本文將兩種樹模型(tree models)，一種蒙地卡羅模擬方法(Monte Carol method)和兩種靜態避險組合方法(static hedging portfolio methods)拓展並運用于歐式連續式雙障礙選擇權(European continuous double barrier options)及其報償(rebates)的評價。通過數值試驗我們發現在短計算時間下BTT樹模型(the bino-trinomial tree model of Dai and Lyuu (2010))最為精確，對精度有更高要求時Ritchken三項樹模型計算最快(the modified trinomial tree model of Ritchken (1995))。同時我們發現有使用歐式cash-or-nothing二元選擇權(European cash-or-nothing binary options)的改進靜態避險組合方法(modified static hedging portfolio methods)相比Derman等人的靜態避險組合方法(static hedging portfolio methods of Derman et al.(1995))效率更好。

In this work, we extend and apply the modified trinomial tree model of Ritchken (1995), the bino-trinomial tree model of Dai and Lyuu (2010), the new Monte Carol method of Moon (2008), the static hedging portfolio method of Derman et al. (1995) and a new static hedging portfolio method to price European continuous double barrier options and their rebates. Numerical experiments show that all the above-mentioned methods generate satisfying convergence results, among them the bino-trinomial tree model of Dai and Lyuu (2010) is most efficient under tight time constraints and the modified trinomial tree model of Ritchken (1995) behaves most efficient when high accuracy is needed. The modified static hedging portfolio method with European cash-or-nothing binary options behaves better than the static hedging portfolio method of Dermal et al. (1995) for both pricing European continuous double barrier options and their rebates.

論文審定書 i
誌謝 ii
摘要 iii
ABSTRACT iv
1. INTRODUCTION 1
2. EUROPEAN CONTINUOUS DOUBLE BARRIER OPTION 5
2.1 MODEL ASSUMPTIONS 5
2.2 DEFINITIONS 5
2.3 IN-OUT PARITY 7
2.4 TRIVIAL AND NON-TRIVIAL CASES 7
2.5 REBATES 8
2.6 ANALYTICAL APPROXIMATION FORMULAS 9
3. TREE MODELS 12
3.1 BASIC BINOMIAL AND TRINOMIAL TREE MODELS 12
3.2 MODIFIED TRINOMIAL TREE MODEL OF RITCHKEN (1995) 14
3.2.1 Review of the Model 14
3.2.2 Algorithms and Numerical Results 17
3.3 BINO-TRINOMIAL TREE MODEL OF DAI AND LYUU (2010) 19
3.3.1 Review of the Model 19
3.3.2 Reflection Principles and Combinatorial Algorithm 21
3.3.3 Algorithms and Numerical Results 22
4. MONTE CAROL METHODS 25
4.1 STANDARD MONTE CAROL METHOD 25
4.2 MOON’S NEW MONTE CAROL METHOD (2008) 26
4.2.1 Review of the Model 26
4.2.2 Algorithms and Numerical Results 27
5. STATIC HEDGING PORTFOLIO METHODS 30
5.1 DERMAN’S SHP METHOD (1995) 30
5.1.1 Review of the Model 30
5.1.2 Algorithms and Numerical Results 32
5.2 THE NEW SHP METHOD 33
5.2.1 Review of the Model 33
5.2.2 Algorithms and Numerical Results 35
6. A COMPARISON OF NUMERICAL EFFICIENCY BETWEEN MODELS 38
6.1 NUMERICAL EXPERIMENTS BASED ON EUROPEAN DOUBLE BARRIER OPTIONS 38
6.2 NUMERICAL EXPERIMENTS BASED ON REBATES 42
7. CONCLUSION 44
REFERENCES 45
APPENDIX 96

[1] Andricopoulos, A. D., Widdicks, M., Duck, P. W., & Newton, D. P. (2003). Universal option valuation using quadrature methods. Journal of Financial Economics, 67(3), 447-471.
[2] Andricopoulos, A. D., Widdicks, M., Newton, D. P., & Duck, P. W. (2007). Extending quadrature methods to value multi-asset and complex path dependent options. Journal of Financial Economics, 83(2), 471-499.
[3] Carr, P., Ellis, K., & Gupta, V. (1998). Static hedging of exotic options. The Journal of Finance, 53(3), 1165-1190.
[4] Carr, P., & Wu, L. (2014). Static hedging of standard options. Journal of Financial Econometrics, 12(1), 3-46.
[5] Chen, D., Härkönen, H. J., & Newton, D. P. (2014). Advancing the universality of quadrature methods to any underlying process for option pricing. Journal of Financial Economics, 114(3), 600-612.
[6] Cheuk, T. H., & Vorst, T. C. (1996). Complex barrier options. The Journal of Derivatives, 4(1), 8-22.
[7] Choe, G. H., & Koo, K. H. (2014). Probability of multiple crossings and pricing of double barrier options. The North American Journal of Economics and Finance, 29, 156-184.
[8] Chung, S. L., Shih, P. T., & Tsai, W. C. (2013). Static hedging and pricing American knock-in put options. Journal of Banking & Finance, 37(1), 191-205.
[9] Derman, E., Ergener, D., & Kani, I. (1995). Static options replication. The Journal of Derivatives, 2(4), 78-95.
[10] Dai, T. S., Liu, L. M., & Lyuu, Y. D. (2008). Linear-time option pricing algorithms by combinatorics. Computers & Mathematics with Applications, 55(9), 2142-2157.
[11] Dai, T. S., & Lyuu, Y. D. (2010). The bino-trinomial tree: A simple model for efficient and accurate option pricing. The Journal of Derivatives, 17(4), 7-24.
[12] Figlewski, S., & Gao, B. (1999). The adaptive mesh model: a new approach to efficient option pricing. Journal of Financial Economics, 53(3), 313-351.
[13] Geman, H., & Yor, M. (1996). PRICING AND HEDGING DOUBLE‐BARRIER OPTIONS: A PROBABILISTIC APPROACH. Mathematical finance, 6(4), 365-378.
[14] Hui, C. H. (1996). One-touch double barrier binary option values. Applied Financial Economics, 6(4), 343-346.
[15] Kunitomo, N., & Ikeda, M. (1992). Pricing options with curved boundaries. Mathematical finance, 2(4), 275-298.
[16] Li, A. (1999). The pricing of double barrier options and their variations. Advances in Futures and Options Research, 10, 17-41.
[17] Lyuu, Y. D. (1998). Very fast algorithms for barrier option pricing and the ballot problem. The Journal of Derivatives, 5(3), 68-79.
[18] Moon, K. S. (2008). Efficient Monte Carlo algorithm for pricing barrier options. Communications of the Korean Mathematical Society, 23(2), 285-294.
[19] Nouri, K., & Abbasi, B. (2015). Implementation of the modified Monte Carlo simulation for evaluate the barrier option prices. Journal of Taibah University for Science.
[20] Pelsser, A. A. J. (1997). Pricing double barrier options: An analytical approach (No. TI 97-015/2). Tinbergen Institute Discussion Paper Series.
[21] Ritchken, P. (1995). On pricing barrier options. Currency Derivatives: Pricing Theory, Exotic Options, and Hedging Applications, 275-289.