摘要
隨著台灣金融市場的開放，各式各樣的新金融商品逐漸誕生。而所謂的彩虹選擇權 (the rainbow option)，是指這種選擇權在到期日的報酬是與多個標的資產有關，就如同彩虹是由好幾個不同的顏色所組成。 彩虹選擇權中的一個例子是極大值選擇權 (the maximum option)，到期日的報酬為 max (max (S1,S2, … , Sn ) – K , 0)，其中標的資產數目為n，執行價為K；對於兩個標的物的極大值選擇權解析解1982年由Stulz解出，而1987年由Johnson推廣至對於任n個標的物。然而，這個解析解，必須計算n維多元常態累積分配；而多元常態累積分配涉及的多重積分，至今仍無法有效的計算。因此，我的論文的目的便是要找出一般的彩虹選擇權的定價方式，以及她的避險係數 (the Greeks) 的計算方式。 據無套利理論，彩虹選擇權的價格是風險中立下期望值的折現值。因此，以蒙地卡羅模擬的值估計期望值，並提出了兩個加速收斂方法，反向變異法 (antithetic variates)及控制變異法 (control variates)；在控制變異法中，幾何平均選擇權 (the geometric average basket option, GABO)，成為為一個很好的媒介。此外，避險係數的估計，也是在設計新金融商品不可缺少的一環，這裡提出了一個新的方法，可以有效的估計並且減少約一半的計算時間。
並列摘要
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.