Tham khảo tài liệu 'an introduction to financial option valuation: mathematics, stochastics and computation_11', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Notes and references 197 Chapter 13 of Bjork 1998 deals with barriers and lookbacks from a martingale risk-neutral perspective. The use of the binomial method for barriers lookbacks and Asians is discussed in Hull 2000 Kwok 1998 . There are many ways in which the features discussed in this chapter have been extended and combined to produce ever more exotic varieties. In particular early exercise can be built into almost any option. Examples can be found in Hull 2000 Kwok 1998 Taleb 1997 Wilmott 1998 . Practical issues in the use of the Monte Carlo and binomial methods for exotic options are treated in Clewlow and Strickland 1998 . From a trader s perspective how to hedge is more important than how to value . The hedging issue is covered in Taleb 1997 Wilmott 1998 . EXERCISES . Suppose that the function V S t satisfies the Black-Scholes PDE . Let V S t S1 -2 V Show that ff 1 - 2 S2 iff rS If-dt 2 dS2 ds - rV s1 -2 2 - 2 . X d V r-2-- s d s rV Deduce that V S t solves the Black-Scholes PDE. . Using Exercise deduce that CB in satisfies the Black-Scholes PDE . Confirm also that CB satisfies the conditions and when B E. . Explain why holds for all down and up barrier options. . Why does it not make sense to have B E in an up-and-in call option . The value of an up-and-out call option should approach zero as s approaches the barrier B from below. Verify that setting s B in returns the value zero. . Consider the geometric average price Asian call option with payoff max fjscti E 0 J 1 198 Exotic options where the points ti 1 are equally spaced with ti i kt and nkt T. By writing ns t. 5 tn S tn-1 2 S tn-2 V 1 1 1 S tn-1 S tn-2 J S tn-3 J S t3 y-2 S t2 V-1 S t1 V S t2 J S t1V So 0 and using the additive mean and variance property of independent normal random variables mentioned as item iii at the end of Section show that for the asset model under risk neutrality we have log N r - 1 ơ2 n- 1 T a2 n 1
