The Complex Side of Financial Derivatives
Team Latte
27^{th} October 2013
In the world of
financial derivatives complex numbers appear via the probability function. As
we all know that valuation of financial derivatives involve estimation of the
probability – the risk neutral probability – of a certain event happening, such
as the stock price finishing in the money. Option pricing models require the
estimation of probability density function (pdf) of the natural logarithm of
the stock price, i.e. ,
where, is
the stock price. However, most often it is much easier to estimate the
characteristic function of than
the probability density function.
It is much more
desirable to work with characteristic functions rather than the probability
density functions as far as valuation of financial derivatives are concerned.
Why is this so? In derivatives valuation we need to know the dynamic process
for the underlying asset for hedging purposes. If we specify a dynamic process
for , the
natural logarithm of stock price and its initial conditions then we can compute
the probability distribution of over
the time interval .
However, the reverse is not true. If we are given a conditional distribution
for over
a particular time interval then
in essence there can be many processes that can generate this distribution.
This is the reason we need to work with the characteristic function because a
characteristic function uniquely and completely defines the distribution of a
random variable.
The
characteristic function of the natural logarithm of the stock price, is
given by
Where, is
the imaginary number and is
the probability density function of .
This is a complex integral, involving the imaginary number, . The
trick is that if we invert this characteristic function we get the probability
density function. Inversion of gives
us
As we can see
that the characteristic function is the Fourier transform of the probability
distribution function. The cumulative distribution function for
the logarithm of the stock price can be easily obtained from the above and is
given as follows:
Where, and is
the strike price of the option. Essentially, the above formula gives the risk
neutral probability that the call option will finish in the money, i.e. at
maturity the following holds: . In
the above formula, means
the real part of the complex function.
Reference:
Vainberg,
Gregory and Douglas, Fabrice, Option Pricing Models & Volatility, John
Wiley & Sons (2007).
Wu, Liuren From
Characteristic Functions and Fourier Transforms to PDFs/CDFs and Option Prices Zicklin
School of Business, Baruch College
Bhattacharya,
Rahul, The Book of Greeks, CFE School Publishing, Hong Kong
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