Backus, D., S. Foresi, K. Lai, and L. Wu, 1997, #Accounting for Biases in Black-Scholes," mimeo, New York University.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....spaced closer, than far apart. The market perception of the price of jump risk 14 is embedded in the evolution of the implied volatility curve #Rubinstein #1994##. The following result#that relates the implied volatility function to the risk neutral moments#is borrowed with some modi#cation from Backus, Foresi, Lai, and Wu #1997#. As in Longsta# #1995#, it hinges on an approximate representation of any risk neutral density in terms of the Gaussian. Theorem 4 Let ##y; t; # # denote Black Scholes implied volatility #as recovered by solving #28##. Then, for a given moneyness, the implied volatility is a#ne in the ....

....set ##0 and reexamine #63#. In sum, while leverage generates negativeskew, its implications for index skewness are diametrically opposite to those originating from risk aversion and fat tailed physical distributions. 2 34 Proof of Equation #29# in Theorem 4 Although the proof is available in Backus, Foresi, Lai, and Wu #1997#, wesketch the basic steps to make our analysis self contained. To justify the functional form #29#, standardize stock returns so that they have mean zero and unit variance. Accordingly, let x # R#t;#### # # , where, as before, # #E # t #R#t; # ##, and # # # p E # t fR#t; ## #E # t ....

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Backus, D., S. Foresi, K. Lai, and L. Wu, 1997, #Accounting for Biases in Black-Scholes," mimeo, New York University.

....uncorrelated with the underlying asset price, they show that the price of a European option is the BS price integrated over the probability distribution of the average variance during the life of the option 4 . Unfortunately, however, this framework 3 Alternatively, Corrado and Su (1996) and Backus, Foresi, Li and Wu (1997) adapt a GramCharlier series expansion of the normal density function to obtain skewness and kurtosis adjustment terms for the BS formula. Eberlein, Keller and Prause (1998) introduce the hyperbolic density to account for excess kurtosis and skewness, and they are even able to obtain a closed ....

Backus, D., Foresi, S., Li, K, and L. Wu (1997). "Accounting for biases in BlackScholes ", Working Paper, Stern School of Business, New York University.

....reflects asymmetry in the risk neutral distribution of the underlying stock return, as measured by skewness, and the 1 See, for example, Ait Sahalia and Lo (1998) Jackwerth and Rubinstein (1996) and Rubinstein (1994) for empirical documentation of this phenomenon in S P 500 Index options. 2 Backus, Foresi, and Wu (1997) document this pattern for major currency options, although the flattening occurs at a much slower rate than implied by i.i.d innovations. 3 For example, Ait Sahalia and Lo (1998) use the ratio of strike over futures on the horizontal axis of their plots of the implied volatility smirk. ....

....which is a standard example of a Levy process, i.e. a process with stationary independent increments. It is well known that the Black Scholes model has a free scale parameter # which controls the width of the risk neutral distribution. Our Levy # stable process is also a Levy 4 See, for example, Backus, Foresi, and Wu (1997) and Das and Sundaram (1999) on the convergence to normality under stochastic volatilities. 2 process, but in addition to a free scale parameter #, the process also has an additional free parameter # # [0, 2] which controls the tails of the distribution. In particular, setting # = 2 causes our ....

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Backus, David, Silverio Foresi, and Liuren Wu, 1997, Accounting for biases in black-scholes, Working paper, New York University.

....23 Conversely, given a time series of option prices it is possible to infer a time series of volatilities implied by a particular option pricing model, see, e.g. Dumas, Fleming and Whaley (1998) for a recent discussion of S P500 implied volatilities. More closely related to the present paper, Backus, Foresi, Li and Wu (1998) provide an intriguing characterization of the salient biases in the Black Scholes pricing formula, along with an analysis of the corresponding implied volatilities for a set of foreign currency options. 24 Equally significant short run autocorrelations are evident in the implied daily FX ....

....characterization of the salient biases in the Black Scholes pricing formula, along with an analysis of the corresponding implied volatilities for a set of foreign currency options. 24 Equally significant short run autocorrelations are evident in the implied daily FX volatilities analyzed by Backus, Foresi, Li and Wu (1998). 14 signs of the corresponding two daily returns. The distributions on negative return days are almost indistinguishable from those on positive return days. Similarly, the distributions of corr t conditional on the signs of the two daily returns, which we show in the bottom panel of Figure ....

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Backus, D., S. Foresi, K. Li, and L. Wu (1998), "Accounting for Biases in Black-Scholes," Manuscript, Stern School, New York University.

....excess kurtosis for all maturities. Still, it provides a useful starting point for thinking about the role of jumps in pricing fixed income derivatives. In related work on currencies, we have found a Gram Charlier expansion to be a more tractable non normal distribution for pricing options; see Backus, Foresi, Li, and Wu (1998). Most of that work can be translated directly to fixed income. 8 Multifactor Models We turn now to multifactor models, in which bond yields are governed by the movements in two or more state variables. The motivation for such models should be clear from Section 4.4: single factor models cannot ....

....precise estimates of their values. We could use, for example, estimates of the conditional variance or unconditional higher moments of the short rate. None of the alternatives are easy, and none work very well. Interested readers might consult Chen and Scott (1993) Duffie and Singleton (1997) or Backus, Foresi, Mozumdar, and Wu (1998). The following models can be viewed as attempts to restrict the two factor affine model further, thereby simplifying the estimation process. Longstaff and Schwartz The Longstaff and Schwartz (1992) model is a special case of the two factor CIR model in which one of the risk parameters has been ....

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Backus, David, Silverio Foresi, Kai Li, and Liuren Wu, 1998, "Accounting for biases in Black-Scholes," manuscript, Stern School of Business, New York University.

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