Abstract Essay 1: Enhancing the Computational Efficiency for the Monte Carlo Simulation Approach Abstract…………………………………………………………………………………………………….1 1. Introduction………………………………………………………………………….…………………..2 2. Monte Carlo simulation…………………………………………………………………………………3 3. The Extended Grant, Vora and Weeks method………………………………………………………….5 4. Numerical results………………………………………………………………………………………..9 5. Conclusion..…………………………………………………………………………………………….11 References………………………………………………………………….…………………………….13 Figures and Tables………………………………………………………………………………………...14 Essay 2: Valuing Moving-Average Reset Options Abstract……………………………………………………………………………….…………………..22 1. Introduction…………………………………………………………………………………………….23 2. Monte Carlo Simulation………………………………………………………………………………..24 3. The GVW Method……………………………………………………………………………………..25 4. Valuing Moving-Average Reset Options with GVW…………………………………………………..27 5. Optimal Reset ratio and Liquidity Cost………………………………………………………………..28 6. Numerical analysis……………………………………………………………………………………..29 7. Conclusion……………………………………………………………………………………………..32 Appendix: Empirical martingale simulation……………………………………….……………………..33 References………………………………………………………………………….……………………..34 Figures and Tables………………………………………………………………………………………...35 Essay 3: Valuing Euro-Convertible Bond Abstract……………………………………………………………………………….…………………..51 1. Introduction………………………………………..………………….………………………………..52 2. The ECB…………………………………………..……………...…………………………………….52 3. Monte Carlo Simulation.……………………………………..…….……………..……………………55 4. Numerical analysis…….……………………………..………………….……………………………..58 5. Conclusions………………..………………………………..………….……..….…………………….59 References………………………………………………………..….…….……….……………………..60 Figures and Tables……………………………………………………..……………..…………………...61 List of Figures Essay 1: Enhancing the Computational Efficiency for the Monte Carlo Simulation Approach Figure 1 The regions of early exercise for the minimum put options……………………………………14 Figure 2 The small square regions in two-dimension for early exercise decisions………………………14 Figure 3 The volatility change in the case of call options on maximum of two risky assets…………….15 Figure 4 The volatility change in the case of put options on minimum of two risky assets……………...15 Figure 5 The change of correlation in the case of call option on maximum of two risky assets…………16 Figure 6 The change of correlation in the case of put option on maximum of two risky assets………….16 Essay 2: Valuing Moving-Average Reset Options Figure 1 Reset ratio and values of call options…………………………………………………………...35 Figure 2 Reset ratio and values of put options……………………………………………………………35 Figure 3 Reset ratio and values of basket reset options…………………………………………………..36 Figure 4 The effects of different moving average days and volatility to values of call options………….36 Figure 5 The effects of different moving average days and volatility to values of put options…………..37 Figure 6 The effects of different moving average days and volatility to values of basket reset options…37 Figure 7 Reset frequencies effects to values of call and put reset options………………………………..38 Figure 8 Reset frequencies effects to values of basket reset options……………………………………..38 Figure 9 Reset ratio and values of basket reset options…………………………………………………..39 Figure 10 Expected liquidity cost of different parameters and optimal reset ratio— Call options case…...39 List of Tables Essay 1: Enhancing the Computational Efficiency for the Monte Carlo Simulation Approach Table 1 American Put Option Prices using different methods……………………………………………17 Table 2 American Put Option on the Minimum of Two assets…………………………………………...18 Table 3 The Critical Price Sets for American Put Options on Minimum of Two Risky Assets………….18 Table 4 Sensitivity analysis: the case of different strike prices…………………………………………...19 Table 5 The Sensitivity Analysis: the case of volatility effects…………………………………………..19 Table 6 Correlation effects………………………………………………………………………………..20 Table 7 Hedge Ratio for American Put Options………………………………………………………….21 Essay 2: Valuing Moving-Average Reset Options Table 1 Comparisons between standard simulation and empirical martingale simulation….……………40 Table 2 Call options prices and probability with/without reset one time and different reset ratio……….41 Table 3 Put option price and probability with/without reset one time and different reset ratio………….42 Table 4 Basket call option prices and probabilities with/without reset one time and different reset ratios………………………………………………………………………………………………..43 Table 5 Call options prices and probabilities with different stock price volatilities and different moving average stock price………………………………………………………..……………………..44 Table 6 Put options prices and probabilities with different stock price volatilities and different moving average stock price…………………………………………...…………………………………..45 Table 7 Basket call options prices and probabilities with different stock price volatilities and different moving average stock price…………………………………………….……………………….46 Table 8 The effects of reset frequencies to call options prices and reset probabilities…………………...47 Table 9 The effects of reset frequencies to put options prices and reset probabilities……………………48 Table 10 The effects of reset frequencies and coefficients to basket call options prices and reset probabilities……………………………………………………..………………………………….49 Table 11 Basket call option prices and probabilities with/without reset one time and different reset ratios………………………………………………………………………………………………………50 Essay 3: Valuing Euro-Convertible Bond Table 1Pure American option value: Comparisons between tree method and LSM……………………..61 Table 2 Benchmark parameters…………………………………………………………………………..62 Table 3 The effect of exchange rate volatility and stock price volatility to the value of ECB…………...63 Table 4 The values of conversion option and embedded currency option in ECB……………………….64 Table 5 Effects of state variable correlations to the values of conversion option and embedded currency option in ECB…………………………………………………………………………………...65