在一篇過去發表的文章中,Broadie and Kaya (2007), 作者們發表
了如何用二元樹的方式去評價一家公司的資本結構,也就是該公司的權益價值、債
權價值、以及公司價值。該篇文章的二元樹方法放寬了在更早的一篇文章,Leland
(1994)的限制, 該篇文章提出了一個架構, 該架構可以提供上述的三個公司資本結
構價值的封閉解, 不過該架構有限制, 就是該公司必須發行無限到期日的永續年金
債券。所以在用二元樹評價時就可以解決這種限制,甚至可以考慮到美國聯邦破產
法條第十一章的允許公司宣告破產後的重整。我們的模型最主要是把Broadie and
Kaya (2007)的模型再次放寬到隨機利率的狀態下, 因為原作者的模型是單變數二
元樹模型, 也就是把權益、債權、公司價值視為公司資產價值的衍生性商品, 也就
是該衍生性商品的狀態變數只有資產價值; 我們的模型放寬成兩個狀態變數, 資產
價值以及市場利率, 也就是雙變數二元樹模型。動機是我們認為短期上可不考慮利
率的隨機變動, 可是當整個二元樹的模型(債權的存續) 期間越長, 利率的隨機性
所帶來的影響就越大, 在第五章節我們即將紀錄該時間相較於忽略利率變動性的
影響。此外我們的模型也有考慮到公司資產價值與市場利率在隨機上的連動性,也
就是說如果該公司是屬於類似金融產業等等資產價值與市場利率相關性較強的廠
商, 那我們的模型所提出的評價方式將更為適用在該廠商上。綜合我們新加上的變
化以及之前學者所發展出來的架構,我們的模型將是在考慮了利率隨機性、違約風
險、股權稀釋償債、股東有限負債、破產清算、有效稅率、稅盾、美國破產法第七
章、以及公司營收能力之下的精確合理價值, 在權益、債權、公司價值上的合理價
值。

In the past research, Broadie and Kaya (2007), the authors have
derived a lattice to price the capital structure under constant interest rates.
But the pricing is determined by the states of debt, the randomness of interest
rates should not be ignored. Accordingly, we extend the pricing model of
capital structure to a model simultaneously considering stochastic interest
rates and its correlation with asset value, following the past research, Hilliard
et al. (1996). Finally, the contributions to our model are that we particularly
take some important factors into consideration, including the default risk, the
stochastic interest rates, and the asset value varied with market rates.

Boyle, Phelim P. (1988), “A lattice framework for option pricing with two
state variables”, Journal of Financial and Quantitative Analysis, 23(1),
1–12.
Broadie, Mark, Chernov, Mikhail, and Sundaresan, Suresh (2007), “Optimal
debt and equity values in the presence of chapter 7 and chapter 11”,
Journal of Finance, 62, 1339–1375.
Broadie, Mark and Kaya, Ozgur (2007), “A binomial lattice method for
pricing corporate debt and modeling chapter 11 proceedings”, Journal of
Financial and Quantitative Analysis, 42(2).
Cox, John C., Ingersoll, Jonathan E., and Ross, Stephen A. (1985), “A theory
of the term structure of interest rates”, Econometrica, 53(2).
Cox, John C., Ross, Stephen A., and Rubinstein, Mark (1979), “Option
pricing: A simplified approach”, Journal of Financial Economics, (3).
Dixit, Avinash K. and Pindyck, Robert S. (1994),
Investment Under Uncertainty, Princeton.
Fran¸cois, Pascal and Morellec, Erwan (2004), “Capital structure and asset
prices: Some effects of bankruptcy procedures”, Journal of Business, 77,
387–441.
Hilliard, Jimmy E., Schwartz, Adam L., and Tucker, Allan A. (1996), “Bivariate
binomial options pricing with generalized interest rate processes”,
Journal of Financial Research, XIX(4).
Hull, John (2006), Option, Futures, And Other Derivatives, Pearson Education,
6th edition.
Hull, John and White, Alan (1994), “Numerical procedures for implementing
term structure models i: Single factor models”, Journal of Derivatives, (3).
Leland, Hayne E. (1994), “Corporate debt value, bond covenants, and optimal
capital structure”, Journal of Finance, XLIX(4).
Longstaff, Francis A. and Schwartz, Eduardo S. (2001), “Valuing american
options by simulation: A simple least-squares approach”, Journal of Financial
Studies, 14(1), 113–147.
Merton, Robert C. (1973), “Theory of rational option pricing”, Bell Journal
of Economics and Management Science, 141–183.
Nelson, Daniel B. and Ramaswamy, Krishna (1990), “Simple binomial processes
as diffusion approximation in financial models”, Review of Financal
Studies.
Vasicek, Oldrich (1977), “An equilibrium characterization of the term structure”,
Journal of Financial Economics, 5, 177–188.
46