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Security price process models : do these have the correct properties for understanding options values?

By Robert G. Tompkins


It is well known that the market price of options are inconsistent with option\ud pricing models that assume the innovation of the underlying price follows Geometric\ud Brownian Motion. What is not clear is why this occurs.\ud The testing of option pricing models requires a joint hypothesis to be tested\ud that the options pricing models are correct and that the options markets are efficient.\ud To test the option pricing models, we will examine the relationship between the\ud objective and risk neutral dispersion processes for twelve financial futures markets.\ud These markets have been selected so that we can investigate the dynamics of equity,\ud fixed income and foreign exchange asset classes.\ud Our analysis of the objective dispersion processes allows us to reject the\ud hypothesis that the prices of these twelve markets follow Geometric Brownian\ud motion. For all twelve markets and for various sub-periods of analysis, we find that\ud the optimal models for capturing the dynamics of the objective dispersion process\ud include jump diffusion and stochastic volatility.\ud For the risk neutral dispersion process, we chose to examine the implied\ud volatility surfaces from the closing prices of options (on these same futures markets).\ud It appears that both within and between markets similar dynamics determine the\ud shapes of the implied volatility surfaces. By employing an Analysis of Covariance\ud approach, we found that important consistencies exist within asset classes and\ud between markets. The first order strike price effect (skewness) differs widely among\ud markets but is fairly consistent within the same asset class. The second order strike\ud price effect (kurtosis) is consistent among all markets. The dependency of both strike\ud price effects on the time to expiration is also similar across all markets and suggests\ud that both jump diffusion and stochastic volatility play a role.\ud A comparison of option prices, which are consistent with the objective\ud dispersion process, with actual options prices suggests that significant divergences\ud exist. The actual smile patterns display greater variation in the amplitude compared to\ud those from the objective function. The least degree of discrepancy exists for the\ud foreign exchange markets. Both the stock index and fixed income options dispersion\ud processes display behaviours that diverge considerably. This divergence is primarily\ud due to the existence of negative skews that are not justified by the objective dispersion\ud processes. This suggests that other mechanisms are at work for the risk neutral\ud dispersion processes for these asset classes. The most likely explanations are the\ud existence of risk premia associated with stochastic volatility and non-diversifiable\ud jumps or that transaction costs are relevant

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