交易成本與利率選擇權之評價及複製策略

Pricing and Replication of Interest Rate Options with Transaction Costs

何博文(Po-Wen Ho)

指導教授 ： 巫和懋

國立臺灣大學/管理學院/國際企業學研究所/碩士(2004年)

https://doi.org/10.6342/NTU.2004.02207

摘要

論文摘要

Leland (1985)提出考量交易成本下之股票選擇權複製策略，其依照Black-Scholes-Merton選擇權定價公式和複製方法，在其餘參數不變的情況下，只調整了計算選擇權價格時使用的股價報酬率波動度，藉此推導出避險誤差與動態調整避險組合時間間隔之關係式，並證明當動態調整投資組合的時間間隔很小時，即在近似連續時間下，依照調整後波動度之Delta避險，可使避險誤差趨近於零。而在Leland之後，有釵h的文獻接續探討交易成本相關的複製策略與評價方法，但幾乎所有文獻皆集中在股票選擇權市場內的研究，在利率選擇權市場的討論卻付之闕如。

本篇論文則試著將交易成本的問題導入利率選擇權市場，在Miltersen, Sandmann, and Sondermann (1997)等人提出的市場模型架構下，探討交易成本對利率上、下限選擇權的評價與複製策略。首先我們在無交易成本的情況下，將連續時間之評價與複製模型推廣至離散時間，以做為放寬交易成本條件之基礎。進而當市場存在交易成本時，我們亦推導出經調整後之利率上、下限選擇權評價與複製策略，並利用蒙地卡羅模擬之數值方法對避險誤差與模型相關參數進行分析。

由文本之研究結果我們可得到下列幾點結論：（1）在無交易成本情況下，當動態調整避險組合時間間隔越小，即調整次數越頻繁時，總避險誤差之期望值和變異數會趨近於0。（2）當市場存在交易成本時，利率上、下限選擇權複製策略的調整與交易成本、動態調整避險組合時間間隔及債券遠期合約之遠期價格有關。若依此避險策略，則當調整避險組合次數越頻繁時，總避險誤差之期望值和變異數亦會趨近於0。（3）經由蒙地卡羅模擬有交易成本之情況後我們發現，當交易成本佔成交金額比例愈高或選擇權處於愈價外時，要精確的複製利率上、下限選擇權其困難度是較高且成本較大的。（4）當市場存在交易成本時，我們亦提出了利率上、下限選擇權價格之上下界。

關鍵字

市場模型 ； 交易成本 ； 利率上(下)限選擇權

並列摘要

Abstract

Leland has proposed a replication strategy of stock option with transaction costs in 1985. He gave a pricing formula and replication strategy as same as Black-Scholes-Merton’s except employing a volatility modified by transaction costs and rebalance time step. He also derive the relationship between hedge error and dynamic rebalance time step, and has proved that when rebalance time step approaches zero, i.e. almost doing continuous time hedge, hedge error will approach zero too if executing rebalances with the modified volatility. After Leland (1985), there were many articles kept studying the replication strategy and pricing method of stock option when transaction costs being considered, but it seems that there were almost not any other articles to consider interest rate options with transaction costs.

This article tries to include transaction costs in pricing method and replication strategy of the most popular interest rate derivatives in market, Cap and Floor. The research is under Miltersen, Sandmann, and Sondermann (1997) framework. Firstly we extend the continuous time replication strategy to discrete time with no transaction costs existing as foundation for further studies, and then try to derive the replication strategy of Cap and Floor when there are transaction costs needed for hedging. And finally use Monte-Carlo simulation method to analysis the relationship of hedge error and model parameters.

We have some conclusions in this research:
1.If there is no transaction costs needed for hedging, we found with the smaller the rebalance time step and the higher the rebalance frequency, then the closer to zero the expectation and variance of total hedge error will be.
2.With transaction costs in market, the replication strategy of Cap and Floor has close connection with transaction costs、rebalance time step and forward price of the forward contracts. Following the strategy proposed in this article, expectation and variance of total hedge error would approach zero when hedge frequency approach infinity.
3.After doing Monte-Carlo simulation with transaction costs included, we found it’s much harder and more costly to replicate Cap and Floor when transaction costs are higher or the options are more out-of-the-money.
4.There is upper and lower bound of Cap and Floor been proposed here when transaction costs are included.

並列關鍵字

Cap&Floor ； Market Model ； Transaction Costs

參考文獻

Black, F. and M. Scholes (1973)：Pricing of Options and Corporate Liabilities, Journal of Political Economy 81, 637-654.

Boyle, P. and D. Emanuel (1980)：Discretely Adjusted Option Hedges, Journal of Financial Economics 8, 259-282.

Boyle, P. and T. Vorst (1992)：Option Replication in Discrete Time with Transaction Costs, Journal of Finance 47, 271-293.

Brace, A., D. Gatarek, and M. Musiela (1997)：The Market Model of Interest Rate Dynamics, Mathematical Finance 7, 127-155.

Cox, J., J. Ingersoll, and S. Ross (1985)：A Theory of the Term Structure of Interest Rates, Econometrica 53, 363-384.