公債期貨暨其隱含交割品質選擇權之評價

The Valuation of Bond Futures and Implied Quality Options

陳宗健(Tsung-Chien Chen)

指導教授 ： 李賢源 ； 陳業寧

國立臺灣大學/管理學院/財務金融學研究所/碩士(2004年)

https://doi.org/10.6342/NTU.2004.10163

摘要

本篇論文主要以交換選擇權的概念，利用機率平賭評價理論(Martingale Pricing Theory)重新推導在風險中立測度下公債期貨價格與其隱含交割品質選擇權之理論價格，並且選用Hull-White二因子利率模型，介紹如何在立體九元利率樹上演算公債期貨與交割品質選擇權之價值。
自從Ritchken, and Sankarasubramanian(1992)以HJM之利率模型來評價交割品質選擇權價值，已經改正傳統理論忽略利率隨機波動的缺點，但因HJM利率模型本身不具備馬可夫性質(Non- Markovian)，操作彈性受到侷限，因此林丙煇與周建輝於1997年採用Hull-White單因子模型進行評價，其缺陷在於其利率模型中將短期利率隨機過程之波動度參數( )設定為固定值( )，造成模型本身無法配適現實市場上較複雜之波動度期間結構(volatility structure)；此外，在其論文中採用Kijima and Nagayama(1994)在其建構之利率樹上所推導的零息債券價格公式，並且搭配Hull-White三元利率樹之數值分析來進行品質選擇權評價，但此二者在理論上不具一致性，其惟有在短期利率隨機過程之波動度參數( )設為固定值( )下，此二者搭配之評價架構才屬正確；換言之，當Hull-White利率模型中之波動性參數設為固定值時，Kijima and Nagayama(1994)在其建構之利率樹下所推導的零息債券價格公式正好為Hull-White單因子利率模型下零息債券價格封閉解(李賢源與謝承熹)之特例(當 )；
在利率模型方面，Hull-White單因子利率模型允陬u期利率之波動度參數為時間之函數，雖然可以符合初期市場實際之波動度期間結構，但是隨著時間的演進，反而導致波動度期間結構不具備穩定性質(Non- Stationary)，有鑑於此，為了同時符合市場利率與其波動度期間結構且具備穩定性質(Stationary)下，因此採用Hull-White二因子利率模型來進行交割品質選擇權之評價；本文結論有兩點：

1.根據第三章第二節中之定理一與定理二以及數字範例之結果，皆可証明Kijima and Nagayama(1994)所推導之零息債券價格公式並非一般化之結果，事實上，其公式乃為允釭i動性參數為時間函數(σ(t))之Hull-White單因子模型下的零息債券價格公式(李賢源與謝承熹)的特例；
2.由於利率模型可以同時配適利率及波動度期間結構且兼具穩定性質(Non- Stationary)，並在均數復歸的特性下，可大幅提升評價之效率性與準確性；在敏感性分析方面，利率模型中之第二個利率因子(u)之參數(b與σ2)對於品質選擇權之價值有較顯著的影響。

關鍵字

品質選擇權 ； 公債期貨 ； Hull-White利率模型

並列摘要

The thesis mainly based on the concept of exchange options combining the martingale pricing theory to deduce the valuation of bond futures and quality options under risk neutral probability, and then adopted the Hull-White two-factor model to introduce the pricing process of the bond futures and quality options on interest rate tree.
Since the paper published by Ritchken, and Sankarasubramanian(1992) which used HJM model to compute the value of the quality options, the restriction of constant interest rate has been removed. However, the operating elasticity is limited due to the lack of Markovian property with HJM model. Therefore, in 1997, Lin and Chou chose Hull-White single factor model to implement the computation of the quality options. The drawbacks of this method is that the single factor model can not fit some sophisticated volatility term structures. In addition, it is not proper to use the zero coupon bond formula which derived by Kijima and Nagayama(1994) combining the Hull-White single-factor model. Only when the volatility parameter( ) is set to be constant, the framework would be correct. In other words, the zero coupon bond formula which derived by Kijima and Nagayama is the special case of the one(Lee and Hsieh) in the Hull-White single-factor model when the volatility parameter ( ) is set to be constant.
On the interest rate model, the Hull-White single factor model with time-dependent volatility parameter can fit the initial volatility term structure, but it causes the non-stationary problem instead. Hence, this study opts the Hull-White two-factor model to proceed the valuation under the conditions of the stationary property and the abilities of fitting the term structures of interest rate and volatility.
To sum up, there are two conclusions in this study:

1.According to the results of proposition 1 and 2 in section 3.2, it can show that the zero coupon bond formula derived by Kijima and Nagayama(1994) is not the general case. In fact, it is the special case of the one in the Hull-White single- factor model when the volatility parameter( ) is set to be constant.
2.Because the Hull-White two-factor model possess the stationary property and the abilities of fitting the term structures of interest rate and volatility, it can improve the accuracy and efficiency of the pricing; furthermore, in the sensitivity analysis, it found that the parameters of the second interest rate factor in Hull-White two- factor model have more significant effects on the value of the quality options than the first one.

並列關鍵字

Hull-White Interest Rate Model ； bond futures ； quality options

參考文獻

“Pricing and Quality Option in Japanese Government Bond

The relation between forward prices and futures price,

Journal of Financial Economics, 9, 321± 46.

theory of the term structure of interest rate,

Econometrica, 53, 385± 407.