Complex Numbers in Finance (Excepts from CFE Lectures) Complex numbers are made up of imaginary numbers (there is a real part and an imaginary part in a complex number). An example of a complex number is 3+2i where, i is the imaginary part and is equal to √(-1) (square root of minus one). An imaginary number, such as square root of minus one (√(-1)) or square root of minus 5 (√(-5)) is about as abstract as it gets in mathematics. No one can find the square root of a negative number – it does not exist – and hence they are called “imaginary”. If complex numbers are such abstract entities then how come they are used so pervasively in engineering and physics? Nature is teeming with complex numbers. In fact, most of engineering and physics – disciplines that deal with the real world around us – would not be possible without the use of complex numbers. A remarkable property of complex numbers is that they can be used in algebraic calculations in such a way that they produce very tangible solutions. When operated on a complex number produces a solution that is in the real space. What about Finance? Well, finance is no exception. Even here we come across complex numbers. It may sound strange but it’s true that the world of dollars and cents contains something as abstract as complex numbers. Complex numbers enter finance via the probability distribution. Option pricing problem entails working with the probability density function of the logarithm of the stock price, X=ln(S), where S is the price of the stock on which the option is written. Probability density function enters into the option pricing model – such as the Black-Scholes model – and this function needs to be evaluated to arrive at the price of an option or a financial derivative. This is the classical approach, one that was taken by the early pioneers, such as Fischer Black and Myron Scholes. However, often times probability density functions are not easy to obtain. In such cases, working with characteristic function of a probability distribution becomes much easier. Even otherwise, in most cases, characteristic function of a probability distribution is easier to calculate. A characteristic function completely defines a probability distribution. A characteristic function of the logarithm of the stock price is easy to obtain. This is where complex number enters. To evaluate the characteristic function of X=ln(S) we need to work with complex numbers as you can see from the adjoining equation. Once a characteristic function for the probability distribution is defined, we can easily find the density function by taking the inverse of the characteristic function. It may sound daunting and as you can see from the equation that it involves evaluating complex integrals, it is actually quite simple and easily doable in Excel™. This approach of taking the characteristic function of the distribution of the logarithm of stock price to calculate the option price has been taken by many prominent derivatives practitioners and academics over the past decade and a half, especially when it has come to evaluating closed form solutions of stochastic volatility models, such as the Heston model. The Fourier transform method (FFT method) for evaluation of option prices also involves working with complex integrals and is now universally accepted as an efficient and fast method for option pricing.
Posted on: Tue, 06 Aug 2013 03:52:23 +0000
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