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

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The price, at t = 0 of the claim, expressed in the foreign currency is thus givenby the formulaΠ (0; Ξ f ) = e cT I 0 Y 0 N[d 1 ] ¡ Y 0 N[d 2 ]+Y 0 , (17)d 1 = ln (I 0)+ ¡ c + 1 2 σ2¢ Tσ p Td 2 = d 1 ¡ σ p T.Finally, the price at t = 0 in domestic terms is given byΠ (0; Ξ d )=X 0 Π (0; Ξ f )=e cT I 0 N[d 1 ] ¡ N[d 2 ]+1. (18)Remark 4.1 For practical purposes it may be more convenient to model Y andI asdY t = Y t (r f ¡ r d )dt + Y t σ Y dWt Y ,dI t = I t α I dt + I t σ I dWt I ,where now σ Y and σ Y are constant scalars, whereas W Y and W I are scalarWiener processes with local correlation given by dWt Y dWtI = ρdt.In this model (which of course is logically equivalent to the one above) we havethe pricing formulas (17)-(18), but now with the notationc = α I + ρσ Y σ I ,qσ = σY 2 + σ2 I +2ρσ Y σ I,5 Pricing convertible bondsStandard pricing models of convertible bonds concentrate on pricing the bondand its conversion option at date t = 0 (see, for example, Brennan-Schwartz1977, Bardhan et.al. 1993). A somewhat less-standard problem is the pricing ofthe bond at some date 0
² The bond matures at a fixed date T 1 .² The underlying stock pays no dividends² At a fixed date T 0 ,withT 0
Page 1 and 2: ON THE USE OF NUMERAIRES IN OPTIONP
Page 3 and 4: Contents1 Introduction 32 The chang
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: where Q f denotes the risk neutral
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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