Stochastic Calculus

Stochastic Calculus
Steve Lalley
http://www.stat.uchicago.edu/ lalley/Courses/390/
Stochastic Calculus – p. 1/27

Tonight —
Foreign Exchange & Exchange Rate Fluctuations
Linear Stochastic Differential Equations
Cameron-Martin-Girsanov Formula
Stochastic Calculus – p. 2/27

Foreign Exchange
Stochastic Models for Exchange Rates
Interest Rates and Exchange Rates
Options on Currency Exchange
Stochastic Calculus – p. 3/27

Basic Principles
Share price processes of tradeable assets are
martingales under any risk-neutral probability
measure.
Risk-neutrality of a probability measure depends on
the numeraire.
Currencies are not tradeable assets!
Money market shares are!
Stochastic Calculus – p. 4/27

In a more realistic model, the drift and/or diffusion
coefﬁcients might be time-varying but deterministic:
dYt = µtYt dt + σtYt dWt
Exchange Rate Model
Let Yt denote the exchange rate at time t between US
Dollars $ and UK Pounds Sterling £, i.e., the number of
pounds that one dollar will buy. A simple model:
dYt = µYt dt + σYt dWt
where Wt is a standard Wiener process under the risk
neutral measure for £ investors, and µ and σ are
constants.
Stochastic Calculus – p. 5/27

Exchange Rate Model
Let Yt denote the exchange rate at time t between US
Dollars $ and UK Pounds Sterling £, i.e., the number of
pounds that one dollar will buy. A simple model:
dYt = µYt dt + σYt dWt
where Wt is a standard Wiener process under the risk
neutral measure for £ investors, and µ and σ are
constants.
In a more realistic model, the drift and/or diffusion
coefﬁcients might be time-varying but deterministic:
dYt = µtYt dt + σtYt dWt
Stochastic Calculus – p. 5/27

Here Wt is a standard Wiener process (Brownian
motion), and At, Bt are adapted process, that is,
processes such that for any time t, the current values
At, Bt are independent of the future increments of the
Wiener process.
The local quadratic variation of the Itô process Zt is
deﬁned by
d[Z, Z]t = B2t dt
Itô Processes
An Itô process is a stochastic process that satisﬁes a
stochastic differential equation of the form
dZt = At dt + Bt dWt
Stochastic Calculus – p. 6/27

The local quadratic variation of the Itô process Zt is
deﬁned by
d[Z, Z]t = B2t dt
Itô Processes
An Itô process is a stochastic process that satisﬁes a
stochastic differential equation of the form
dZt = At dt + Bt dWt
Here Wt is a standard Wiener process (Brownian
motion), and At, Bt are adapted process, that is,
processes such that for any time t, the current values
At, Bt are independent of the future increments of the
Wiener process.
Stochastic Calculus – p. 6/27

Itô Processes
An Itô process is a stochastic process that satisﬁes a
stochastic differential equation of the form
dZt = At dt + Bt dWt
Here Wt is a standard Wiener process (Brownian
motion), and At, Bt are adapted process, that is,
processes such that for any time t, the current values
At, Bt are independent of the future increments of the
Wiener process.
The local quadratic variation of the Itô process Zt is
deﬁned by
d[Z, Z]t = B2t dt
Stochastic Calculus – p. 6/27

Itô’s formula has a number of important generalizations.
Here is one which is sometimes useful in solving SDEs
with time-dependent coefﬁcients: If u(x, t) is a smooth
function of two variables, then
1
du(Zt, t) = ut dt + ux dZt + u
2 xx d[Z, Z]t
Itô’s Formula
If Zt is an Itô process, and if f (x) is a smooth function,
then f (Zt) is also an Itô process whose Itô SDE is
1
df (Zt) = f (Zt) dZt + f (Z
2
t) d[Z, Z ]t
Stochastic Calculus – p. 7/27

Itô’s Formula
If Zt is an Itô process, and if f (x) is a smooth function,
then f (Zt) is also an Itô process whose Itô SDE is
1
df (Zt) = f (Zt) dZt + f (Z
2
t) d[Z, Z ]t
Itô’s formula has a number of important generalizations.
Here is one which is sometimes useful in solving SDEs
with time-dependent coefﬁcients: If u(x, t) is a smooth
function of two variables, then
1
du(Zt, t) = ut dt + ux dZt + u
2 xx d[Z, Z]t
Stochastic Calculus – p. 7/27

Try Itô with f (x) = log x:
d log(Yt) = µ dt + σ dWt − (σ2/2) dt
Since µ and σ are constants, this is easily integrated to
give the general solution to the SDE:
Yt = Y0 exp{(µ − σ2/2)t + σWt}
Solving the SDE
The idea is to guess a solution by applying the Itô
formula to the right process. Assume that under the
probability measure P the exchange rate Yt satisﬁes
dYt = µYt dt + σYt dWt
Stochastic Calculus – p. 8/27

Since µ and σ are constants, this is easily integrated to
give the general solution to the SDE:
Yt = Y0 exp{(µ − σ2/2)t + σWt}
Solving the SDE
The idea is to guess a solution by applying the Itô
formula to the right process. Assume that under the
probability measure P the exchange rate Yt satisﬁes
dYt = µYt dt + σYt dWt
Try Itô with f (x) = log x:
d log(Yt) = µ dt + σ dWt − (σ2/2) dt
Stochastic Calculus – p. 8/27

Solving the SDE
The idea is to guess a solution by applying the Itô
formula to the right process. Assume that under the
probability measure P the exchange rate Yt satisﬁes
dYt = µYt dt + σYt dWt
Try Itô with f (x) = log x:
d log(Yt) = µ dt + σ dWt − (σ2/2) dt
Since µ and σ are constants, this is easily integrated to
give the general solution to the SDE:
Yt = Y0 exp{(µ − σ2/2)t + σWt}
Stochastic Calculus – p. 8/27

Itô:
d log(Yt) = µt dt + σ dWt − (σ2/2) dt
and so
Yt = Y0 exp{(¯
µt − σ2/2)t + σWt}
where
1
t
¯
µt =
µ
t
s ds
0
Time-Dependent SDEs
A similar strategy works for equations with
time-dependent coefﬁcients, for example:
dYt = µYt dt + σYt dWt
Stochastic Calculus – p. 9/27

and so
Yt = Y0 exp{(¯
µt − σ2/2)t + σWt}
where
1
t
¯
µt =
µ
t
s ds
0
Time-Dependent SDEs
A similar strategy works for equations with
time-dependent coefﬁcients, for example:
dYt = µYt dt + σYt dWt
Itô:
d log(Yt) = µt dt + σ dWt − (σ2/2) dt
Stochastic Calculus – p. 9/27

Time-Dependent SDEs
A similar strategy works for equations with
time-dependent coefﬁcients, for example:
dYt = µYt dt + σYt dWt
Itô:
d log(Yt) = µt dt + σ dWt − (σ2/2) dt
and so
Yt = Y0 exp{(¯
µt − σ2/2)t + σWt}
where
1
t
¯
µt =
µ
t
s ds
0
Stochastic Calculus – p. 9/27

Assume that the riskless rates of return rA, rB in the two
currencies are constant, but not necessarily equal. Then
At = exp{rAt} dollars
Bt = exp{rBt} pounds
Interest Rates
Assume that for each of the two currencies US Dollar
and UK Pound Sterling there is a riskless Money Market.
Let At and Bt be the “share prices” of US Money Market
and UK Money Market, respectively, and for simplicity
assume that the time-zero share prices are both 1.
Stochastic Calculus – p. 10/27

Interest Rates
Assume that for each of the two currencies US Dollar
and UK Pound Sterling there is a riskless Money Market.
Let At and Bt be the “share prices” of US Money Market
and UK Money Market, respectively, and for simplicity
assume that the time-zero share prices are both 1.
Assume that the riskless rates of return rA, rB in the two
currencies are constant, but not necessarily equal. Then
At = exp{rAt} dollars
Bt = exp{rBt} pounds
Stochastic Calculus – p. 10/27

Theorem: µ = rB − rA.
Exchange and Interest Rates
The asset US Money Market is riskless to a Dollar
investor, but not to a Pound Sterling investor. Evaluated
in Pounds Sterling, the share price of the US Money
Market asset is
AtYt = Y0 exp{rAt + µt − σ2t/2 + σWt}
where Wt is a standard Wiener Process under the risk
neutral probability measure QB for Pound investors.
Stochastic Calculus – p. 11/27

Exchange and Interest Rates
The asset US Money Market is riskless to a Dollar
investor, but not to a Pound Sterling investor. Evaluated
in Pounds Sterling, the share price of the US Money
Market asset is
AtYt = Y0 exp{rAt + µt − σ2t/2 + σWt}
where Wt is a standard Wiener Process under the risk
neutral probability measure QB for Pound investors.
Theorem: µ = rB − rA.
Stochastic Calculus – p. 11/27

the expectation is taken under QB.
Thus
Y0 = EQ e−rBtA
B
tYt
= EQ e−rBtY
B
0 exp{rAt + µt − σ2t + σWt}
= Y0 exp{(rA − rB + µ − σ2/2)t}EQ exp{σW
B
t}
= Y0 exp{(rA − rB + µ)t}
Proof
Since US Money Market is a tradeable asset, its share
price Y0 at time t = 0 must be the expected value of its
discounted share price AtYt (in £) at time t, where
the discount rate is rB, and
Stochastic Calculus – p. 12/27

Thus
Y0 = EQ e−rBtA
B
tYt
= EQ e−rBtY
B
0 exp{rAt + µt − σ2t + σWt}
= Y0 exp{(rA − rB + µ − σ2/2)t}EQ exp{σW
B
t}
= Y0 exp{(rA − rB + µ)t}
Proof
Since US Money Market is a tradeable asset, its share
price Y0 at time t = 0 must be the expected value of its
discounted share price AtYt (in £) at time t, where
the discount rate is rB, and
the expectation is taken under QB.
Stochastic Calculus – p. 12/27

= EQ e−rBtY
B
0 exp{rAt + µt − σ2t + σWt}
= Y0 exp{(rA − rB + µ − σ2/2)t}EQ exp{σW
B
t}
= Y0 exp{(rA − rB + µ)t}
Proof
Since US Money Market is a tradeable asset, its share
price Y0 at time t = 0 must be the expected value of its
discounted share price AtYt (in £) at time t, where
the discount rate is rB, and
the expectation is taken under QB.
Thus
Y0 = EQ e−rBtA
B
tYt
Stochastic Calculus – p. 12/27

= Y0 exp{(rA − rB + µ − σ2/2)t}EQ exp{σW
B
t}
= Y0 exp{(rA − rB + µ)t}
Proof
Since US Money Market is a tradeable asset, its share
price Y0 at time t = 0 must be the expected value of its
discounted share price AtYt (in £) at time t, where
the discount rate is rB, and
the expectation is taken under QB.
Thus
Y0 = EQ e−rBtA
B
tYt
= EQ e−rBtY
B
0 exp{rAt + µt − σ2t + σWt}
Stochastic Calculus – p. 12/27

= Y0 exp{(rA − rB + µ)t}
Proof
Since US Money Market is a tradeable asset, its share
price Y0 at time t = 0 must be the expected value of its
discounted share price AtYt (in £) at time t, where
the discount rate is rB, and
the expectation is taken under QB.
Thus
Y0 = EQ e−rBtA
B
tYt
= EQ e−rBtY
B
0 exp{rAt + µt − σ2t + σWt}
= Y0 exp{(rA − rB + µ − σ2/2)t}EQ exp{σW
B
t}
Stochastic Calculus – p. 12/27

Proof
Since US Money Market is a tradeable asset, its share
price Y0 at time t = 0 must be the expected value of its
discounted share price AtYt (in £) at time t, where
the discount rate is rB, and
the expectation is taken under QB.
Thus
Y0 = EQ e−rBtA
B
tYt
= EQ e−rBtY
B
0 exp{rAt + µt − σ2t + σWt}
= Y0 exp{(rA − rB + µ − σ2/2)t}EQ exp{σW
B
t}
= Y0 exp{(rA − rB + µ)t}
Stochastic Calculus – p. 12/27

Solution: The option is identical to a call on e−rAT
shares of the US Money Market. To a £ investor, the US
Money Market is a risky asset with price process e−rAtYt.
Thus, the call option may be priced using the
Black-Sholes Formula.
Exercise: Do it! While you’re at it, show how to hedge
the option.
Currency Options
Consider an option Call that gives the owner the right to
buy $1 for £K at time T . What is the arbitrage price at
time 0?
Stochastic Calculus – p. 13/27

Exercise: Do it! While you’re at it, show how to hedge
the option.
Currency Options
Consider an option Call that gives the owner the right to
buy $1 for £K at time T . What is the arbitrage price at
time 0?
Solution: The option is identical to a call on e−rAT
shares of the US Money Market. To a £ investor, the US
Money Market is a risky asset with price process e−rAtYt.
Thus, the call option may be priced using the
Black-Sholes Formula.
Stochastic Calculus – p. 13/27

Currency Options
Consider an option Call that gives the owner the right to
buy $1 for £K at time T . What is the arbitrage price at
time 0?
Solution: The option is identical to a call on e−rAT
shares of the US Money Market. To a £ investor, the US
Money Market is a risky asset with price process e−rAtYt.
Thus, the call option may be priced using the
Black-Sholes Formula.
Exercise: Do it! While you’re at it, show how to hedge
the option.
Stochastic Calculus – p. 13/27

This is a special case of a more general phenomenon:
Risk-Neutral Measure for $
Theorem: Let QA be the risk-neutral probability
measure for the US Dollar investor, and QB the
risk-neutral measure for the UK Pound Sterling investor.
Unless σ = 0 (that is, unless the exchange rate is purely
deterministic), it must be the case that
QA = QB
Stochastic Calculus – p. 14/27

Risk-Neutral Measure for $
Theorem: Let QA be the risk-neutral probability
measure for the US Dollar investor, and QB the
risk-neutral measure for the UK Pound Sterling investor.
Unless σ = 0 (that is, unless the exchange rate is purely
deterministic), it must be the case that
QA = QB
This is a special case of a more general phenomenon:
Stochastic Calculus – p. 14/27

Theorem: QA = QB if and only if SA/SB
t
t
is a constant
random variable. Furthermore, in general, for any ﬁnite
time T ,
dQB
SB
SA
=
T
0
dQA
SA
SB
FT
T
0
Numeraire Change
Suppose that a market has tradeable assets A, B with
share price processes SA
t and SB
t
(evaluated in a common
numeraire C). Let QA and QB be risk-neutral measures
for numeraires A, B, respectively.
Stochastic Calculus – p. 15/27

Numeraire Change
Suppose that a market has tradeable assets A, B with
share price processes SA
t and SB
t
(evaluated in a common
numeraire C). Let QA and QB be risk-neutral measures
for numeraires A, B, respectively.
Theorem: QA = QB if and only if SA/SB
t
t
is a constant
random variable. Furthermore, in general, for any ﬁnite
time T ,
dQB
SB
SA
=
T
0
dQA
SA
SB
FT
T
0
Stochastic Calculus – p. 15/27

Therefore, the likelihood ratio between the risk-neutral
measures for £ and $ investors is
dQB
Y
−1
=
T
exp{(r
dQA
Y
B − rA)T }
F
0
T
Consequence
In the foreign exchange context, the riskless assets for
the two numeraires are US Money Market and UK
Money Market, with share prices (in $)
At = exp{rAt}
Bt = exp{rBt}/Yt
Stochastic Calculus – p. 16/27

Consequence
In the foreign exchange context, the riskless assets for
the two numeraires are US Money Market and UK
Money Market, with share prices (in $)
At = exp{rAt}
Bt = exp{rBt}/Yt
Therefore, the likelihood ratio between the risk-neutral
measures for £ and $ investors is
dQB
Y
−1
=
T
exp{(r
dQA
Y
B − rA)T }
F
0
T
Stochastic Calculus – p. 16/27

V A
t
= V C
t /SA
t
V B
t
= V C
t /SB
t
The time-zero share price is the discounted expected
value of the time−t share price for each of the
numeraires A, B. The discount factors are 1, so
V A = V C/SA = EAV C
0
0
0
t /SA
t
V B = V C/SB = EBV C/SB
0
0
0
t
t
Likelihood Ratio Identity
Let V it be the time-t share price of any contingent claim
in numeraire i = A, B, C. These share prices satisfy:
Stochastic Calculus – p. 17/27

The time-zero share price is the discounted expected
value of the time−t share price for each of the
numeraires A, B. The discount factors are 1, so
V A = V C/SA = EAV C
0
0
0
t /SA
t
V B = V C/SB = EBV C/SB
0
0
0
t
t
Likelihood Ratio Identity
Let V it be the time-t share price of any contingent claim
in numeraire i = A, B, C. These share prices satisfy:
V A
t
= V C
t /SA
t
V B
t
= V C
t /SB
t
Stochastic Calculus – p. 17/27

V A = V C/SA = EAV C
0
0
0
t /SA
t
V B = V C/SB = EBV C/SB
0
0
0
t
t
Likelihood Ratio Identity
Let V it be the time-t share price of any contingent claim
in numeraire i = A, B, C. These share prices satisfy:
V A
t
= V C
t /SA
t
V B
t
= V C
t /SB
t
The time-zero share price is the discounted expected
value of the time−t share price for each of the
numeraires A, B. The discount factors are 1, so
Stochastic Calculus – p. 17/27

Likelihood Ratio Identity
Let V it be the time-t share price of any contingent claim
in numeraire i = A, B, C. These share prices satisfy:
V A
t
= V C
t /SA
t
V B
t
= V C
t /SB
t
The time-zero share price is the discounted expected
value of the time−t share price for each of the
numeraires A, B. The discount factors are 1, so
V A = V C/SA = EAV C
0
0
0
t /SA
t
V B = V C/SB = EBV C/SB
0
0
0
t
t
Stochastic Calculus – p. 17/27

Apply this to the contingent claim with payoff V CSB at
T
T
time T to obtain the following identity, valid for all
nonnegative random variables V C measurable F
T
T :
SBSA
EBV C
T
0
T
= EAV C
T
SASB
T
0
This is the deﬁning property of a likelihood ratio.
Likelihood Ratio Identity
It follows that for every contingent claim V with share
price V C
t
(in numeraire C),
SAEA(V C
EB(V C
0
t /SA
t ) = SB
0
t /SB
t )
Stochastic Calculus – p. 18/27

This is the deﬁning property of a likelihood ratio.
Likelihood Ratio Identity
It follows that for every contingent claim V with share
price V C
t
(in numeraire C),
SAEA(V C
EB(V C
0
t /SA
t ) = SB
0
t /SB
t )
Apply this to the contingent claim with payoff V CSB at
T
T
time T to obtain the following identity, valid for all
nonnegative random variables V C measurable F
T
T :
SBSA
EBV C
T
0
T
= EAV C
T
SASB
T
0
Stochastic Calculus – p. 18/27

Likelihood Ratio Identity
It follows that for every contingent claim V with share
price V C
t
(in numeraire C),
SAEA(V C
EB(V C
0
t /SA
t ) = SB
0
t /SB
t )
Apply this to the contingent claim with payoff V CSB at
T
T
time T to obtain the following identity, valid for all
nonnegative random variables V C measurable F
T
T :
SBSA
EBV C
T
0
T
= EAV C
T
SASB
T
0
This is the deﬁning property of a likelihood ratio.
Stochastic Calculus – p. 18/27

Fact: Zt is a positive martingale. Proof: Itô!
dZt = Ztθt dWt − Ztθ2t dt/2+Ztθ2t dt/2
= Ztθt dWt
=⇒
t
Zt = Z0 +
Zsθs dWs
0
Exponential Martingales
Let Wt be a standard Wiener process, wth Brownian
ﬁltration Ft, and let θt be a bounded, adapted process.
Deﬁne
t
t
Zt = exp
θs dWs −
θ2s ds/2
0
0
Stochastic Calculus – p. 19/27

Proof: Itô!
dZt = Ztθt dWt − Ztθ2t dt/2+Ztθ2t dt/2
= Ztθt dWt
=⇒
t
Zt = Z0 +
Zsθs dWs
0
Exponential Martingales
Let Wt be a standard Wiener process, wth Brownian
ﬁltration Ft, and let θt be a bounded, adapted process.
Deﬁne
t
t
Zt = exp
θs dWs −
θ2s ds/2
0
0
Fact: Zt is a positive martingale.
Stochastic Calculus – p. 19/27

dZt = Ztθt dWt − Ztθ2t dt/2+Ztθ2t dt/2
= Ztθt dWt
=⇒
t
Zt = Z0 +
Zsθs dWs
0
Exponential Martingales
Let Wt be a standard Wiener process, wth Brownian
ﬁltration Ft, and let θt be a bounded, adapted process.
Deﬁne
t
t
Zt = exp
θs dWs −
θ2s ds/2
0
0
Fact: Zt is a positive martingale. Proof: Itô!
Stochastic Calculus – p. 19/27

Exponential Martingales
Let Wt be a standard Wiener process, wth Brownian
ﬁltration Ft, and let θt be a bounded, adapted process.
Deﬁne
t
t
Zt = exp
θs dWs −
θ2s ds/2
0
0
Fact: Zt is a positive martingale. Proof: Itô!
dZt = Ztθt dWt − Ztθ2t dt/2+Ztθ2t dt/2
= Ztθt dWt
=⇒
t
Zt = Z0 +
Zsθs dWs
0
Stochastic Calculus – p. 19/27

Theorem: Under the measure Q, the process
{Wt − t θ
0
sds}0≤t≤T is a standard Wiener process.
Girsanov’s Theorem
Because Zt is a positive martingale under P with initial
value Z0 = 1, for every ﬁxed time T the random variable
ZT is a likelihood ratio: that is,
Q(F ) := EP (IF ZT )
deﬁnes a new probability measure on the σ−algebra FT
of events F that are observable by time T .
Stochastic Calculus – p. 20/27

Girsanov’s Theorem
Because Zt is a positive martingale under P with initial
value Z0 = 1, for every ﬁxed time T the random variable
ZT is a likelihood ratio: that is,
Q(F ) := EP (IF ZT )
deﬁnes a new probability measure on the σ−algebra FT
of events F that are observable by time T .
Theorem: Under the measure Q, the process
{Wt − t θ
0
sds}0≤t≤T is a standard Wiener process.
Stochastic Calculus – p. 20/27

Exchange Rates
Consider again the $ and £ currencies. Assume that each
has a riskless Money Market, and that the rates of return
rA, rB are constant. Assume that the exchange rate Yt
obeys
dYt = (rB − rA)Yt dt + σYt dWt
where Wt is a standard Wiener process under the
risk-neutral probability QB for £ investors. Thus,
Yt = Y0 exp{(rB − rA − σ2/2)t + σWt}.
Stochastic Calculus – p. 21/27

Girsanov implies that under QA the process Wt is a
Wiener process with drift σ. Thus, to the $ investor, it
appears that the exchange rate obeys
dYt = (rB − rA − σ2)Yt dt + σYt d ˜
Wt
where ˜
Wt is a standard Wiener process under QA.
Exchange Rates
Since
dQA
Y
=
T
exp{−(r
dQB
Y
B − rA)T }
F
0
T
= exp{σWT − σ2T /2}
Stochastic Calculus – p. 22/27

Exchange Rates
Since
dQA
Y
=
T
exp{−(r
dQB
Y
B − rA)T }
F
0
T
= exp{σWT − σ2T /2}
Girsanov implies that under QA the process Wt is a
Wiener process with drift σ. Thus, to the $ investor, it
appears that the exchange rate obeys
dYt = (rB − rA − σ2)Yt dt + σYt d ˜
Wt
where ˜
Wt is a standard Wiener process under QA.
Stochastic Calculus – p. 22/27

Stochastic Calculus – p. 23/27

For this, it
sufﬁces to show that for any real λ,
EQ exp{λ(WT − ΘT )} = exp{λ2T /2}
To evaluate the expectation, change measure:
EQ exp{λ(WT − ΘT )} = EP exp{λ(WT − ΘT )}ZT
Proof of Girsanov 1
The statement that X is a standard Wiener process is an
assertion that the increments of X are independent
Gaussian random variables with the correct variances.
Let’s show that under Q, the distribution of WT − ΘT is
gaussian with var T (where ΘT = T θ
0
s ds).
Stochastic Calculus – p. 24/27

To evaluate the expectation, change measure:
EQ exp{λ(WT − ΘT )} = EP exp{λ(WT − ΘT )}ZT
Proof of Girsanov 1
The statement that X is a standard Wiener process is an
assertion that the increments of X are independent
Gaussian random variables with the correct variances.
Let’s show that under Q, the distribution of WT − ΘT is
gaussian with var T (where ΘT = T θ
0
s ds). For this, it
sufﬁces to show that for any real λ,
EQ exp{λ(WT − ΘT )} = exp{λ2T /2}
Stochastic Calculus – p. 24/27

Proof of Girsanov 1
The statement that X is a standard Wiener process is an
assertion that the increments of X are independent
Gaussian random variables with the correct variances.
Let’s show that under Q, the distribution of WT − ΘT is
gaussian with var T (where ΘT = T θ
0
s ds). For this, it
sufﬁces to show that for any real λ,
EQ exp{λ(WT − ΘT )} = exp{λ2T /2}
To evaluate the expectation, change measure:
EQ exp{λ(WT − ΘT )} = EP exp{λ(WT − ΘT )}ZT
Stochastic Calculus – p. 24/27

t
t
= exp{
(θs + λ) dWs +
(λθs − θ2/2 − λ2/2) ds}
s
0
0
t
t
= exp{
(θs + λ) dWs −
(θs + λ)2 ds/2}
0
0
Thus, Ht is an exponential martingale under P , and so its
expectation is constant over time. A similar calculation
establishes the independence of the increments.
Proof of Girsanov 2
Objective: Show that EP HT = 1, where
Ht = exp{λ(Wt − Θt) − λ2t/2}Zt
Stochastic Calculus – p. 25/27

t
t
= exp{
(θs + λ) dWs −
(θs + λ)2 ds/2}
0
0
Thus, Ht is an exponential martingale under P , and so its
expectation is constant over time. A similar calculation
establishes the independence of the increments.
Proof of Girsanov 2
Objective: Show that EP HT = 1, where
Ht = exp{λ(Wt − Θt) − λ2t/2}Zt
t
t
= exp{
(θs + λ) dWs +
(λθs − θ2/2 − λ2/2) ds}
s
0
0
Stochastic Calculus – p. 25/27

Thus, Ht is an exponential martingale under P , and so its
expectation is constant over time. A similar calculation
establishes the independence of the increments.
Proof of Girsanov 2
Objective: Show that EP HT = 1, where
Ht = exp{λ(Wt − Θt) − λ2t/2}Zt
t
t
= exp{
(θs + λ) dWs +
(λθs − θ2/2 − λ2/2) ds}
s
0
0
t
t
= exp{
(θs + λ) dWs −
(θs + λ)2 ds/2}
0
0
Stochastic Calculus – p. 25/27

A similar
calculation establishes the independence of the
increments.
Proof of Girsanov 2
Objective: Show that EP HT = 1, where
Ht = exp{λ(Wt − Θt) − λ2t/2}Zt
t
t
= exp{
(θs + λ) dWs +
(λθs − θ2/2 − λ2/2) ds}
s
0
0
t
t
= exp{
(θs + λ) dWs −
(θs + λ)2 ds/2}
0
0
Thus, Ht is an exponential martingale under P , and so
its expectation is constant over time.
Stochastic Calculus – p. 25/27

Proof of Girsanov 2
Objective: Show that EP HT = 1, where
Ht = exp{λ(Wt − Θt) − λ2t/2}Zt
t
t
= exp{
(θs + λ) dWs +
(λθs − θ2/2 − λ2/2) ds}
s
0
0
t
t
= exp{
(θs + λ) dWs −
(θs + λ)2 ds/2}
0
0
Thus, Ht is an exponential martingale under P , and so
its expectation is constant over time.
A similar
calculation establishes the independence of the
increments.
Stochastic Calculus – p. 25/27

Scratch
Stochastic Calculus – p. 26/27

Scratch
Stochastic Calculus – p. 27/27

Stochastic Calculus

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