The Finite Moment Logstable Process And Option Pricing

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 flatten 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. Ever since the stock market crash of 1987, the U.S. stock index options market has been exhibiting a consistent pattern documented by academics and practitioners alike. At a given maturity level, the Black and Scholes (1973) implied volatilities for out-of-the-money puts are much higher than those of out-of-the-money calls.1 This phenomenon is commonly referred to as the “volatility smirk.” It is, however, less well known that these implied volatilities also exhibit a strong empirical regularity in the maturity direction.2 When implied volatilities are graphed against a standard measure of “moneyness,” we document that the resulting implied volatility smirk does not flatten out as maturity increases up to the observable horizon of two years. The measure of moneyness for which this observation holds is the logarithm of the strike over the forward, normalized by the square root of maturity. This maturity pattern of the volatility smirk is somewhat surprising as it ignores the implications of the central limit theorem. It is widely appreciated that the implied volatility smirk is a direct result of conditional non-normality in stock returns. In particular, the downward slope of the smirk reflects asymmetry (negative skewness) in the risk-neutral distribution of the underlying index return, while the positive curvature of the smirk reflects the fat-tails (leptokurtosis) of this distribution. Yet, the central limit theorem implies that under fairly general conditions, the conditional return distribution should converge to normality as the maturity increases. As a result, the volatility smirk should flatten out accordingly. To account for the volatility smirk at a certain maturity, a large stream of the option pricing literature models the log return as a Lévy process; i.e., a process with stationary independent increments. Prominent examples include the Poisson jump model of Merton (1976), the variance-gamma model of Carr, Madan, and Chang (1998), the log-gamma model of Heston (1993b), and the CGMY model of Carr, Geman, Madan, and Yor (2002). However, for all of these models, the central limit theorem implies that the absolute value of skewness decreases like the reciprocal of the square-root of maturity, while the kurtosis decreases with the reciprocal of maturity (Konikov and Madan (2000)). As a result, the implied volatility smirk obtained from these models flattens out very quickly as maturity increases. Incorporating a persistent stochastic volatility process, e.g. Heston (1993), slows down the speed of convergence, but does not stop it, so long as the volatility process is stationary.3 To prevent the flattening of the volatility smirk, we develop a parsimonious option pricing model that deliberately violates the conditions leading to the validity of the central limit theorem. One of the key conditions for the central limit theorem to hold is that the return distribution has finite second moments. In our model, the return distribution of the underlying index has infinite moments for any order of two or greater. As a result, the central limit theorem no longer applies. Nevertheless, our model guarantees that all moments of the index level itself are finite. The finiteness of these price moments guarantees the existence of an equivalent martingale measure and the finiteness of option prices at all maturities. To combine infinite return moments with finite price moments, we model returns as driven by an α-stable motion with maximum negative skewness. The α-stable motion is a Lévy process whose departure from Brownian motion is controlled by the tail index α∈ (0,2]. Setting α = 2 degenerates the α-stable motion into a Brownian motion and our model into the Black-Scholes model. Setting α below two induces a pure jump processes with fat-tails in the return distribution. In contrast to a standard Poisson or compound Poisson process, this pure jump process has an infinite number of jumps over any time interval, allowing it to capture the extreme activity traditionally handled by diffusion processes. Most of the jumps are small and may be regarded as approximating the transition from one decimalized price to another one nearby. We allow α to be a free parameter, whose exact value is determined by calibrating to market option prices. Like the Brownian motion, the α-stable motion exhibits a self-similarity or stability property. This property means that the distribution of the α-stable motion over any horizon has the same shape, upon scaling. As a result, with α < 2, the risk-neutral distribution has fat tails at all horizons. Thus, our model can generate the maturity pattern of the volatility smirk observed in the S&P index options market. Our specification also has a simple analytical form for the characteristic function of the return. Many standard contingent claims can then be readily priced by the fast Fourier transform (FFT) method of Carr and Madan (1999). The relevance of α-stable motions for option pricing has been recognized previously. For example, Janicki, Popova, Ritchken, and Woyczynski (1997); Popova and Ritchken (1998); and Hurst, Platen, and Rachev (1999) have worked on option pricing in a symmetric α-stable security market. However, modeling log returns by symmetric α-stable motions generates infinite price moments, and hence po-