考量發行券商隨機資產與負債下脆弱選擇權之評價

Abstract

本文提出封閉解與二項樹演算法以評價脆弱衍生性商品。報償函數主要延續Klein (1996) 與 Ammann (2001) 的信用風險結構。三個隨機過程 -- 標的股價，發行券商資產，與發行券商負債 -- 在模型中將被適當的建構。在所提出的報償函數下，我們推導出脆弱歐式選擇權的封閉解。利用Liu and Liu (2006) 期望內含價值的想法，配合 dimension reduction 的技巧，我們適當地為兩個具相關性的隨機過程 -- 標的股價與發行券商資產負債比 -- 建立了條件二項樹演算法。此外，根據 Rubinstein (1994) 的作法，我們亦建立了一個具一般性的二項金字塔演算法。這兩個演算法可同時作為脆弱歐式選擇權與脆弱美式選擇權評價的近似評價法。同時，解析證明與數值分析說明了所提出的二項樹模型會收斂到對應的封閉解。本文亦包含脆弱選擇權的敏感度分析。

This paper presents both closed-form formulas and binomial tree algorithms to evaluate vulnerable derivatives. The payoff function extends mainly from the Klein (1996) and the Ammann (2001) credit risk frameworks. Three stochastic processes -- the underlying stock price, the assets value of the option writer, and the liabilities value of the option writer -- are suitably modeled. Closed-form solutions are derived for vulnerable European options under the suggested payoff function. Adopting the innovation of expected intrinsic value with a trick of dimension reduction by Liu and Liu (2006), a conditional binomial tree (CBT) algorithm for two correlated stochastic processes, the underlying stock price and the asset-to-debt ratio process, are properly established. Moreover, following Rubinstein (1994), a general binomial pyramid (BP) algorithm is set up. Both algorithms serve as discrete approximations for vulnerable European and vulnerable American options evaluation. It is analytically verified and numerically illustrated that the proposed binomial tree model contains the closed-form formula as a limiting case. Some sensitivity analyses for the discussed vulnerable options are also included.