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k l f $

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Risk-Neutral Measure for $

Theorem: Let QA be the risk-neutral probabilitymeasure for the US Dollar investor, and QB therisk-neutral measure for the UK Pound Sterling investor.Unless = 0 (that is, unless the exchange rate is purelydeterministic), it must be the case that

QA = QB

Stochastic Calculus p. 14/2

Ri k N l M f $

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Risk-Neutral Measure for $

Theorem: Let QA be the risk-neutral probabilitymeasure for the US Dollar investor, and QB therisk-neutral measure for the UK Pound Sterling investor.Unless = 0 (that is, unless the exchange rate is purelydeterministic), it must be the case that

QA = QB

This is a special case of a more general phenomenon:


N i Ch g

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Numeraire Change

Suppose that a market has tradeable assets A, B withshare price processes S At and S Bt (evaluated in a common

numeraire C ). Let QA

and QB

be risk-neutral measuresfor numeraires A, B , respectively.


N meraire Change

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Numeraire Change

Suppose that a market has tradeable assets A, B withshare price processes S At and S Bt (evaluated in a common

numeraire C ). Let QA

and QB

be risk-neutral measuresfor numeraires A, B , respectively.

Theorem: QA = QB if and only if S At /S Bt is a constantrandom variable. Furthermore, in general, for any nitetime T ,

dQB

dQA F T =S BT S AT

S A0S B0


Consequence

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Consequence

In the foreign exchange context, the riskless assets forthe two numeraires are US Money Market and UKMoney Market, with share prices (in $)

At = exp {r A t}B t = exp {r B t}/Y t


Consequence

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Consequence



Therefore, the likelihood ratio between the risk-neutralmeasures for and $ investors is

dQB

dQA F T =

Y T Y 0

1

exp{(r B r A)T }


Consequence

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Consequence



Therefore, the likelihood ratio between the risk-neutralmeasures for and $ investors is

dQB

dQA F T =

Y T Y 0

1

exp{(r B r A)T }


Likelihood Ratio Identity

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Let V it be the time- t share price of any contingent claimin numeraire i = A,B,C . These share prices satisfy:



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V At = V C t /S AtV Bt = V

C t /S

Bt



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C t /S

Bt

The time-zero share price is the discounted expectedvalue of the time t share price for each of thenumeraires A, B . The discount factors are 1, so



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C t /S

Bt

The time-zero share price is the discounted expectedvalue of the time t share price for each of thenumeraires A, B . The discount factors are 1, so

V A0 = V C 0 /S A0 = E AV C t /S At

V B

0 = V C

0 /S B0 = E

B

V C

t /S Bt



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It follows that for every contingent claim V with shareprice V C t (in numeraire C ),

S A0 E A(V C t /S At ) = S B0 E B (V C t /S Bt )



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Apply this to the contingent claim with payoff V C T S BT attime T to obtain the following identity, valid for allnonnegative random variables V C T measurable F T :

E B V C T = E AV C T S

BT S A0S AT S B0



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Apply this to the contingent claim with payoff V C T S BT attime T to obtain the following identity, valid for allnonnegative random variables V C T measurable F T :

E B V C T = E AV C T S

BT S A0S AT S B0

This is the dening property of a likelihood ratio.Stochastic Calculus p. 18/2

Exponential Martingales

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Let W t be a standard Wiener process, wth Brownianltration F t , and let t be a bounded, adapted process.Dene

Z t = exp t

0s dW s

t

02s ds/ 2



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Z t = exp t

0s dW s

t

02s ds/ 2

Fact: Z t is a positive martingale.



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po e t a a t ga es


Z t = exp t

0s dW s

t

02s ds/ 2

Fact: Z t is a positive martingale. Proof: It!



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p g


Z t = exp t

0s dW s

t

02s ds/ 2

Fact: Z t is a positive martingale. Proof: It!

dZ t = Z t t dW t Z t 2

t dt/ 2+ Z t 2

t dt/ 2= Z t t dW t =

Z t = Z 0 +

t

0 Z s s dW sStochastic Calculus p. 19/2

Girsanovs Theorem

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Girsanov s Theorem

Because Z t is a positive martingale under P with initialvalue Z 0 = 1 , for every xed time T the random variableZ T is a likelihood ratio: that is,

Q(F ) := E P (I F Z T )

denes a new probability measure on the algebra F T of events F that are observable by time T .


Girsanovs Theorem

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Girsanov s Theorem

Because Z t is a positive martingale under P with initialvalue Z 0 = 1 , for every xed time T the random variableZ T is a likelihood ratio: that is,

Q(F ) := E P (I F Z T )

denes a new probability measure on the algebra F T of events F that are observable by time T .Theorem: Under the measure Q, the process

{W t t

0 s ds}0 t T is a standard Wiener process.


Exchange Rates

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g

Consider again the $ and currencies. Assume that eachhas a riskless Money Market, and that the rates of returnr A , r B are constant. Assume that the exchange rate Y tobeys

dY t = ( r B r A)Y t dt + Y t dW t

where W t is a standard Wiener process under therisk-neutral probability QB for investors. Thus,

Y t = Y 0

exp{(r B r A 2

/ 2)t + W t }.


Exchange Rates

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g

Since

dQA

dQB F T =Y T Y 0 exp{ (r B r A)T }

= exp {W T 2 T/ 2}


Exchange Rates

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Since

dQA

dQB F T =Y T Y 0 exp{ (r B r A)T }

= exp {W T 2 T/ 2}

Girsanov implies that under QA the process W t is aWiener process with drift . Thus, to the $ investor, itappears that the exchange rate obeys

dY t = ( r B r A 2 )Y t dt + Y t dW t

where W t is a standard Wiener process under QA .Stochastic Calculus p. 22/2

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Proof of Girsanov 1

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The statement that X is a standard Wiener process is anassertion that the increments of X are independentGaussian random variables with the correct variances.Lets show that under Q, the distribution of W T T isgaussian with var T (where T =

T 0 s ds).


Proof of Girsanov 1

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T 0 s ds). For this, it

sufces to show that for any real ,E Q exp{(W T T )} = exp {

2 T/ 2}


Proof of Girsanov 1

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T 0 s ds). For this, it

sufces to show that for any real ,E Q exp{(W T T )} = exp {

2 T/ 2}

To evaluate the expectation, change measure:

E Q exp{(W T T )} = E P exp{(W T T )}Z T


Proof of Girsanov 2

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Objective: Show that E P H T = 1 , where

H t = exp {(W t t ) 2 t/ 2}Z t


Proof of Girsanov 2

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H t = exp {(W t t ) 2 t/ 2}Z t

= exp { t

0(s + ) dW s +

t

0(s

2s / 2

2 / 2) ds}


Proof of Girsanov 2

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H t = exp {(W t t ) 2 t/ 2}Z t

= exp { t

0(s + ) dW s +

t

0(s

2s / 2

2 / 2) ds}

= exp { t0 (s + ) dW s t0 (s + )2 ds/ 2}


Proof of Girsanov 2

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H t = exp {(W t t ) 2 t/ 2}Z t

= exp { t

0(s + ) dW s +

t

0(s

2s / 2

2 / 2) ds}

= exp { t0 (s + ) dW s t0 (s + )2 ds/ 2}Thus, H t is an exponential martingale under P , and soits expectation is constant over time.


Proof of Girsanov 2

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H t = exp {(W t t ) 2 t/ 2}Z t

= exp { t

0(s + ) dW s +

t

0(s

2s / 2

2 / 2) ds}

= exp { t0 (s + ) dW s t0 (s + )2 ds/ 2}Thus, H t is an exponential martingale under P , and soits expectation is constant over time. A similar

calculation establishes the independence of the

increments.Stochastic Calculus p. 25/2

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