By Jianwei Zhu

The sound modeling of the smile impression is a crucial factor in quantitative finance as, for greater than a decade, the Fourier remodel has demonstrated itself because the best software for deriving closed-form alternative pricing formulation in numerous version sessions. This e-book describes the purposes of the Fourier rework to the modeling of volatility smile, by way of a complete therapy of alternative valuation in a unified framework, overlaying stochastic volatilities and rates of interest, Poisson and Levy jumps, together with numerous asset sessions comparable to fairness, FX and rates of interest, in addition to a number of numberical examples and prototype programming codes. Readers will make the most of this ebook not just by means of gaining an outline of the complex conception and the giant diversity of literature on those issues, but additionally by means of receiving first-hand suggestions at the useful purposes and implementations of the idea. The e-book is aimed toward monetary engineers, danger managers, graduate scholars and researchers.

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

**Sample text**

This phenomena called the volatility smile contradicts the assumption in the Black-Scholes model that volatility is constant over all strikes and maturities, and at the same time, indicates that the constant volatility in the Black-Scholes model should be too restrictive to capture the true market movements. Implied volatility is then a “wrong” quantity generated by a “wrong” pricing tool with a right market price. Generally, volatility smile has two basic patterns: symmetric smile and skew. Symmetric smile is referred to as the case where implied volatility increases if strike moves away from the stock spot price.

52) with which the CF f1 (φ ) may be derived for initial condition. If the risk-free interest rate r is constant as in the Heston model, we have shown that Q2 is identical to Q, this means that x(t) and V (t) in the Heston model under the measures Q2 are governed by the original risk-neutral processes respectively and 1 dx(t) = r − V (t) dt + 2 V (t)dW1Q2 dV (t) = κ [θ −V (t)]dt + σ V (t)dW2Q2 . It follows the corresponding Kolmogorov’s backward equation under the measure Q2 , 1 ∂ 2 f2 1 ∂ f2 ∂ 2 f2 ∂ 2 f2 = V + σ 2V + ρσ V 2 ∂T 2 ∂x ∂ x∂ V 2 ∂V 2 ∂ f2 1 ∂ f2 + κ (θ −V ) .

1 Properties of Characteristic Functions The characteristic function of a random variable is essentially a Fourier transform mapping the real random variable into a frequency domain. Therefore, CFs f j (φ ) derived above share any property of Fourier transform, and also any property of integral operator. We now briefly examine some special features of CF with respect to probability. The first useful property of CF lies in dealing with independent random variables. Let X1 , X2 , · · · , Xn , be some independent, but not necessarily identical random variables, and Yn = ∑nj=1 X j is the sum of these variables, then the CF of Yn is just the product of the CF of each X j , namely n f (φ ;Yn ) = ∏ f (φ , X j ).