數位選擇權結構簡單，且具有支付固定數額的特點。因此，數位選擇權常被應用於避險，或是被用來評價結構複雜的金融商品。基於數位選擇權的重要性，我們將幾個重要的數位選擇權評價模型以擴散過程(diffusion processes)與跳躍擴散過程(jump-diffusion processes)做分類，討論為何假設不同波動過程其背後的理由，並整理各模型的評價結果。
此外，在數值方法的部份，我們利用傅利葉轉換與拉普拉斯轉換之間可相互替代的關係，運用快速傅利葉逆轉換，將其套用在陳等於2007 年提出的模型所算出觸及數位選擇權的拉普拉斯轉換上，來求算觸及數位選擇權的價格。

Digital options, the basic building blocks for valuing complex financial assets, they play an important role in options valuation and hedging. We survey the digital options pricing formula under diffusion processes and jump-diffusion processes.
Since the existent first-touch digital options pricing formulas with jump-diffusion processes are all in their Laplace transform of the option value. To inverse the Laplace transforms is critical when doing options valuation. Therefore, we adopt a phase-type jump-diffusion model which is developed by Chen, Lee and Sheu [2007] as our main model, and use FFT inversion to get the first-touch digital option price under
(2,2)-factor exponential jump-diffusion processes.

Abstract .................................................................................................... ii
Acknowledgements ................................................................................. iv
Contents .................................................................................................... v
List of Tables ........................................................................................... vi
List of Figures ........................................................................................ vii
1 Introduction ............................................................................................ 1
2 Introduction to Digital Options and Preliminaries in Lévy Process ...... 3
2.1 Exotic Options and Digital Options ........................................................ 3
2.2 Lévy Process ........................................................................................... 4
2.2.1 Poisson Process .........................................................................................5
2.2.2 Compound Poisson Process ......................................................................6
2.2.3 Lévy-Khinchin formula ............................................................................7
3 Digital Options Pricing Models ............................................................. 9
3.1 Pricing Model under Diffusion Process .................................................. 9
3.1.1 The Ingersoll Model ..................................................................................9
3.2 Pricing Model under Jump-Diffusion Process ...................................... 11
3.2.1 Normal Jump-Diffusion Model ............................................................... 11
3.2.2 Double Exponential Jump-Diffusion Model ...........................................12
3.3 Main Model ........................................................................................... 13
4 Numerical Method ............................................................................... 16
4.1 Chen et al. ODE Approach for the First-Touch Digital Option ............ 16
4.2 Numerical Method ................................................................................ 18
4.2.1 Connection between the Laplace Transform and the Fourier Transform18
4.3 Valuation of First-Touch Digital Options.............................................. 19
4.3.1 The Characteristic Exponent of
(2,2)-Factor Exponential Jump-Diffusion Processes .............................20
4.3.2 Inverse of Discrete Fourier Transform ..................................................20
4.4 Result ............................................................................................................. 21
5Summary and Future Research ............................................................. 34
Reference ................................................................................................ 35

Andersen, L., Andreasen, J. and Eliezer, D., 2002, Static replication of barrier options:
some general results, Journal of Computational Finance 5, 1-25.
Barndoroff-Nielsen, O. E. 1977, Exponentially decreasing distributions for the
logarithm of particle size, Proc. Roy. Soc. LondonA 353, 401-419.
Bates, D. S., 1996, Jumps and statistic volatility: exchange rate processes implicit in
deutsche market options, Review of Financial Studies 9, 69-107.
Bertoin, J., 1996, Lévy processes, Cambridge Tracts in Mathematics, vol. 121.
Cambridge express.
Black, F. and M. Scholes, 1973, The pricing of options and corporate liabilities,
Journal of Political Economy 81, 637-659.
Carr, P. and Chou, A. 1997, Hedging complex barrier options, Working Paper,
Morgan Stanley and MIT Computer Science.
Carr, P., Ellis, K. and Gupta, V., 1998, Static hedging of exotic options, Journal of
Finance 53, 1165-1191.
Carr, P. and L. Wu, 2001, A simple robust test for the presence of jumps in assets
prices, Working Paper, Fordham University.
Chen, Y. T., Lee, C. F. and Sheu, Y. C., 2007, An ODE approach for the expected
discounted penalty at ruin in a jump-diffusion model, Finance and Stochastics 11,
323-355.
Chernov, M. and E. Ghysels, 2000, A study towards a unified approach to the joint
estimation of objective and risk neutral measures for the purpose of options
valuation, Journal of Financial Economics 56, 407-458.
Derman, E., Ergener, D. and Kani, I, 1995, Static option replication, Journal of
Derivatives 2, 78-95.
Duffie, D., J. Pan, and K. Singleton, 2000, Transform analysis and asset pricing for
Affine jump-diffusions, Econometrica 68, 1343-1376.
Fama, E. F., 1965, The behavior of stock market prices, Journal of Business 38,
34-105.
Hull, J. C. and A. D. White, 1987, The pricing of options on assets with stochastic
volatilities, Journal of Finance 42, 281-300.
Ingersoll, Jonathan E. Jr., 2000, Digital contracts: simple tools for pricing complex
derivatives, Journal of Business 73, 67-88.
Jackwerth, J. C. and M. Rubinstein, 1996, Recovering probability distributions from
option prices, Journal of Finance 51, 1611-1632.
Jackwerth, J. C. and M. Rubinstein, 2001, Recovering stochastic processes from
option prices, Working Paper, University of Wisconsin.
Kou, S. G. and H. Wang, 2003, First passage time of a jump diffusion process,
36
Advance in Applied Probability 35, 504-531.
Kou, S. G. and H. Wang, 2004, Option pricing under a double exponential jump
diffusion model, Management Science 50, 1178-1192.
Mandelbort, B., 1963, The valuation of certain speculative price, Journal of Business
36, 394-419.
Merton, R.C., 1976, Option pricing when underlying stock returns are discontinuous,
Journal of Financial Economics 3, 125-144.
Oleg Kudryavtsv and Sergei LevendorskiˇI, 2006, Priding of first touch digitals under
normal inverse Gaussian processes, International Journal of Theoretical and
Applied Finance 9, 915-949
Ramezani, C. A., and Y. Zeng, 1999, Maximum likelihood estimation of asymmetric
jump-diffusion process: application to security prices. Working Paper, Wixconsin
University.
Ramezani, C. A., and Y. Zeng, 2007, Maximum likelihood estimation of the double
exponential jump-diffusion process, Annals of Finance 3, 487-507.
S. I. Boyarchenko and S.Z. Levendorskii, 2002, Barrier opyions and touch-and-out
options under regular processes of exponential type, Annals of Applied
Probability12, 1261-1298.
Sato, K., 1999, Lévy processes and Infinitely Divisible Distributions, Cambridge
Studies in Advanced Mathematics, vol.68. Cambridge University Press.
Vanden, J. M., 2005, Digital contracts and price manipulation, Journal of Business 78,
1891-1915.
Zhou, C., 2001, The term structure of credit spreads with jump risk, Journal of
Banking and Finance 25, 2015-2040.