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Exchange Rate process and Interest RateParity

Mauricio Bedoya

javierma36@gmail.com

September 2014

To understand this blog, we must know:

1. Ito Product Rule.

2. Interest Rate Parity.

3. Ito Calculus.

Definition: Ito Product Rule.

This rule state, that if you have the product of two stochastic process1 X and Y, then:

d(X Y)X Y =dX Y + X dY + dX dY (1)

for the last term, we must consider the multiplication table (stochastic calculus), that state:dt dt= 0; dt dw= 0 and dw dw= dt.

Defnition: Interest Rate Parity (Wikipedia).Is a no-arbitrage condition representing an equilibrium state under which investors will be

indifferent to interest rates available on bank deposits in two countries. Mathematically, theevolution of the exchange rate can be expressed

X(t) = X(0) e(rfrd)t (2)with rd (domestic interest rate), rf (foreign interest rate) constant. Now, lets try to unders-tand this equation with an example. Imagine that we are a Colombian (COP) citizen thatis going to invest 1000 COP in Europe (e). In this case, X(t) characterize an

e

COP relations-

hip. If we capitalize the numerator and denominator by their corresponding interest rate, we get

1Check any Stochastic process or search in google for a definition.

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$1000COP X(t) 1+r(e)

1+r(COP)

Assuming that both interest rate are close to zero, we can assume that the previous equationis an Euler approximation of

$1000COP X(t) er(e)r(COP)

The previous expression allow to characterize the evolution of an investment in the foreignmarket, were randomness comes only from the exchange rate evolution.

Because the exchange rate can NOT be negative, we can use the Geometric Brownian Motionto characterize the relative change.

dX(t)

X(t) = dt + dw (3)

with (mean rate), (volatility) constant, and dw N[0,dt].

Now, lets apply the Ito product rule to equation 2

dX(t)=dX(0) e(rfrd)t + X(0) (rf rd) dt e(rfrd)t + dX(0) (rf rd) dt e(rfrd)t

replacing dX(0) with equation 3, we get

dX(t)

X(t)= ( + rf rd) dt + dw

to make this Martingala (drift-less), we make a change of variable

dX(t)

X(t)= [

( + rf rd) dt

+ dw] dw

(4)

Then, to eliminate the drift: = rdrf. Replacing in equation 3 the value of and integratingin the interval [0,T], we get:

T0

dX(t)

X(t) Lebesgue Integral

=

T0

(rd rf) dt Lebesgue Integral

+

T0

dw Ito Integral

(5)

The Ito integral requires some knowledge that you can found in any Stochastic process book.The primary difference between Lebesgue and Ito calculus is the quadratic variation. Lets solvethe Ito integral:

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d( w(t)) = dw+1

22 dt

T0

dw= T0

d( w(t)) T0

122 dt

T0

dw= w(T) 1

22 T

(6)

Solving the Lebesgue integrals and remplacing equation 6 in 5, we get

X(T)=X(0) e(rdrf12

2)T+ w(T) (7)

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