ON THE USE OF NUMERAIRES IN OPTION PRICING by Simon ...

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On the Use of Numeraires in Option Pricing<strong>Simon</strong> Benninga ∗Faculty of ManagementTel-Aviv University ISRAELe-mail: benninga@post.tau.ac.ilTomas BjörkDepartment of FinanceStockholm School of EconomicsSWEDENe-mail: fintb@hhs.seZvi Wiener †School of BusinessHebrew University of JerusalemISRAELe-mail: mswiener@mscc.huji.ac.ilDecember 1, 2001AbstractIn this paper we discuss the significant computational simplificationthat occurs when option pricing is approached through the change ofnumeraire technique. The original impetus was a paper (Hoang, Powell,Shi 1999) on endowment options recently published in this journal; in thepresent paper we extend these results to the case of stochastic interestrates. We also discuss four additional option pricing problems within theframework of a change of numeraire:• Pricing savings plans which incorporate a choice of linkage.• Pricing convertible bonds.• Pricing employee stock ownership plans• Pricing options where the strike price is in a currency different fromthe stock price.∗ Benninga acknowledges a grant from the Israel Institute of Business Research at Tel-AvivUniversity.† Wiener’s research was funded <strong>by</strong> the Krueger Center at the Hebrew University.1
Contents1 Introduction 32 The change of numeraire approach 43 Endowment warrants 73.1 Institutionalsetup .......................... 73.2 Mathematicalmodel ......................... 74 Pricing savings plans with choice of linkage 104.1 Institutionalsetup .......................... 104.2 Mathematicalmodel ......................... 115 Pricing convertible bonds 135.1 Institutionalsetup .......................... 135.2 Mathematicalmodel ......................... 146 Employee stock ownership plans 166.1 Institutionalsetup .......................... 166.2 Mathematicalmodel ......................... 167 Options with a foreign-currency strike price 197.1 Institutionalsetup .......................... 197.2 Mathematicalmodel ......................... 207.3 Pricingtheoptionindollars..................... 217.4 Pricingtheoptiondirectlyinpounds................ 228 Conclusion 242
Page 1: ON THE USE OF NUMERAIRES IN OPTIONP
Page 5 and 6: 2 The change of numeraire approachT
Page 7 and 8: ² The Q 0 -dynamics of the Z-proce
Page 9 and 10: In order to analyze this option we
Page 11 and 12: our model in two ways: The Q-dynami
Page 13 and 14: where Q f denotes the risk neutral
Page 15 and 16: ² The bond matures at a fixed date
Page 17 and 18: whereσ Z (t) =σ S + Σ p (t, T 1
Page 19 and 20: It is now time to perform a change
Page 21 and 22: 7.2 Mathematical modelWe model the
Page 23 and 24: The dollar price (expressed in doll
Page 25 and 26: After simplification this reduces t

ON THE USE OF NUMERAIRES IN OPTION PRICING by Simon ...

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