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Change of Measure (Domestic Currencyas Numeraire)

Mauricio Bedoya

javierma36@gmail.com

September 2014

To understand this blog, we must known:

1. Ito Quotien Rule.

2. Martingala Process.

3. Ito Calculus.

Definition: Ito Quotien Rule

This rule state, that if you have the quotient of two stochastic process 1 X and Y, then:

d(XY

)X

Y

=dX

X dY

Y +

dY2

Y2 dX dY

X Y (1)

Equation 1 can be understood like:

We are long X and short Y.

We are long Y volatility.

We are short

(XY

)

2

.

Definition: Martingala

A process is Martingala if:

E[X(u)|F(t)] =X(t) with u > t. (2)In English, a process is Martingala if the old (known) information, doesnt help in forecastingthe future value ofX(t).

1

Check any Stochastic process or search in google for a definition.2Correlation between X and Y.

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Now, explaining Itos Calculus in a blog is not possible. However, we will try to implementevery step, until we get the final desired expression.

Now assume that S(t) follow a Geometric Brownian Motion, then:

d(S(t))

S(t)= dt + dw (3)

with dw N[0,t], and constant mean and volatility respectively. Next, assume thatC(t) = e

rt; called the money market capitalization process. The inverse ofC(t) is D(t); calledthe money market discount process.

Our goal is to make the discount stock price process (domestic) Martingala. Mathematically

E[D(u)S(u)

|F(t)] =E[

S(u)

C(u) |F(t)] =S(t) with u >t. (4)

To proceed, we have two options: use the Ito Product Rule or Ito Quotien rule. Here, we aregoing to use the quotien rule3.

From equation 4, we have

d(S(t)

C(t))

S(t)

C(t)

=dS(t)

S(t) dC(t)

C(t)+d(C(t), C(t))

C(t) d(S(t), C(t))

S(t) C(t)

=S(t)

(

dt +

dw)

S(t) C(t)

r

dt

C(t) + 0 0= ( r) dt + dw= (

( r) dt

+ dw) dw

= dw

(5)

Now, a few explanation. The first zero comes from the quadratic variation of a non randomprocess (dt * dt = 0). The second zero comes from the product of (dt * dw = 0) and (dt *

dt = 0). Next, we make a change of variable to make the process Martingala (if no dt in theequation, the process is Martingala, remember this). Now, lets integrate between 0 and T.

T

0

d(S(t)

C(t))

S(t)

C(t) Lebesgue Integral.

=

T

0

dw Ito Integral.

(6)

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:

3Implement the process using the product rule. You should arrive to the same result.

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d( w(t)) = dw+122 dt

T0

dw= T0

d( w(t)) T0

122 dt

T

0

dw= w(T) 122 T

(7)

Integrating equation 7 in 6 and solving the Lebesgue integral, we get

Ln[S(t)

C(t)]T0 = w(T)

1

22 T

S(T)

C(T) =

S(0)

C(0) ew(T)

1

2

2T

S(T) = S(0) C(T) ew(T)122TS(T)=S(0) e(r122)T+w(T) .

(8)

Ok, this is it. Replacing equation 8 in equation 4, we can found that effectively S(t)is Martingale.

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