The Variance Gamma Process and Option Pricing

A three parameter stochastic process, termed the variance gammaprocess, that generalizes Brownian motion is developed as amodel for the dynamics of log stock prices. Theprocess is obtainedby evaluating Brownian motion with drift at a random time givenby a gamma process. The two additional parameters are the driftof the Brownian motion and the volatility of the time change.These additional parameters provide control over the skewnessand kurtosis of the return distribution. Closed forms are obtainedfor the return density and the prices of European options.Thestatistical and risk neutral densities are estimated for dataon the S&P500 Index and the prices of options on this Index.It is observed that the statistical density is symmetric withsome kurtosis, while the risk neutral density is negativelyskewed with a larger kurtosis. The additional parameters alsocorrect for pricing biases of the Black Scholes model that isa parametric special case of the option pricing model developedhere.     

On a continuous time stock price model with regime switching,delay, and threshold

Motivated by the need to describe bear-bull market regime switching in stock prices, we introduce and study a stochastic process in continuous time with two regimes, threshold and delay, given by a stochastic differential equation. When the difference between the regimes is simply given by a different set of real valued parameters for the drift and diffusion coefficients, with changes between regimes depending only on these parameters, we show that if the delay is known there are consistent estimators for the threshold as long we know how to classify a given observation of the process as belonging to one of the two regimes. When the drift and diffusion coefficients are of geometric Brownian motion type we obtain a model with parameters that can be estimated in a satisfactory way, a model that allows differentiating regimes in some of the NYSE 21 stocks analyzed and also, that gives very satisfactory results when compared to the usual Black–Scholes model for pricing call options.     

Momentum and reversion in risk neutral martingale probabilities

A time homogeneous, purely discontinuous, parsimonous Markov martingale model is proposed for the risk neutral dynamics of equity forward prices. Transition probabilities are in the variance gamma class with spot dependent parameters. Markov chain approximations give access to option prices. The model is estimated on option prices across strike and maturity for five days at a time. Properties of the estimated processes are described via an analysis of return quantiles, momentum functions that measure the response of tail probabilities to such moves. Momentum and reversion are also addressed via the construction of reverse conditional expectations. Term structures for the moments of marginal distributions support a decay in skewness and excess kurtosis with maturity at rates slower than those implied by Lévy processes. Out of sample performance is additionally reported. It is observed that risk neutral dynamics by and large reflect the presence of momentum in numerous probabilities. However, there is some reversion in the upper quantiles of risk neutral return distributions.     

A one-factor volatility smile model with closed-form solutions for European options

The common practice of using different volatilities for options of different strikes in the Black-Scholes (1973) model imposes inconsistent assumptions on underlying securities. The phenomenon is referred to as the volatility smile. This paper addresses this problem by replacing the Brownian motion or, alternatively, the Geometric Brownian motion in the Black-Scholes model with a two-piece quadratic or linear function of the Brownian motion. By selecting appropriate parameters of this function we obtain a wide range of shapes of implied volatility curves with respect to option strikes. The model has closed-form solutions for European options, which enables fast calibration of the model to market option prices. The model can also be efficiently implemented in discrete time for pricing complex options.
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When are Options Overpriced? The Black--Scholes Model and Alternative Characterisations of the Pricing Kernel

An important determinant of option prices is the elasticityof the pricing kernel used to price all claims in the economy.In this paper, we first show that for a given forward priceof the underlying asset, option prices are higher when the elasticityof the pricing kernel is declining than when it is constant.We then investigate the implications of the elasticity of thepricing kernel for the stochastic process followed by the underlyingasset. Given that the underlying information process followsa geometric Brownian motion, we demonstrate that constant elasticityof the pricing kernel is equivalent to a Brownian motion forthe forward price of the underlying asset, so that the Black–Scholesformula correctly prices options on the asset. In contrast,declining elasticity implies that the forward price processis no longer a Brownian motion: it has higher volatility andexhibits autocorrelation. In this case, the Black–Scholesformula underprices all options.     

The square-root process and Asian options

Although the square-root process has long been used as an alternative to the Black–Scholes geometric Brownian motion model for option valuation, the pricing of Asian options on this diffusion model has never been studied analytically. However, the additivity property of the square-root process makes it a very suitable model for the analysis of Asian options. In this paper, we develop explicit prices for digital and regular Asian options. We also obtain distributional results concerning the square-root process and its average over time, including analytic formulae for their joint density and moments. We also show that the distribution is actually determined by those moments.     

OPTIONS ON U.S. TREASURY COUPON ISSUES

Pricing models for options on default-free coupon bonds are developed and tested under the assumption that the bond prices, rather than interest rates, are the underlying stochastic factors. Under the assumption that coupon bond prices, excluding accrued interest, follow a generalized Brownian bridge process, preference-free, continuous-time pricing models are developed for European put and call options, and a discrete-time model is developed for American puts and calls. The empirical validity of the models is assessed using a six-moth sample of daily closing prices.     

Risk-Neutral Parameter Shifts and Derivatives Pricing in Discrete Time

We obtain a large class of discrete‐time risk‐neutral valuation relationships, or “preference‐free” derivatives pricing models, by imposing a simple restriction on the state‐price density process. The risk‐neutral stock‐return and forward‐rate dynamics are obtained by changing only a location parameter, which can be determined independent of the preference and true location parameters. The Gaussian models of Rubinstein (1976) , Brennan (1979) , and Câmera (2003) , and the gamma model of Heston (1993) are all special cases. The model provides simple relationships between expected returns and state‐price density parameters analogous to the diffusion case.     

Drift Estimation of Generalized Security Price Processes from High Frequency Derivative Prices

This paper presents a framework for using high frequency derivative prices to estimate the drift of generalized security price processes. This work may be seen more generally as a quasi-likelihood approach to estimating continuous-time parameters of derivative pricing models using discrete option data. We develop a generalized derivative-based estimator for the drift where the underlying security price process follows any arbitrary state-time separable diffusion process (including arithmetic and geometric Brownian motion as special cases). The framework provides a method to measure premia in derivative prices, test for risk-neutral pricing and leads to a new empirical approach to pricing derivative contingent claims. A sufficient condition for the asymptotic consistency of the generalized estimator is also obtained. A study based on generating the S&P500 index and calls shows that the estimator can correctly estimate the drift parameter. This revised version was published online in November 2006 with corrections to the Cover Date.