An introduction to quasirandom numbers （介绍quasirandom数字）.pdfVIP

An introduction to quasi-random numbers By George Levy, Numerical Algorithms Group Ltd. Introduction Monte-Carlo simulation and random number generation are techniques that are widely used in financial engineering as a means of assessing the level of exposure to risk. Typical applications include the pricing of financial derivatives and senario generation in portfolio management. In fact many of the financial applications that use Monte-Carlo simulation involve the evaluation of various stochastic integrals which are related to the probabilities of particular events occurring. A case in point is the pricing of a simple European option, where the value of a call option is _ rT _ T rT T V e E[max(( S X )),0)] while the value of a put is V e E[max(( X S ),0)] . Here c p X is the strike price,T is the maturity of the option, r is the risk free interest rate, S T is the market _ value of the asset at maturity and E[] denotes the expectation operator. The value of a European put is therefore, V e rT p (S T ) max(( X S T ),0) dS T where p (S T ) is p the probability that the asset will have market value S T at maturity. If it is assumed that the value of the asset follows geometric Brownian motion andp (S T ) is the lognormal distribution then the Black-Scholes formula [1] can be used to price the options as follows: V S 0 N (d ) e rT X N (d ), c 1 2 V S 0 N (d ) e rT X N (d )), p