Risk neutral measure -the continuous case
Change of measure-reminder
Theorem
Let (Ω, F, P) be a probability space and let Z be an almost
surely positive random variable with E[Z ] = 1. For A ∈ F ,
define
Z
Z (w)dP(w)
P̃(A) = E[1A Z ] =
A
Then P̃ is a probability measure . Furthermore if X is a
nonnegative random variable then
Ẽ[X ] = E[ZX ]
If Z is strictly posititive with probability 1 , we have
E[Y ] = Ẽ[
Y
]
Z
Remark The theorem can be generalized to a general random
variable when E|XZ | exists.
Equivalent probability measures
Definition
Let Ω be nonempty set and F be a σ-algebra of subsets of Ω.
Two probability measures P and P̃ are equivalent if they agree
on which sets of F that have probability 0. I.e.
P(A) = 0 ⇔ P̃(A) = 0
Theorem
Let P and P̃ be equivalent probability measures defined on
(Ω, P). Then there exist an almost surely positive random
variable Z , such that E[Z ] = 1 and
Z
P̃(A) =
Z (w)dP(w)
A
Z is called the Radon-Nikodym derivative of P̃ with respect to P.
Example
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Let X be a standard Normal random variable, with densidty
x2
1
f (x) = √ e− 2
2π
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Y = X + θ.
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Define Z = eθX − 2 θ
Z is positive and E[Z ] = 1
1 2
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Z y
1 2
1
2
P̃(Y ≤ y ) = √
eθx− 2 θ e−(x−θ) /2 dx
2π −∞
Z y
1
2
=√
e−x /2 dx
2π −∞
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Thus, under P̃, Y has standard Normal distribution.
Radon Nikodym process-Introduction
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Objective: To extend the concept change of measure to a
stochastic process.
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(Ω, F, P) with filtration Ft , 0 ≤ t ≤ T .
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Let Z = ZT be a positive random variable measurable with
respect to FT ,with E[Z ] = 1.
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Z
P̃(A) = E[1A Z ] =
Z (w)dP(w)
A
defines a new probability measure.
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For a random variable Y which is F measurable
Ẽ[Y ] = E[ZY ].
Radon-Nikodym process
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Define Zt = E[Z |Ft ].
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Zt is Ft measurable.
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Zt is a martingale. For s < t
E[Zt |Fs ] = E[E[Z |Ft |Fs ] = E[Z |Fs ] = Zs
The first equality follows from the definition of Zt and the
second from tower property of conditional expectation.
Expectation under "tilde" measure
Lemma
Let 0 ≤ t ≤ T and let Y be Ft measurable. Then
Ẽ[Y ] = E[YZt ]
Proof
Ẽ[Y ] = E[YZ ] = E[E[YZ |Ft ] = E[YE[Z |Ft ]] = E[YZt ]
The first equality follows from the definition of Ẽ, the second
from the tower property of conditional expectation, the fourth
from the property "taking out wht is known" of conditional
expectation, since Y is Ft measurable,and the last from the
definition of Zt .
No contradiction
Let Y be Fs measurable (and hence, also Ft measurable). Let
s<t
E[YZt ] = E[E[YZt |Fs ]] = E[YE[Zt |Fs ]] = E[YZs ]
We call the process Z (t), 0 ≤ t ≤ T the Radon-Nikodym
process.
Conditional expectation for the "tilde" measure
Lemma
Let 0 ≤ s < t ≤ T . Let Y be Ft measurable. For s < t:
Ẽ[Y |Fs ] =
1
E[YZt |Fs ]
Zs
Proof
Need to show that for A ∈ Fs
Ẽ[1A
1
E[YZt |Fs ]] = Ẽ[1A Y ]
Zs
1
E[YZt |Fs ]] = E[1A E[YZt |Fs ]]]
Zs
1.
= E[E[1A YZt |Fs ] = E[1A YZt ] = Ẽ[1A Y ]
Ẽ[1A
2.
3.
4.
1. From Lemma 2.1 since
1
Zs E[YZt |Fs ]
is Fs measurable.
2. Since 1A is Fs measurable, and the property "taking out
what is known" of conditional expectation.
3. Follows from the tower property of conditional expectation.
4. From the definition of Ẽ in Lemma 2.1.
Compare to the Binomial model
In the binomial model we find a risk neutral measure
−d
u−1−r
p̃ = 1+r
u−d , q̃ = u−d . Then we show that under this measure:
1. The discounted stock price is martingale.
2. There is a self-financing portfolio with stocks and money
where at each period its value is the same value as the
option.
3. The value of the option at time n is Ẽ[ (1+rVN)N−n ] where Ẽ is
the expectation under the risk neutral measure.
We want to define similar quantities in the case that the stock
value is geometrical brownian motion i.e.
dS(t) = α(t)S(t)dt + σ(t)S(t)dW (t)
or
S(t) = S(0)e
Rt
Rt
1 2
0 (α(s)− 2 σ (s))ds+ 0
σ(s)dW (s)
Define risk neutral measure-Girsanov Theorem
Theorem
Let (Ω, F, P) be a probability space, with filtration Ft . Let W (t)
be a Brownian motion and Θ(t) an adapted process, 0 ≤ t ≤ T .
Define
t
Z
Z (t) = exp(−
0
1
Θ(s)dW (s) −
2
and
Z
W̃ (t) = W (t) +
Z
t
Θ2 (s)ds)
0
t
Θ(u)du
0
Assume that:
Z
E[
T
Θ2 (u)Z 2 (u)du] < ∞.
0
Let Z = Z (T ). Then E[Z ] = 1. Define P̃ by P̃(A) = E[1A Z ].
Then the process W̃ is a Brownian motion with respect to P̃.
Proof of Girsanov theorem I
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Z (t) is a Radon-Nikodym process.
Z
Z t
1 t 2
Θ (s)ds
X (t) = −
Θ(s)dW (s) −
2 0
0
Then Z (t) = eX (t) .
Let f (x) = ex , f 0 (x) = f (x),f ”(x) = f (x).
I
1
dZ (t) = Z (t)dX (t) + Z (t)dX (t)dX (t)
2
1
1
= −Z (t)Θ(t)dW (t) − Θ2 (t)dt + Θ2 (t)dt
2
2
= −Z (t)Θ(t)dW (t)
Z
Z (t) = Z (0) −
t
Z (s)Θ(s)dW (s)
0
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is a martingale with respect to P(since it is Ito integral) .
E[Z (t)] = Z (0) = 1, Since Z is positive, E[Z ] = 1
Z (t) = E[Z (T )|F(t)], thus, a Radon-Nikodym process.
W̃ is a Brownian motion under P̃
Apply the Levy chractiriztion theorem of Brownian motion.
Need to show that:
1. W̃ (0) = 0.
2. W̃ has continuous sample paths.
3. At each time t, W̃ has quadratic variation equal to t.
4. Under the measure P̃, W̃ is a martingale.
Clearly 1 and 2 hold. Next we show 3:
d W̃ (t)d W̃ (t) = (Θ(t)dt + dW (t))2 = dt
In order to show that under P̃, W̃ is a martingale we will use the
product rule:
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Let X (t) and Y (t) two Ito processes:
dX (t) = α1 (t)dt + σ1 (t)dW (t),
dY (t) = α2 (t)dt + σ2 (t)dW (t)
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d(X (t)Y (t)) = dX (t)Y (t) + X (t)dY (t) + dX (t)dY (t)
Z (t)W̃ (t) is a martingale under P
We saw that dZ (t) = −Z (t)Θ(t)dW (t).
Z t
d(Z (t)W̃ (t)) = −Z (t)Θ(t)dW (t)( Θ(s)ds + W (t))
0
+Z (t)(Θ(t)dt + dW (t)) − Z (t)Θ(t)dW (t)(Θ(t)dt + dW (t))
Z t
= −Z (t)Θ(t)dW (t)( Θ(s)ds + W (t))
0
+Z (t)(Θ(t)dt + dW (t)) − Z (t)Θ(t)dt
Z t
= Z (t)dW (t)(1 −
Θ(s)ds − W (t)) = Z (t)dW (t)(W̃ (t) − 1)
0
Thus Z (t)W̃ (t) is an Ito integral and therefore martingale under
P.
W̃ is a martingale under P̃
Let s < t, by Lemma 5
Ẽ[W̃ (t)|Fs ] =
=
1
E[W̃ (t)Z (t)|Fs ]
Z (s)
1
W̃ (s)Z (s) = W̃ (s)
Z (s)
Stock price under P̃
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The stock price behaves as a Geometric Brownian motion
dS(t) = α(t)S(t)dt + σ(t)dW (t)
S(t) = S(0)e
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Rt
0 (α(s)−
σ(s)dW (s)
the interest rate at time t is an adapted process R(t), thus
the discount process is:
D(t) = e−
.
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R
σ 2 (s)
ds+ 0t
2
When R(t) = r , D(t) = e−rt .
Rt
0
R(s)ds
The discounted stock price under the risk neutral
measure
d(D(t)S(t)) = dD(t)S(t) + D(t)dS(t) + dS(t)dD(t)
= −R(t)D(t)S(t)dt + D(t)α(t)S(t)dt + D(t)σ(t)S(t)dW (t)
α(t) − R(t)
dt + dW (t)
= D(t)σ(t)S(t)
σ(t)
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Θ(t) =
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α(t) − R(t)
σ(t)
(1)
Under the measure defined in Girsanov theorem
Z t
θ(s)ds
W̃ (t) = W (t) +
0
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is a Brownian motion.
Under the risk neutral measure D(t)S(t) is a martingale.
Stock price under risk neutral measure
The solution to the stochastic differential equaiton
dD(t)S(t) = D(t)σ(t)S(t)d W̃ (t)
is
0
R
σ 2 (s)
ds+ 0t
2
σ(s)d W̃ (s)
0 (R(s)−
R
σ 2 (s)
)ds+ 0t
2
σ(s)d W̃ (s)
D(t)S(t) = S(0)e−
Rt
Thus
S(t) = S(0)e
Rt
α(t) does not appear in the above formula. Under the risk
neutral measure P̃ defined by Girsanov theorem the yield from
the stock is the same as the yield from interest.
Self-financing portfolio
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X (t)- A self financing portfolio with stocks and money.
X (0) is the value of the portfolio at time 0, then
X (0) = ∆(0)S(0) + (X (0) − ∆(0)S(0))
i.e. the portfolio has ∆(0) stocks and invest
(X (0) − ∆(0)S(0)) in money.
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e
R t+h
t
R(u)du
= 1 + hR(t) + o(h)
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X (t + dt) = ∆(t)S(t + dt) + (1 + R(t)dt)(X (t) − ∆(t)S(t))
= X (t) + ∆(t)(S(t + dt) − S(t)) + R(t)(X (t) − ∆(t)S(t))dt
X (t + dt) − X (t)
= ∆(t)(S(t + dt) − S(t)) + R(t)(X (t) − ∆(t)S(t))dt
Self-financing porfolio
dX (t) = ∆(t)dS(t) + R(t)(X (t) − ∆(t)S(t))dt
= ∆(t)(α(t)S(t)dt + σ(t)S(t)dW (t)) + R(t)(X (t) − ∆(t)S(t))dt
Thus
d(D(t)X (t)) = dD(t)X (t) + D(t)dX (t) = −D(t)R(t)X (t)dt
+D(t)(∆(t)(α(t)S(t)dt + σ(t)S(t)dW (t))
+R(t)(X (t)dt − ∆(t)S(t)))
α(t) − R(t)
dt + dW (t)
= D(t)∆(t)S(t)σ(t)
σ(t)
= D(t)∆(t)S(t)σ(t)d W̃ (t)
Thus under the risk neutral measure the discounted value of
the portfolio is a martingale.
The option price
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The option’s value at time T , V (T ) is known for each
scenario e.g. V (T ) = (S(T ) − K )+ .
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Want to obtain the option price-c(t, x) at time t when the
S(t) = x.
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Interst rate is fixed equals R(t) = r . α(t) = α,σ(t) = σ.
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S(t) = S(0)e(α−
P.
σ2
)t+σW (t)
2
W (t) a Brownian motion under
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d(e−rt S(t)) = −re−rt S(t)dt + e−rt (αS(t)dt + σS(t)dW (t))
α−r
dt + dW (t)) = S(t)e−rt σd W̃ (t)
= S(t)e−rt σ(
σ
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S(t) = S(0)e(r −
σ2
)t+W̃ (t)
2
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Consider a self financing portfolio that at any time equals
the value of the option e.g. X (T ) = V (T ),
e−rt X (t) = e−rt V (t)
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e−rt X (t) is a martingale under P̃.
X (t)e−rt = Ẽ[X (T )e−rT )|F(t)]
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Thus defne
V (t) = Ẽ[e−r (T −t) V (T )|F(t)]
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e−rt V (t) is a martingale (under P̃).
Black-Schols again
c(t, x) = Ẽ[e−(T −t)r (S(T ) − K )+ ]|Ft ]
S(T ) = S(t) exp(σ(W̃ (T ) − W̃ (t)) + (r −
Define τ = T − t, Y =
σ2
)(T − t))
2
W̃ (T )−W̃ (t)
√
τ
c(t, x) = e−r τ Ẽ(S(T ) − K )+ )|Ft ]
Z ∞
√
y2
σ2
−r τ 1
√
=e
(xeσ τ y +(r − 2 )τ − K )+ e− 2 dy
2π −∞
Z
2
2
√
1
− y2
σ τ y +(r − σ2 )τ
−
K
)e
dy
= e−r τ √
(xe
2π y > σ√1 τ (ln(K /x)−(r − σ22 )τ )
Difference of two integrals
1
√
2π
Z
2
xeσ
√
2
2
τ y − σ2 τ − y2
e
dy
1
y > σ√
(ln(K /x)−(r − σ2 )τ )
τ
1
= x√
2π
Z
1
= x√
2π
Z
x(1 − Φ(
1
2
e− 2 (y
2 −2σ √τ y +σ 2 τ )
1
(ln(K /x)−(r − σ2 )τ )
y > σ√
τ
1
y > σ√
τ
1
2
(ln(K /x)−(r − σ2 )τ )
e− 2 (y −σ
√
τ )2
y
ln(K /x) − (r + σ 2 /2)τ
)) = xΦ(d+ (τ, x))
στ
where
d+ (τ, x) =
(r + σ 2 /2)τ + ln(x/K ))
√
σ τ
dy
The second integral is evaluated similarly and equals:
Z
1
2
K√
e−y /2 dy
2
2π y > σ√1 τ (ln(K /x)−(r − σ2 )τ )
1
σ2
)τ )))
= K (1 − Φ( √ (ln(K /x) − (r −
2
σ τ
1
σ2
= K Φ( √ (ln(x/K ) + (r −
)τ )) = K Φ(d− (τ, x))
2
σ τ
e−r τ K Φ(
ln(x/K ) + (r − σ 2 /2)τ
√
) = e−r τ K Φ(d− (τ, x))
σ τ
where
d− (τ, x) = d+ (τ, x) − σ 2 τ