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

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standard Black-Scholes model where, under the usual risk neutral measure Qwe have the dynamicsdS t = rS t dt + σS t dW t , (23)dB t = rB t dt, (24)with a deterministic and constant short rate r. The price Π (t; X ) of the optioncan obviously be writtenwhere the T 1 -claim Y is defined byΠ (t; X )=S t ¡ βΠ (t; Y)Y =min[S T1 ,S T0 ] .In order to compute the price of Y we now basically want to do as follows.² Perform a suitable change of numeraire.² Use a standard version of some well known option pricing formula.The problem with carrying out this small program is that, at the exercise timeT 1 ,thetermS T0 does not have a natural interpretation as a spot price of atraded asset. In order to overcome this difficulty we therefore introduce a newasset S 0 defined by½St 0 S=t , 0 · t · T 0 ,S T0 e r(t−T0) , T 0 · t · T 1 .In other words, S 0 can be thought of as the value of a self financing portfoliowhere you at t = 0 buy one share of the underlying stock and keep it untilt = T 0 .Att = T 0 you then sell the share and put all the money into the bankaccount.We then have ST 0 1= S T0 e r(T1−T0) so we can now express Y in terms of ST 0 1asY =min £ S T1 ,K¢ ST 0 ¤1(25)whereK = e −r(T1−T0) (26)The point of this is that ST 0 1in (25) can formally be treated as the price at T 1 ofatradedasset. Infact,fromthedefinition above we have the following trivialQ-dynamics for S 0dSt 0 = rSt 0 dt + σt 0 St 0 dW twhere the deterministic volatility is defined byσ 0 t =½ σ, 0 · t · T0 ,0, T 0 · t · T 1 .(27)17
It is now time to perform a change of numeraire, and we can choose either Sor S 0 as the numeraire. From a logical point of view the choice is irrelevant,but the computations become somewhat easier if we choose S 0 .WithS 0 as thenumeraire we obtain (always with t
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 and 14: where Q f denotes the risk neutral
Page 15 and 16: ² The bond matures at a fixed date
Page 17: whereσ Z (t) =σ S + Σ p (t, T 1
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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