| Multi-asset derivative pricing using quasi-random numbers (2008) | |||||||||||||
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| In a previous Financial Engineering News article [1] the author gave introductory details concerning the use of quasi-random numbers for Monte Carlo simulation. The benefits to be gained by using quasirandom numbers instead of pseudo-random numbers were illustrated by using different random sequences to estimate the known value of a six-dimensional integral. The sequences were obtained using both the NAG pseudo-random number generators and the NAG quasi-random generators [2][3], and the integral was estimated by uniformly sampling the six-dimensional unit hypercube. The results illustrated that quasi-random numbers can give more accurate integral estimates. In this article we present results of using NAG software to estimate the value of some (simple) financial derivatives. Here instead of generating uniform pseudo-random and uniform quasi-random sequences we generate multivariate Normal pseudo-random and multivariate Normal quasi-random sequences with a given mean and covariance matrix. The current price of the financial derivative is estimated by evaluating an integral which represents the expected (discounted) value of the derivative’s pay-off at maturity. First we will briefly consider the Black Scholes option model. This assumes that the process followed by the underlying asset, S, is: dS = r S dt + σ S dX, where dS is the change in the asset price over the time interval dt, r is the risk free interest rate, σ is the volatility of the asset S, and dX is drawn from a Normal distribution with mean zero and variance dt. Using Ito’s lemma [4] we then find that the process followed by Y = log(S) is: dY = ( r − σ 2 / 2) dt + σ dX, where dY is the change in value of log(S) over the time interval dt. It can also be shown that the value of an option, V, written on S, satisfies the following (Black Scholes) partial differential equation: | |||||||||||||
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