Replicating Portfolios for Lookback Options in Discrete Time with Transaction Costs

我們考慮回顧式選擇權的價格當市場有交易成本的時候。此時傳統Black - Scholes的選擇權訂價公式不能再被使用。在二元樹的架構下，我們證明回顧式買權(浮動式履約價)的買方部位仍然存在唯一的複製組合。我們根據以下三篇文章來證明這個結果。Boyle and Vorst (1992)假設買賣股票時要付成交金額的一定比例的手續費，他們導出歐式買權和賣權的複製策略。Palmer (2001)在單期模型下討論複製組合及超複製組合。Cheuk and Vorst (1997)推導出回顧式選擇權有一個單一狀態變數的二元樹,且計算複雜度跟傳統的二元樹相同。在本文中，我們用上述的結果加上新導出的結果證明,在離散時間及有一定比例交易成本下,回顧式買權(浮動式履約價)的買方部位存在唯一的複製組合。We consider the price of the lookback options in the imperfect market where transaction costs are present. The standard Black - Scholes option pricing methodology is no longer valid since the market is imperfect. We prove that there still exists a unique replicating portfolio for the long lookback call option with floating stike in a binomial framework. We follow three articles to prove this result. Boyle and Vorst (1992) derive self-financing strategies perfectly replicating the final payoffs to long positions in European call and put options in a binomial framework assuming proportional transaction costs on trades in the stocks. Next, Palmer (2001) derives some results about replicating portfolios and super-replicating portfolios in the one-period model. Further, Cheuk and Vorst (1997) show that a one-state variable binomial model for lookback options can be constructed with the same computational complexity as the standard binomial model. In this paper, we use these results and new lemmas of our own to prove that there exists a unique replicating portfolio for a long lookback call option with floating strike in discrete time with proportional transaction costs.Contents ii Acknowledgements v Abstract in Chinese vi Abstract vii 1 Introduction 1 2 The Binomial Models 4 3 Some Useful Lemmas for Lookback Call Options 8 3.1 Replication of Contingent Claims: the One-Period Case 9 3.2 Replication of Contingent Claims: theMulti-Period Case 17 4 The Uniqueness of the Replicating Portfolios for the Long Floating Strike Lookback Call Option 28 4.1 Settled by 28 4.2 Cash Settlement 31 Appendix 35 References 3