A simulation technique known as empirical martingale simulation (EMS) was proposed to improve simulation accuracy. By an adjustment to the standard Monte Carlo simulation, EMS ensures that the simulated price satisfies the rational option pricing bounds and that the estimated derivative contract price is strongly consistent with payoffs that satisfy Lipschitz condition. However, for some currently used contracts such as self-quanto options and asymmetric or symmetric power options, it is open whether the above asymptotic result holds. In this paper, we prove that the strong consistency of the EMS option price estimator holds for a wider class of univariate payoffs than those restricted by Lipschitz condition. Numerical experiments demonstrate that EMS can also substantially increase simulation accuracy in the extended setting.

Empirical martingale simulation (EMS) was proposed by Duan and Simonato (Duan, J.-C., J.-G. Simonato. 1998. Empirical martingale simulation for asset prices. Management Sci. 44(9) 1218-1233) as an adjustment to the standard Monte Carlo simulation to reduce simulation errors. The EMS price estimator of derivative contracts was shown to be asymptotically normally distributed in Duan et al. (Duan, J.-C., G. Gauthier, J.-G. Simonato. 2001. Asymptotic distribution of the EMS option price estimator. Management Sci. 47(8) 1122-1132) when the payoffs are piecewise linear and continuous. In this paper, we extend the asymptotic normality result to more general continuous payoffs, and for discontinuous payoffs we make a conjecture.empirical martingale simulation, Monte Carlo, Black-Scholes, GARCH, options, regression analysis, asymptotic normality, coverage rate

Inspired by the negative price of WTI crude oil observed during the COVID-19 pandemic, we develop a new model for commodity pricing which allows structural change between price normality and lognormality under a Markov regime-switching (RS) framework. We augment the Extended Kalman Filter to calibrate the structural changing model. The model performance in calibration is compared to that of the common RS model with historical WTI spots, various futures and hypothetical scenarios. We conclude that our model is superior in capturing price dynamics especially in the oil market downturns. Encouragingly, the regime probabilities estimated with the new model indicate that during severe events including the 2008–2010 financial crisis, 2014–2016 oil crash and the outbreak of COVID-19 in 2020, WTI spot itself follows normal rather than the widely assumed lognormal process. This finding is consistent with our empirical studies. In addition, we assess the probability density of spot prices with the new model. Finally, we present the PDE finite difference and Monte Carlo approaches to price commodity options under the new model.