Displaced Diffusion Option Pricing.
ABSTRACT This paper develops a new option pricing formula that pushes the underlying source of risk back to the risk of individual assets of the firm. The formula simultaneously encompasses differential riskiness of the assets of the firm, their relative weights in determining the value of the firm, the effects of firm debt, and the effects of a dividend policy with both constant and random components. Although this setting considerably generalizes the Black‐Scholes  analysis, it nonetheless produces a formula via riskless arbitrage arguments that, given estimated inputs, is as easy to use as the Black‐Scholes formula.
- SourceAvailable from: Claude Martini[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.06/2010;
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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.Working paper, London Business School. 07/2012;
Article: The Quanto Adjustment and The Smile[Show abstract] [Hide abstract]
ABSTRACT: European quanto derivatives are usually priced using the well-known quanto adjustment corresponding to the forward of the quantoed asset under the assumptions of the Black–Scholes model. In this article, I present the quanto adjustment corresponding to the local volatility model that allows pricing quanto derivatives consistently with the observed market equity skew and exchange rate smile. I then examine the model risk arising in the standard quanto adjustment by fitting the local volatility model to market data and then comparing the prices of European quanto euro derivatives on the Nikkei 225 index with those generated by the standard quanto adjustment. The results show that the standard quanto adjustment can be subject to significant pricing errors when compared with the local volatility model. I also compare the pricing performance of the local volatility model with a multivariate stochastic volatility model. The results show that when the correlation between the instantaneous variances associated with the underlying asset and the exchange rate is close to one, as it is the case when we consider historical data, there is little evidence of model risk for the local volatility model in the pricing of European quanto euro derivatives on the Nikkei 225 index.Journal of Futures Markets 01/2011; · 0.46 Impact Factor