Risk Latte - The Complex Side of Financial Derivatives

The Complex Side of Financial Derivatives

Team Latte
27th 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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