sigal's paradox

8/10/2019 Sigal's Paradox

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9.3.4 Siegels Exchange Rate Paradox

In (9.3.16), the mean rate of change for the exchangerate Q(t) is R(t) - Rf(t) under the domestic risk-neutral

measure .

From the foreign perspective, the exchange rate is 1/Q(t),

and one should expect the mean rate of change of 1/Q(t)

to be Rf(t) - R(t).

This turns out not to be as straight forward as one might

expect because of the convexity of the function f(x) = 1/x.

22 1 2 2

2 3

1

9.3.16

f

f

dQ t Q t R t R t dt t t dW t t dW t

Q t R t R t dt t dW t

P


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Example:

Exchange rate of 0.9 euros to the dollar:

1 euro 1.1111 dollars

If the dollar price of euro falls by 5%:

1 euro 0.95

1.1111 = 1.0556 dollarsThis is an exchange rate of 1/1.0556=0.9474 euros to

dollars. The change from 0.9 euros to the dollar to

0.9474 euros to the dollar is a 5.26% ( = 1/0.95 - 1)

increase in the euro price of the dollar, not a 5% increase.


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2 3

2

2 3 2 3 3

2

2 2 3

1 1 2We take so that and . Using 9.3.16 , we obtain

1

1

2

1 1

1

f

f

f x f x f x

x x x

d df QQ

f Q dQ f Q dQdQ

R R dt dW dW dWQ Q

R R dt dWQ t

2

2

9.3.24

The mean rate of change under the domestic risk-neutral measure is ,

not .

f

f

R R

R R


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1

However, the asymmetry introduced by the convexity of is resolved if we

switch to the foreign risk-neutal measure, which is the appropriate one for derivative

security pricing in the foreign cur

f x x

3 2 3 3 2 3

3

2 3

rency. First recall the relaptionship (9.3.20)

In terms of , we may rewrite 9.3.24 as

1 1

f f

f

ff

dW t t dt dW t dW t t dt dW t

W t

d R R dt dW Q Q

(9.3.25)

1Under the foreign risk-neutral measure, the mean rate of change for is ,

as expect.

fR RQ


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Under the actual probability measure , however, the asymmetry remains.

When we begin with (9.3.4), which shows the mean rate of change of theexchange rate to be under , and then use the Ito-Dot

P

P\

2 3

2

2 2 3

eblin formula, we

obtain the formula 9.3.27 below

9.3.4

1 1 1 9.3.27

1Both and have t

dQ t t Q t dt t Q t dW t

d t t dt t dW t Q t Q t Q t

QQ

he same volatility. However, the mean rates of change

1of and are not negatives of one another.Q

Q


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9.3.5 Forward Exchange RatesWe assume in this subsection that the domestic and foreign interest rates are constant

and denote these constants by and , respectively. Recall that is units of domestic

currency per unit of for

fr r Q

21 22 2

eign currency. The exchange rate from the viewpoint

is governed by the stochastic differential equation (9.3.16)

1

Therefore

f

domestic

dQ Q t r r dt t t dW t t t dW t

32

32

is a martingale under , the risk-neutral measure.

:

f

f f

f f

f

fr r

r r t

r r t r r t f

r r t r r t f f

r r t

t

proof

d e Q t

r r e dt Q t e dQ t

e Q t

d

r r e Q t dt e Q t r r dt

o

t dW t

e Q t t d

mestic

W

P

t

money market account

money market account


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At time zero, the (domestic currency) forward price for a unit of foreign

currency, to be delivered at time , is determined by the equation

0

T

rT

F

T

e Q T F E

he left-hand side is the risk-neutral pricing formula applied to the derivative

security that pays in exchange for at time . Setting this equal to zero

determines the forward price. We may solve

Q T F T

this equation for by observing

that it implies

0

which gives the -forward (domestic per unit of foreign) exchange rate

ff fr r TrT rT r T r T

F

e F e Q T e e Q T e Q

T

E E

0

fr r TF e Q


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22 1 2 2

The exchange rate from the foreign viewpoint is given by the stochastic differential equation (9.3.25)

1 1 1

Therefore,

f ffd r r dt t t dW t t t dW t

Q t Q t

1

is a martingale under , the risk-neutral measure.

At time zero, the (foreign currency) forward price for a unit of domestic curre

fr r t

f

f

eQ t

foreign

F

P

ncy to be delivered

at time is determined by the equation

1 0

The left-hand side is the risk-neutral pricing formula applie

ff r T f

T

e FQ T

E

1

d to the derivative security that pays

in exchange for (both denominates in foreign currency) at time . Setting this equal to zero

determines the forward price. We may solve this equation for

f

Q T

F T

by observing that it implies

1 1 1

0

which gives the -forward (foreign per unit of foreign) exchange rate

ff f

f

f f r r Tr T f r T rT rT

F

e F e e e eQ T Q T Q

T

E E

1 1

0

fr r T

fF e Q F


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9.3.6 Garman-Kohlhagen Formula

2We assume the domestic and foreign interest rates and and the volatility

are constant. Consider a call on a unit of foreign currency whose payoff in

domestic currency is . At time zero, th

fr r

Q t K

2 3

e value of this is

In this case, (9.3.16) becomes

from which we conclude that

rT

f

e Q T K

dQ t Q t r r dt dW t

Q

E

22 3 21

0 exp2

fT Q W T r r T


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3

2

2 2

Define

so is a standard normal random variable under . Then the price of the call is

1 0 exp

2

rT

rT f

W TY

T

Y

e Q T K

e Q TY r r

P

E

E

This expression is just like (5.5.10) with , with 0 in place of , and with in place

of the dividend rate . According to (5.5.12), the call price is

f

T K

T Q x r

a

22

2

0 9.3.28

where

01 1 log

2

and is the cumulative standard normal distribution function. Equation (9.3.28) is

frT r T rT

f

e Q T K e Q N d e KN d

Qd r r T

KT

N

E

calles the

- .Garman Kohlhagen formula

2

1, exp (5.5.10)

2

, , , (5.5.12)

r

aT rT

c t x e x Y r a K

c t x xe N d x e KN d x

E


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9.3.7 Exchange Rate Put-Call Duality

Recall the numeraire , which is the domestic price of the foreign money

market account. The Radon-Nikodym derivative of the foreign risk-neutral measure

with respect to the domestic risk-neutral m

fM t Q t

1 for all (9.3.17

easure is

0

Thus, for any random variable ,

)0

f f

ff

f f

A

D T M T Q TdQd

XD T M

A D T M T Q T d AQ

T Q TX

F

PP

E E

P P

0X

Q


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A call struck at on a unit of domestic currency denominated in the foreign currency

1

pays off units of foreign currency at expiration time . The foreign

currency value of this at time

K

K TQ T

zero, which is the foreign risk-neutral expected value of

the discounted payoff, is

1

1

0

ff

f

f

D T KQ T

D T M T Q TD T K

Q Q T

E

E

1

10

1

0

D T KQ TQ

KD T Q T

Q K

E

E


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This is the time-zero value in domestic currency of puts on the foreign

0

1exchange rate. More specifically, a put struck at on a unit of foreign

currency denominated in the domestic currency pays

K

Q

K

1

off units

of domestic currency at expiration time . The domestic currency value of this

put at time zero, which is the domestic risk-neutral expected value of the

dicounted payoff, is

Q TK

T

1

The call we began with is worth of these puts.0

D T Q TK

K

Q

E


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1The foreign currency price of the put struck at on a unit of foreign currency is

1 1

Q 0

The call we began with has a value times this amount. When we

K

D T Q TK

K

E

denominate

both the call and the put this way in foreign currency, we can then understand thefinal result. Indeed, we have seen that the option to exchange units of foreign

currency for one unit of

K

domestic currency (the call) is the same as option to

1exchange units of domestic currency for one unit of foreign currency (the put).

Stated in this way, the result is almost obvius.

K

K

sigal's paradox

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