[show abstract][hide abstract] ABSTRACT: How do stock prices evolve over time? The standard assumption of geometric Brownian motion, questionable as it has been right along, is even more doubtful in light of the stock market crash of 1987 and the subsequent prices of U.S. index options. With the development of rich and deep markets in these options, it is now possible to use options prices to make inferences about the risk-neutral stochastic process governing the underlying index. We compare the ability of models including Black-Scholes, naive volatility smile predictions of traders, constant elasticity of variance, displaced diffusion, jump diffusion, stochastic volatility, and implied binomial trees to explain otherwise identical observed option prices that differ by strike prices, times-toexpiration, or times. The latter amounts to examining predictions of future implied volatilities. Certain naive predictive models used by traders seem to perform best, although some academic models are not far behind. We find that the better performing models all incorporate the negative correlation between index level and volatility. Further improvements to the models seem to require predicting the future at-the-money implied volatility. However, an efficient markets result makes these forecasts difficult, and improvements to the option pricing models might then be limited.
[show abstract][hide abstract] ABSTRACT: We study the term structure of the implied volatility in a situation where the smile is symmetric. Starting from the result by Tehranchi that a symmetric smile generated by a continuous martingale necessarily comes from a mixture of normal distributions, we derive representation formulae for the at-the-money (ATM) implied volatility level and curvature in a general symmetric model. As a result, the ATM curve is directly related to the Laplace transform of the quadratic variation of the log price. To deal with the remaining part of the volatility surface, we build a time dependent SVI-type approximation which matches the ATM and extreme moneyness structure. As an instance of a symmetric model, we consider uncorrelated Heston: in this framework, our representation of the ATM volatility takes semiclosed (and easy to implement) form and the time-dependent SVI approximation displays considerable performances in a wide range of maturities and strikes. In addition, we show how to apply our results to a skewed smile by considering a displaced model. Finally, a noteworthy fact is that all along the paper we will deal only with Laplace transforms and not with Fourier transforms, thus avoiding any complex-valued function.
[show abstract][hide abstract] ABSTRACT: Calibrating option pricing models to market prices often leads to optimisation problems to which standard methods (like such based on gradients) cannot be applied. We investigate two models: Heston’s stochastic volatility model, and Bates’s model which also includes jumps. We discuss how to price options under these models, and how to calibrate the parameters of the models with heuristic techniques.
Finance and Stochastics 01/2010; 14(1):129-152. · 1.21 Impact Factor
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