only a signal if it is genuine. Parity violations can be manufactured by a stale quote on a thin strike, by borrow costs and hard-to-locate shorts that break parity mechanically, or by a single fat order skewing one wing for an afternoon. The discipline is to ask, every time the SDI spikes: is this the market telling me something, or is this plumbing? A real signal persists across maturities, survives a liquidity filter, and shows up in the dual RND as a coherent reshaping of the tail — not just a notch on one strike.