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

By Jianwei Zhu

The sound modeling of the smile influence is a crucial factor in quantitative finance as, for greater than a decade, the Fourier remodel has verified itself because the most productive device for deriving closed-form choice pricing formulation in quite a few version periods. This ebook describes the purposes of the Fourier remodel 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 resembling fairness, FX and rates of interest, in addition to numerous numberical examples and prototype programming codes. Readers will reap the benefits of this e-book not just by means of gaining an outline of the complex conception and the substantial diversity of literature on those themes, but in addition by way of receiving first-hand suggestions at the sensible purposes and implementations of the idea. The e-book is aimed toward monetary engineers, hazard managers, graduate scholars and researchers.

Show description

Read or Download Applications of Fourier Transform to Smile Modeling: Theory and Implementation PDF

Similar economy books

The Pursuit of Wealth - The Incredible Story of Money Throughout The Ages

An epic saga that recounts the turbulent background of cash. The Pursuit of Wealth is an interesting 5,000 yr trip during the evolution of cash and making an investment. From probability as opposed to go back in Mesopotamia via trendy rough-and-tumble, high-stakes inventory markets, this behind-the-scenes story will intrigue either monetary historians and history-minded traders alike.

Handbook of Law and Economics, Vol. 2

Legislations might be seen as a physique of principles and felony sanctions that channel habit in socially fascinating instructions &151; for instance, by means of encouraging participants to take right precautions to avoid injuries or through discouraging rivals from colluding to elevate costs. The incentives created through the felony approach are therefore a normal topic of research via economists.

Urban-Rural Interactions: Towns as Focus Points in Rural Development

Sleek Europe has rural roots. Even this day, up to ninety consistent with cent of Europe (EU25) contains rural components within which half the inhabitants lives. whereas diverse rural components frequently face varied demanding situations, the shift from agricultural creation in the direction of a multifunctional panorama and the expanding worth assigned to environmental values have an effect on all rural parts.

Advances in Mathematical Economics Volume 19

Presents mathematicians with new stimuli from monetary theories, and economists with potent mathematical instruments for his or her research
Is released each year less than the auspices of the study middle for Mathematical Economics
Presents a uncomplicated expository evaluate of all difficulties less than discussion

The sequence is designed to compile these mathematicians who're heavily attracted to getting new demanding stimuli from fiscal theories with these economists who're looking powerful mathematical instruments for his or her examine. loads of financial difficulties should be formulated as restricted optimizations and equilibration in their options. a variety of mathematical theories were delivering economists with fundamental machineries for those difficulties coming up in monetary conception. Conversely, mathematicians were inspired through quite a few mathematical problems raised via financial theories.

Topics: online game idea, Economics, Social and Behav. Sciences, chance idea and Stochastic techniques

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

Sample text

In a generalized valuation framework for options, the distribution function of stock price is analytically unknown. To express (quasi-) closed-form exercise probabilities and valuation formula, characteristic functions of the underlying stock returns (logarithms) are proven to be not only a powerful and convenient tool to achieve analytical tractability, but also a large accommodation for different stochastic processes and factors. In first section, we derive two important characteristic functions under Delta measure and forward measure respectively, under which two exercise probabilities can be calculated.

There is a corresponding extension for FX options. 1, we first demonstrate some nice properties of characteristic function. As conditional expected value, characteristic function shares all properties of integration operator and expectation operator. The most important property of characteristic function with respect to setting up a comprehensive option pricing framework is that if stochastic factors are mutually independent, the characteristic function of the sum of stochastic factors is just a product of the characteristic function of each stochastic factor.

2, we follow an approach of Bakshi and Madan (2000) and interpret characteristic functions as Arrow-Debreu prices in a Fourier-transformed space. 3 by some examples, most popular pricing models, particularly, stochastic volatility models, admit analytical solutions for characteristic functions, and therefore, also analytical solutions for valuation formulas in terms of inverse Fourier transform. The Black-Scholes formula can be easily verified with the characteristic functions of normal distribution.

Download PDF sample

Rated 4.51 of 5 – based on 4 votes