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

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 verified itself because the most productive instrument for deriving closed-form choice pricing formulation in a number of version sessions. This e-book describes the purposes of the Fourier rework to the modeling of volatility smile, via 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 periods similar to fairness, FX and rates of interest, in addition to a variety of numberical examples and prototype programming codes. Readers will make the most of this ebook not just by way of gaining an summary of the complex conception and the immense diversity of literature on those issues, but additionally through receiving first-hand suggestions at the useful purposes and implementations of the speculation. The ebook is geared toward monetary engineers, chance managers, graduate scholars and researchers.

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Example text

As seen in the Black-Scholes model, x(t) is Gaussian and therefore has a skewness of zero and a kurtosis of three. It is empirically evident and also supported by smile effect of implied volatilities that the actual distribution of stock returns x(t) exhibits a negative skewness and an excess kurtosis. Therefore, in order to capture a leptokurtic distribution and to recover smile effect, a reasonable stock price process should imply m3 m4 < 0, > 3. 3/2 m22 (m2 ) These two statistics may be checked in advance by employing CF under the riskneutral measure or the original statistical measure.

Obviously, a moment-generating function is very similar to a CF, and there is an interchange between them by setting φ = −iη or η = iφ , f (−iη ) = mg f (η ), mg f (iφ ) = f (φ ). In this sense, the CF f (φ ) is equivalent to the moment-generating function, and therefore can be applied to calculate the moments of X(T ). To compute skewness and kurtosis, the first four central moments mn (x(T )) should be known and can be given by a recursive scheme (Stuard and Ord, 1994): m2 = m∗2 − (m∗1 )2 , m3 = m∗3 − 3m∗2 m∗1 + 2(m∗1 )2 , m4 = m∗4 − 4m∗3 m∗1 + 6m∗2 (m∗1 )2 − 3(m∗1 )4 .

Since the CF of a normal density function n(μ , σ ) is f (φ ) = exp iφ μ − 12 σ 2 φ 2 , two probabilities N(d j ), j = 1, 2, in the Black-Scholes formula are Gaussian with cumulative distribution functions N(rT + x0 ± 12 σ 2 T, 12 σ 2 T ) respectively. 2 Understanding Characteristic Functions 37 (2). The Heston Model (1993). 49) dW1 (t)dW2 (t) = ρ dt. Here the squared volatilities follow a mean-reverting square-root process which was used in finance for the first time by CIR (1985b) to specify stochastic interest rates.

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