Closed Form Solution and Term Structure for SPX Options

Quantitative Finance, Forthcoming

35 Pages Posted: 8 Jul 2016 Last revised: 4 Oct 2016

See all articles by Stephen H.T. Lihn

Stephen H.T. Lihn

Atom Investors LP; Novus Partners, Inc.

Date Written: September 2016


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

Suggested Citation

Lihn, Stephen H.T., Closed Form Solution and Term Structure for SPX Options (September 2016). Quantitative Finance, Forthcoming. Available at SSRN:

Stephen H.T. Lihn (Contact Author)

Atom Investors LP ( email )

3711 S Mopac Expressway
Austin, TX 78746
United States
917-603-4133 (Phone)

Novus Partners, Inc. ( email )

521 5th Ave
29th Floor
New York, NY 10175
United States
917-603-4133 (Phone)

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