By Jianwei Zhu
The sound modeling of the smile impact is a crucial factor in quantitative finance as, for greater than a decade, the Fourier remodel has demonstrated itself because the most productive instrument for deriving closed-form choice pricing formulation in quite a few version periods. This booklet describes the purposes of the Fourier remodel to the modeling of volatility smile, by way of a finished therapy of alternative valuation in a unified framework, overlaying stochastic volatilities and rates of interest, Poisson and Levy jumps, together with a number of asset sessions resembling fairness, FX and rates of interest, in addition to a number of numberical examples and prototype programming codes. Readers will take advantage of this publication not just through gaining an outline of the complicated concept and the great diversity of literature on those subject matters, but in addition by means of receiving first-hand suggestions at the useful purposes and implementations of the idea. The publication is aimed toward monetary engineers, threat managers, graduate scholars and researchers.
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Extra resources for Applications of Fourier Transform to Smile Modeling: Theory and Implementation
The models following this approach are the so-called stochastic volatility models. 1) gives an overview of some representatives of stochastic volatility models. 1 except for model (5) [Sch¨obel and Zhu, 1999], model (6) [Heston, 1993] as well as model (7). Models (1) and (3) perform no mean-reversion property and therefore can not capture the effects of shocks to volatility in the valuation of options. 1 Overview of one-factor stochastic volatility models. of volatility or variance. The so-called 3/2 model of Lewis (2000) is unbounded, and as a result, option prices under the 3/2 model are not martingales, but only local martingales, therefore not appropriate for practical applications.
The foreign interest rate r∗ (t) is governed by a similar process as r(t). In the following, we use a simple functional form of b(v(t),t) for general discussion, namely b(v(t),t) = v(t). A popular choice for b(·) may be b(v(t),t) = V (t) as in the Heston model (1993). Another more complicated and still tractable form for b(·) may be the linear affine structure suggested by Dai and Singleton (2000). 2 Valuation of European-style Options via CFs The general process for stock price given above includes the Black-Scholes world, stochastic interest rates, stochastic volatilities, and can be extended by adding random jumps.
Scaling: f (aφ ) ⇐⇒ • Time reversal: 1 (y1 ∗ y2 )(x). 2π 1 y(x/a), |a| a ∈ R, a = 0. f (−φ ) ⇐⇒ y(−x). This feature can be considered as a special case of the scaling feature. • Time shift: f (φ ) ⇐⇒ y(−x). Note that f1 and f2 here are two arbitrary CFs, not same as the first and second CFs given above in the context of option pricing. The moments of a random variable can also be derived from its characteristic function. Thus, some important statistical parameters such as variance, skewness and kurtosis are automatically available if the characteristic function is known.
Applications of Fourier Transform to Smile Modeling: Theory and Implementation by Jianwei Zhu
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