The Finite Moment Log Stable Process and Option Pricing

Abstract

We document a surprising pattern in S&P 500 option prices. When implied volatilities are graphed against a standard measure of moneyness, the implied volatility smirk does not £atten out as maturity increases up to the observable horizon of two years. This behavior contrasts sharply with the implications of many pricing models and with the asymptotic behavior implied by the central limit theorem (CLT).We develop a parsimonious model which deliberately violates the CLT assumptions and thus captures the observed behavior of the volatility smirk over the maturity horizon. Calibration exercises demonstrate its superior performance against several widely used alternatives.

Stationarity and Ergodicity for an Affine Two-Factor Model

“…The set of affine processes contains a large class of important Markov processes such as continuous state branching processes and OrsteinUhlenbeck processes. Further, a lot of models in financial mathematics are affine such as the Heston model [18], the model of Barndorff-Nielsen and Shephard [3] or the model due to Carr and Wu [6]. A precise mathematical formulation and a complete characterization of regular affine processes are due to Duffie et al [12].…”

Option pricing with Lévy-Stable processes generated by Lévy-Stable integrated variance

“…Financial data suggests that returns are skewed rather than symmetric, see for example [KL76], [CLM97], [CW03]. For instance, since the stock market crash of 1987, the US stock index options market has shown a pronounced skewed implied volatility (volatility smirk) which indicates that, under the risk-neutral measure, log-returns have a negatively skewed distribution.…”

“…In the literature most models have assumed that log-prices, instead of returns, follow a Lévy-Stable process. McCulloch [McC96] assumes that assets are log Lévy-Stable and prices options using a utility maximisation argument; more recently Carr and Wu [CW03] priced European options when the log-stock price follows a maximally skewed Lévy-Stable process.…”

Asset Pricing with Incomplete Information under Stable Shocks

“…We shall see in the empirical section of the paper that this choice for β is consistent with negative skewness in dividends growth data. Carr and Wu (2003) refer to the β = −1 case as the finite moment log-stable process and use it for option pricing. The reason for this terminology is as follows.…”

“…In recent work, Carr and Wu (2003) use this sub-family of α-stable distributions with maximal negative skewness for capturing the observed behavior of the volatility smirk implied by S&P 500 option prices. They too are forced to work with these distributions by imposing 1 β = − in order to ensure finiteness of call option values.…”