Chapter 2
Characteristic Functions in Option Pricing
We consider in this chapter a general It
ˆ
o process for stock prices. In a generalized
valuation framework for options, the distribution function of stock price is analyti-
cally 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 tractabil-
ity, 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. The pricing formula for European-style options may be expressed in a
form of inverse Fourier transform. As a result, we obtain a generalized principle for
valuing options under the risk-neutral measure via characteristic functions. There is
a corresponding extension for FX options.
Next in Section 2.2.1, we first demonstrate some nice properties of characteristic
function. As conditional expected value, characteristic function shares all proper-
ties 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 characteris-
tic function of the sum of stochastic factors is just a product of the characteris-
tic function of each stochastic factor. In Section 2.2.2, we follow an approach of
Bakshi and Madan (2000) and interpret characteristic functions as Arrow-Debreu
prices in a Fourier-transformed space. As shown in Section 2.2.3 by some examples,
most popular pricing models, particularly, stochastic volatility models, admit ana-
lytical 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 distribu-
tion. Additionally, we can establish an equivalence between the Fourier transform
approach and the traditional PDE approach where the Feynman-Kac theorem and
Kolmogorov’s backward equation play a central role. Finally, as one of many advan-
tages of the Fourier transform approach, it allows us a modular valuation for options,
with which all relevant independent stochastic factors such as interest rates, volatil-
ities and jumps may be dealt with as a module in an unified valuation procedure.
J. Zhu, Applications of Fourier Transform to Smile Modeling, Springer Finance,
DOI 10.1007/978-3-642-01808-4
2,
c
Springer-Verlag Berlin Heidelberg 2000, 2010
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