臺灣博碩士論文加值系統

在Black-Scholes 的模型下，股價的波動率假設為已知的常數。但在現實的世界裡波動率則非為常數。許多模型對此現象的解釋為波動率是隨機變動的，因此有許多選擇權模型建立在隨機波動率上。
Hilliard 和Schwartz 在1996年提出了一套bivariate binomial model來評價選擇權，此模型可允許股價與波動率之間有相關性，並且可評價美式選擇權。在本篇論文中，引用了他們的方法並把他們未完成的部分作完。

The volatility smile is frequently observed in options prices. But in the pure Black-Scholes world, there should not be any smile as the volatility should be constant across the strike price and time. The Black-Scholes model makes the strong assumption that stock returns are normally distributed with known variance, but the constant variance assumption is somewhat simplisitc.
Pricing models with stochastic volatility have been addressed in the literature by many authors. The bivariate binomial framework presented by Hilliard and Schwartz [1996] not only allows non-zero correlation between the volatility and the underlying process but can also be used to value American options. This thesis fills that gap by implementing the bivariate binomial tree method to price options.

1 Introduction.....................................1
1.1 Introduction....................................1
1.2 Structures of the Thesis........................2
2 Bivariate Binomial Model.........................3
2.1 Stochastic Volatility Models....................3
2.2 Constructing the Lattice........................3
2.3 Binomial Jumps and Probabilities................5
3 Numerical Results................................7
3.1 Bivariate Binomial Option Pricing...............7
3.2 Evaluating European Put Options................7
3.3 Evaluating American Put Options................17
4 Conclusions.....................................19
Appendix...........................................20
Bibliography.......................................24

[1] S. Heston, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, The Review of Financial Studies, Vol. 6, No. 2 (1993), pp. 327-343.
[2] J. Hilliard and A. Schwartz, Binomial Option Pricing Under Stochastic Volatility and Correlated State Variables, Journal of Derivatives, Fall 1996, pp. 23-39.
[3] J. Hull and A.White, The Pricing of Options on Assets with Stochastic Volatility, Journal of finance, Vol. 42, No. 2 (June 1987), pp. 281-300.
[4] H. Johnson and D. Shanno, Option Pricing when the Variance is Changing, The Journal of Financial and Quantitative Analysis, Vol. 22, No. 2 (June 1987), pp. 143-151.