On Pricing Rainbow Options

The rainbow option is an option whose payoff depends on two or more underlying assets, which has played an important role in the financial innovation. The maximum call option on n assets is an example of the rainbow opion, whose payoff at maturity is max(max(S1*,S2*,...,Sn*)-K,0), where the asterisk
denotes prices at maturity and K denotes the strike price. The exact formula for the maximum option in the case of 2 underlying assets was first derived by Stulz and Johnson independently. The exact formula for the maximum option in the case of n underlying assets was later derived by Johnson (1987). However, the real challenges deploying the formula are to evaluate the multi-variate normal distribution effeciently and to calculate the Greeks for hedging purpose. Therefore, it is necessary to use some numerical methods.

In this thesis, the Monte Carlo simulation is applied to price the rainbow options, and a more efficient Monte Carlo simulation is applied to reduce the variance of the simulation. Two variance-reduction techniques, antithetic variates and control variates are used. In control variates, the geometric average basket option (GABO) is applied as the intermediary in pricing the rainbow option, and GABO is proposed to reduce the variance of the Monte Carlo simulation.

In calculating the Greeks, it is known that the interchange of integration and differentiation is valid under some conditions. However, the problem is to differentiate the discontinuous payoff function. The problem is solved by introducing the idea of generalized functions, and the analysis in differentiating the payoff function is simplified and completed by the introduction of the dirac function. Our methods lead to the the results that the estimate for the Greeks is unbiased and that the computing time can be reduced to half of the computing time of the method by finite difference.

For rainbow options, this more efficient Monte Carlo simulation can be applied to calculate the price and the Greeks with token modifications.