By Jianwei Zhu

The sound modeling of the smile influence is a crucial factor in quantitative finance as, for greater than a decade, the Fourier remodel has verified itself because the most productive device for deriving closed-form choice pricing formulation in quite a few version periods. This ebook describes the purposes of the Fourier remodel to the modeling of volatility smile, through a finished remedy of choice valuation in a unified framework, masking stochastic volatilities and rates of interest, Poisson and Levy jumps, together with a variety of asset sessions resembling fairness, FX and rates of interest, in addition to numerous numberical examples and prototype programming codes. Readers will reap the benefits of this e-book not just by means of gaining an outline of the complex conception and the substantial diversity of literature on those themes, but in addition by way of receiving first-hand suggestions at the sensible purposes and implementations of the idea. The e-book is aimed toward monetary engineers, hazard managers, graduate scholars and researchers.

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**Extra resources for Applications of Fourier Transform to Smile Modeling: Theory and Implementation**

**Sample text**

In a generalized valuation framework for options, the distribution function of stock price is analytically unknown. To express (quasi-) closed-form exercise probabilities and valuation formula, characteristic functions of the underlying stock returns (logarithms) are proven to be not only a powerful and convenient tool to achieve analytical tractability, but also a large accommodation for different stochastic processes and factors. In first section, we derive two important characteristic functions under Delta measure and forward measure respectively, under which two exercise probabilities can be calculated.

There is a corresponding extension for FX options. 1, we first demonstrate some nice properties of characteristic function. As conditional expected value, characteristic function shares all properties of integration operator and expectation operator. The most important property of characteristic function with respect to setting up a comprehensive option pricing framework is that if stochastic factors are mutually independent, the characteristic function of the sum of stochastic factors is just a product of the characteristic function of each stochastic factor.

2, we follow an approach of Bakshi and Madan (2000) and interpret characteristic functions as Arrow-Debreu prices in a Fourier-transformed space. 3 by some examples, most popular pricing models, particularly, stochastic volatility models, admit analytical solutions for characteristic functions, and therefore, also analytical solutions for valuation formulas in terms of inverse Fourier transform. The Black-Scholes formula can be easily verified with the characteristic functions of normal distribution.