ADVANCED NUMERICAL METHODS IN FINANCE

Advanced Numerical Methods in Finance



Advanced Numerical Methods in Finance

Question 1

A forward contract is a contract which obliges the holder to buy an underlying asset at some future time T (the maturity time) for a price K (the delivery price that is xed at contract initiation. Hence, at time T, when the stock price is ST , the contract is worth ST - K (the payo of the forward) to the holder. This payo off is shown in Figure 1.

A futures contract is a rather specialised forward contract, traded on an organised exchange, and such that, if a contract is traded at some time t < T, the delivery price is set to a special value Ft,T , called the futures price of the asset or the forward price of the asset, chosen so that the value of the futures contract at initiation (that is, at time t), is zero.

Question 2

The distribution of maturity values of the option can be obtained from the distribution of the terminal stock value. This fact, together with the assumption of risk-neutrality, is crucial to the numerical price valuation via Monte Carlo simulation. The expected return of the stock is the risk-free rate r, so the present value of the option can be obtained discounting the expected payoff at the risk-free rate of interest. Thus, the value at t is given by

Here, E denotes the expected value in a regime of risk-neutrality. The Monte Carlo method is used to obtain the expected value E[VT ]. A large number J of simulations of Brownian motion for the price of S are carried out starting from the present value. The payoffs for the option are calculated using the final values Sj(T). Compared with the typical rate of convergence of methods like the ones mentioned above or like the binomial method, the rate of convergence of Monte Carlo simulation is quite slow and decelerates in relation to the computational effort (Dempster, 2000). On the other hand, a Monte Carlo simulation is very easy to implement, it is more flexible then methods requiring grids and is robust in that its accuracy depends only on the crudest measure of the complexity of the problem.

Indeed the attractive feature of Monte Carlo simulation is that the rate of convergence is independent of the dimension of the problem. Since the need of valuation of complicated portfolios of options leads to high dimensional mathematical problems, that characteristic has made Monte Carlo simulation one of the most popular numerical methods in this field. Moreover, while when working in a high dimension with grid-based methods it becomes practically impossible to improve the accuracy because of the difficulty of refining the mesh, in principle every new Monte Carlo simulation improves the accuracy. The reason for this is that every simulation is itself an estimate of the correct price. Having said this, it remains of crucial importance to find ways to accelerate the convergence rate of Monte Carlo ...