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Financial calculus Introduction to Financial Option Valuation_8

Tham khảo tài liệu 'financial calculus introduction to financial option valuation_8', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 18.3 Black-Scholes for American options 175 Case 1 A PAm - n r PAm - n . Here the combination PAm - n does better than cash in the bank. We argued that this could be exploited by buying PAm - n that is buying the option and selling n short selling the asset and loaning out the cash . Case 2 A PAm - n r PAm - n . Here the combination PAm - n does worse than cash in the bank. We argued that this could be exploited by selling PAm - n that is selling the option and buying n buying the asset and borrowing the cash . Without the early exercise facility the no arbitrage principle rules out both cases. With early exercise however the story changes. In Case 1 the arbitrageur buys the option and hence controls the exercise facility. This extra freedom can only help the arbitrageur and hence the arbitrage possibility persists. On the other hand in Case 2 the putative arbitrageur sells the option and is at the mercy of the early exercise facility. The arbitrageur may be exercised against at any time and can no longer guarantee to beat the bank risklessly. Overall for an American put the no arbitrage principle rules out Case 1 but not Case 2 and we conclude that 8.15 changes to Am 1 ơ2S2 dYA rS m. - rPAm 0. 18.2 d t 2 d S2 9 S Note that 18.2 is a partial differential inequality. Now at any point S t it will be optimal to either a exercise or b hold on to the option and hence for each S t one of 18.1 and 18.2 is at equality. 18.3 The three components 18.1 18.2 and 18.3 are the key features in the theory of American option valuation. Together they form what is known as a linear complementarity problem. At expiry if the option is still held its payoff matches the European so we have the final time condition PAm S T A S T for all S 0. 18.4 For S 0 the asset always has price zero so a payoff of E is assured. In this case it is optimal to exercise immediately. We may interpret this formally as a boundary condition of the form PAm S t E as S 0 for all 0 t T. 18.5 Similarly if S is .