On Jumps in Common Stock Prices and Their Impact on Call Option Pricing

Abstract

The Black-Scholes call option pricing model exhibits systematic empirical biases. The Merton call option pricing model, which explicitly admits jumps in the underlying security return process, may potentially eliminate these biases. We provide statistical evidence consistent with the existence of lognormally distributed jumps in a majority of the daily returns of a sample of NYSE listed common stocks. However, we find no operationally significant differences between the Black-Scholes and Merton model prices of the call options written on the sampled common stocks. EMPIRICAL EVIDENCE CONFIRMS THE systematic mispricing of theBlack-Scholes call option pricing model. These biases have been documented with respect to the call option's exercise price, its time to expiration, and the underlying common stock's volatility. Black [3] reports that the model overprices deep in-the-money options, while it underprices deep out-of-the-money options. By contrast, MacBeth and Merville [10] state that deep in-the-money options have model prices that are lower than market prices, whereas deep out-of-the-money options have model prices that are higher. These conflicting results may perhaps be reconciled by the fact that the studies examined market prices at different points in time and these systematic biases vary with time (Rubinstein [16 ]). Employing over-the-counter data, Black and Scholes [4] also document that the model underprices both calls written on low volatility stocks and calls near to expiration, while the model overprices calls written on high volatility stocks.A number of explanations for the systematic price bias have been suggested (Geske and Roll [6]). Among these is the fact that the Black-Scholes assumption of a lognormally distributed security price fails to systematically capture important characteristics of the actual security price process. Merton [11, 12] has put forward an option pricing model that explicitly admits jumps in the underlying security return process, and which may resolve these pricing discrepancies. According to the Merton specification, the arrival of normal information leads to price changes which can be modelled as a lognormal diffusion, while the arrival of abnormal information, which can be modelled as a Poisson process, gives rise * Both authors from the Graduate School of Business Administration, The University of Michigan. We thank Phelim Boyle, Robert Jarrow, Eric Kirzner, William Margarabe, Robert Merton, and Stuart Turnbull for helpful suggestions. Michael Jenkins, David Sauer, and Michael Weisbach provided excellent research assistance. The comments of an anonymous referee are gratefully acknowledged. This work was supported by summer research grants from the Graduate School of Business Administration at The University of Michigan. Any remaining errors are the authors' responsibility. 155 156 The Journal of Finance to lognormally distributed jumps in the security return. If the underlying security return follows the Poisson jump-diffusion process, then the resultant equilibrium c...

Oil price dynamics (2002–2006)

“…Nonetheless, Ball and Torous (1985) applied directly the ML method by truncating the number of jumps at n = 10. Ball and Torous (1985) and Jorion (1988) applied the ML method by assuming a Bernoulli process for the jump component.…”

“…Nonetheless, Ball and Torous (1985) applied directly the ML method by truncating the number of jumps at n = 10. Ball and Torous (1985) and Jorion (1988) applied the ML method by assuming a Bernoulli process for the jump component. While the ML estimates achieve the lower bound for Cramer-Rao efficiency criterion, difficulties with the likelihood function arising from computational tractability, un-boundedness over the parameter space, and instability of parameters, have led researchers to explore alternative estimation methods, based essentially on the method of moments.…”

Optimal portfolio choice in the presence of domestic systemic risk: empirical evidence from stock markets

“…We thus have to decide on a cutoff point, N , for practical implementation of the estimation. In a univariate setting, Ball and Torous (1985) used N = 10, corresponding to a maximum number of ten shocks per month for one stock. As systemic shocks occur less frequent than individual shocks (as every systemic shock affects every company, but not vice versa), we use N = 3 in the subsequent analysis.…”

The Relationships between Foreign Exchange Volatility Skew and Jump Risk

“…Such provides theoretical support to the said predictive power of implied volatility skew, and the jump-diffusion model itself finds strong support from a vast body of empirical work that goes beyond the classic diffusion model of Black and Scholes (1973) to prove the importance of including jumps in a diffusion model of asset pricing. These studies include Ball and Torous (1985), Naik and Lee (1990), Bakshi, Cao and Chen (1997), Eraker, Johannes and Polson (2003) and in particular Bates (2000), which succinctly shows that in a post-'87 world of persistent negative skew in S&P 500 implied volatility, the use of a stochastic volatility asset-pricing model would require unrealistic parameters if it were not to incorporate a jump-diffusion model that significantly enhances its fit vis-à-vis observed option prices. That said, drawing from theoretical work to answer why implied volatility and implied volatility skew are able to predict asset returns and price jumps is beyond the scope of this paper.…”