Closed Form Solution and Term Structure for SPX Options

Abstract

Closed form solution for pricing SPX options is developed in the quartic λ pricing model aka λ = 4. The solution fits the term structure of SPX volatility smiles very well, from the shortest expiration dates all the way up to one year, on three distinct market conditions. The ATM volatility, adjusted by diffusion time sqrt(T), is hypothesized to be equal to the standard deviation of the underlying λ distribution. The volatility parameter σ(T) is thus deterministic and the shift parameter ρ(T) can be predictably analyzed. Four universal constants come out of the model, predicting the magnitude of the shift, the power law of the ATM skew, the maximum risk-neutral violation at T = 0, and ratio between minimum volatility and ATM volatility. The ATM skew's dependency on inverse square root of T is a natural outcome from the invariant shape of the normalized volatility smile. Risk-neutral violation is found to have a very rich structure. It strongly influences how shorter-term options behave. When inverted at larger T, it also explains how a V-shape put smile becomes a smirk (back-slash), which is an indication of insufficient risk premium and imbalanced demand. As a side product, the mathematics of tail truncation in the quartic λ model is found to be closely related to that of the asymptotics of scaled error functions.

Keywords: SPX Options, Closed Form Solution, Volatility Smile, Term Structure

JEL Classification: G13, C46, C58