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By Jianwei Zhu

The sound modeling of the smile impression is a vital factor in quantitative finance as, for greater than a decade, the Fourier rework has confirmed itself because the most productive device for deriving closed-form alternative pricing formulation in a variety of version sessions. This ebook describes the functions of the Fourier rework to the modeling of volatility smile, via a accomplished therapy of choice valuation in a unified framework, overlaying stochastic volatilities and rates of interest, Poisson and Levy jumps, together with quite a few asset sessions equivalent to fairness, FX and rates of interest, in addition to numerous numberical examples and prototype programming codes. Readers will make the most of this publication not just via gaining an summary of the complex idea and the immense diversity of literature on those subject matters, but additionally by means of receiving first-hand suggestions at the useful functions and implementations of the idea. The ebook is geared toward monetary engineers, chance managers, graduate scholars and researchers.

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Example text

Given so many Delta volatilities, we can therefore establish a fine FX volatility surface. More interestingly, some data sponsors and brokers do not deliver directly Delta volatilities, but the so-called butterfly volatility σBF , the so-called risk-reversal volatility σBF and ATM volatility σAT M . The market conventions of the butterfly volatility σBF and the risk-reversal volatility σBF are parallel to the trading strategies named butterfly and risk-reversal, and given by σRR = σ15C − σ15P , 1 σBF = (σ15P + σ15C ) − σAT M .

From the point of view of financial engineering, the immediate gain of this modular concept is its convenient implementation. For example, we can define three basic classes for stochastic volatilities, stochastic interest rates and jumps. The particular specification of these stochastic factor may be implemented in the derived classes. Therefore, the implementation of a special new pricing model is just a simple assemble of three concrete classes where the corresponding CFs are implemented. Chapter 3 Stochastic Volatility Models In this chapter we discuss stochastic volatility models that play a crucial role in smile modeling not only on the theoretical side, but also on the practical side.

4) In this general setting, the dynamics of the stock price S(t) are driven by a pure diffusion W (t) as in the simple Black-Scholes model. The essential extensions here are a stochastic interest rate r(t) and a stochastic volatility term b(·), that will be specified in a more correct modeling. As documented in a number of empirical studies, pure diffusive movements are not good enough to capture all statistical and implied characteristics of asset prices. Different types of jumps may be a good complement to the above setting, and will be discussed later.

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