By Jianwei Zhu

ISBN-10: 3642018076

ISBN-13: 9783642018077

The sound modeling of the smile impact is a crucial factor in quantitative finance as, for greater than a decade, the Fourier remodel has confirmed itself because the best instrument for deriving closed-form choice pricing formulation in a number of version periods. This ebook describes the functions of the Fourier rework 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 equivalent to fairness, FX and rates of interest, in addition to quite a few numberical examples and prototype programming codes. Readers will take advantage of this publication not just by way of gaining an outline of the complicated conception and the significant variety of literature on those issues, but additionally via receiving first-hand suggestions at the useful functions and implementations of the speculation. The booklet is geared toward monetary engineers, probability managers, graduate scholars and researchers.

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**Additional info for Applications of Fourier Transform to Smile Modeling: Theory and Implementation**

**Example text**

To expound this interesting feature of CF in the context of option pricing, we define a discounted CF (also called the CF of the remaining uncertainty):6 The contingent claim with payoff cos(φ xT ) or sin(φ xT ) is an “asset” with unlimited liability since cos(φ xT ) and sin(φ xT ) can become negative. 5 It is well-known that normal Arrow-Debreu prices should (must) be positive and smaller than one. Since the transformed spanned space is the complex plane, we can impose the usual L1 -norm on f1 and f2 .

Quotations with implied volatilities clarify market situation and deliver a more clear and transparent picture about market sentiments because implied volatility standardizes option price in year, and is independent of the absolute values of spot stock prices. Usually it is quoted in terms of strike (or moneyness K/S0 ) and maturity. 2. Market participants can compare the relative expenses of different options easily via implied volatility and can check the arbitrage conditions more conveniently.

Since the transformed spanned space is the complex plane, we can impose the usual L1 -norm on f1 and f2 . It follows immediately that | f1 | ≤ 1 and | f2 | ≤ 1. Thus, f1 and f2 are well-defined. 6 See Bakshi and Madan (2000). 2 Understanding Characteristic Functions 33 T f ∗ (φ ; x(T )) = EQ exp − 0 r(t)dt exp(iφ x(T )) . 34) Setting φ = 0, we have T f ∗ (0; x(T )) = EQ exp − r(t)dt = B(0, T ). 13) can be expressed by f2 (φ ) = 1 f ∗ (φ ; x(T )) = f ∗ (φ ; x(T )). 37) where x(T ) = ln S(T ). 28), we can construct the moments of the random variable x(T ) by partially differentiating the CF with respect to φ .

### Applications of Fourier Transform to Smile Modeling: Theory and Implementation by Jianwei Zhu

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