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Financial calculus Introduction to Financial Option Valuation_4

Tham khảo tài liệu 'financial calculus introduction to financial option valuation_4', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 9 More on hedging OUTLINE practical illustration of hedging behaviour of delta near expiry Long-Term Capital Management 9.1 Motivation The hedging idea that was used to derive the Black-Scholes PDE forms the most important concept in this book. In this chapter we therefore take time out to reiterate the steps involved and develop the process into an algorithm that can be illustrated numerically. 9.2 Discrete hedging Having found the explicit formulas 8.19 and 8.24 we may differentiate with respect to s to obtain the required asset holding Aị in 8.10 . This partial derivative d V d s is called the delta of an option and the hedging strategy that we discussed is known as delta hedging. Performing the differentiation leads to dC N d1 delta of a European call ds 9.1 and dP N di 1 delta of a European put d s x x Confirmation of these expressions is deferred until Chapter 10 where various par- tial derivatives are computed. Returning to the delta hedging process we know from 8.7 that ni 1 the value of the portfolio at ti 8t satisfies 9.2 rii 1 Aisi 1 1 r 8t Di. 9.3 87 88 More on hedging The asset holding is rebalanced to Ai 1 and in order to compensate the cash account is altered to Di 1. Since no money enters or leaves the system the new portfolio value Ai 1 Si 1 Di 1 must equal ni 1 in 9.3 so Di 1 1 rSt Di Ai - Ai 1 Si 1. 9.4 We may summarize the overall hedging strategy as follows. Set Ao d Vo d S Do 1 arbitrary n0 Ao So Do For each new time t i 1 St Observe new asset price Si 1 Compute new portfolio value IT 1 in 9.3 Compute Ai 1 dV Compute new cash holding Di 1 in 9.4 New portfolio value is Ai 1 Si 1 Di 1 end More precisely this strategy is discrete hedging as the rebalancing act is done at times iSt. Because we cannot let St 0 in practice there will be some error in the risk elimination. For the purpose of illustration it is possible to simulate an asset path and implement discrete hedging. To write down the resulting algorithm we use Ệi to denote samples from an N