Download 7. Equivalent Martingale Measures So far we have considered

7. Equivalent Martingale Measures
So far we have considered derivative asset
pricing exploiting PDEs implied by arbitragefree portfolios.
Another approach is to change the probability measure to another probability measure
implied by arbitrage-free markets such that
under that the (risk-free return discounted)
prices become martingales.
1
As for background, consider pricing an European call option.
The aim is to find the fair price for the option
given the available information.
To price the option (Ct), we use the best
prediction of the end value in the light of
available information, such that
(1)
Ct = Et [ρ max(ST − K, 0)],
where Et is the conditional expectation given
information up to time t, and ρ is a discount
factor.
2
The no arbitrage theory implies that if the
option is replicable, then the discount factor
will be the riskfree rate, and the probability
measure with respect to which the expectation must be calculated is such that the
discounted price process
(2)
S̃t = e−r tSt,
where r is the riskfree return, is martingale.
3
To illustrate the situation, consider the following single period discrete world.
Example 7.1: Suppose we have a call option C on
stock S and a bank account B. Let the exercise price
of the option be K, and assume that there are two
possible end values S1 > K > S2 of the stock.
So
p
³
³³
³
³³
PP
S0³
PP
P
1 − pPP
S1 = u S0, with u > 1
S2 = d S0, with d < 1,
where p is the probability that the price goes up to
S = u S0 , and S0 is the current price of the stock.
Then the option with initial cost C has the end value
max{S1 − K, 0}. To replicate this with the stock and
bank account with (riskfree) interest rate, r, we may
construct the following strategy:
4
Buy one share of the stock by financing it with cash
and S2 /(1+r) borrowed at rate r from the bank (after
one period the repayment is accordinly S2 ). The value
of the initial position is then
(3)
S0 −
1
S2 .
1+r
The end value of the position is according to the stock
price as
Stock value
Loan repayment
Total payoff
S = S1
S1
−S2
S1 − S2
S = S2
S2
−S2
0
5
We observe that in the case of S = S2 the payoff is
0, the same as with the call options, and in the case
S = S1 the total payoff is S1 − S2 = a(S1 − K), where
a = (S1 − S2 )/(S1 − K). Thus in all, the payoff of the
strategy is exactly the same as the payoff of a call
options.
This implies that in the absence of arbitrage the cost
of the investment must be the same in both cases.
That is, buying a call options must have the same
value as the other strategy based on one stock and
bank loan.
So
aC = S0 −
1
S2
1+r
or
C = (S0 −
1
1
S2 )/a =
p∗ (S1 − K),
1+r
1+r
where
p∗ =
1+r−d
.
u−d
6
We observe that 0 < p∗ < 1 (provided that 1 + r < u
Exercise: What is the rationale that this also holds?),
so that p∗ can be considered as a conditional probability given the initial price S0 of the stock (the probability depends on u and d which are dependent on
S0 , i.e., how much the price should go up to reach
the given value S1 , or to decrease to go down to the
other possible given value S2 ). These probabilities are
called risk neutral, hedging or martingale probabilities
or probability measures.
The last name is because
they make the discounted price process S̃ a martingale.
7
This is seen as follows: We easily find that
(4)
S0 =
1
(p∗ S1 + (1 − p∗)S2 ) = p∗ S̃1 + (1 − p∗ )S̃2 ,
1+r
where S̃ = S/(1 + r) is the discounted price process.
That is, the conditional expectations of the
discounted price process S̃ given the information I0 =
{S0 } is
(5)
E∗ [S̃|I0 ] = p∗ S̃1 + (1 − p∗ )S̃2 = S̃0 = S0 ,
so that S̃ is martingale with respect to the risk neutral
probability measure, as stated above.
8
We observe that the option value does not at all depend on the true probabilities p and q = 1 − p!
However, if we write the above martingale equaton as
(q ∗ = 1 − p∗ )
(6)
p∗
q∗
E [S̃|I0 ] = p S̃1 + q S̃2 = pS̃1 + q S̃2,
p
q
∗
∗
∗
where p∗ /p and q ∗/q can be considered kinds of likelihood ratios or odds ratios, judged by the markets
for the events that the stock price will be S1 and S2 ,
respectively.
So the market expected value of the future stock price
is a kind of likelihood weighted value of the possible
future outcomes.
9
Translations of Probabilities
Probability Measure
As an illustration, consider the probability
density f (z) of a standard normal distribution,
1 − 1 z2
(7)
f (z) = √ e 2 .
2π
Then probability of that the random variable
Z is near a specific value z̄ is
³
(8)
P
1
1
z̄ − ∆ < Z < z̄ + ∆
2
2
Z
´
z̄+ 21 ∆
=
z̄− 21 ∆
1 2
1
√ e− 2 z dz,
2π
which is a real number (between zero and
one).
Thus, the probability associates a real number (in this case between zero and one) to
intervals on real line, or more generally to
(Borel) sets.
Such functions are called measures in mathematics or measure functions.
10
Because ∆ is small
Z z̄+ 1 ∆
2
Z
1
1 − 1 z2
1 − 1 z̄ 2 z̄+ 2 ∆
√ e 2 dz ≈ √ e 2
dz
1
1
z̄− 2 ∆
z̄− 2 ∆
2π
2π
1 − 1 z̄ 2
= √ e 2 ∆.
2π
(9)
For infinitesimal ∆, denoted as dz, we designate the associated measure by symbol dP (z),
or simply by dP .
Thus, in the above case we have
(10)
1 − 1 z2
dP (z) = √ e 2 dz.
2π
11
Generally, if P is a probability measure, we
have
(11)
Z ∞
−∞
dP = 1.
With these notations, e.g.,
(12)
E[X] =
Z ∞
−∞
x dP (x).
So the expected value is mathematically an
integral with respect to probability measure.
dP is called sometimes the density of the
probability measure P .
12
Changing Probability Measure
Martingale model is a central tool for modeling fair prices of derivative securities.
However, generally, if St is a risky asset, then
given information up to time point t, we have
(13)
Et[St+h] > (1 + rf )St,
(h > 0),
because investors want some compensation
for the risk, where Et is the conditional expectation, and rf is the risk-free rate.
13
However, we observed in the PDE approach
that under the arbitrage-free pricing the riskfree rate should be a proper discounting factor in pricing risky derivative assets.
More importantly, the fundamental theorem
of asset pricing establishes the equivalence of
the absence of arbitrage opportunity and existence of martingale measure in (the stochastic model of) financial markets.
14
A probability measure, P̃ , is a martingale measure for the discounted price process S̃t =
e−rtSt, if S̃t is martingale under P̃ , i.e.
(14)
EP̃
t [S̃s ] = S̃t, whenever s ≥ t,
where r is the riskfree rate, and EP̃
t indicates
that the conditional expectation is taken with
respect to probability measure P̃ .
This theory implies that the no-arbitrage price
of a contingent claim with underlying security St and (random) payoff X at maturity T
is obtained by
(15)
Ct =
EP̃
t
h
e−rτ X
i
,
where τ = T − t, and P̃ is the martingale
measure for the discounted price process S̃t.
We say that S̃t is P̃ -martingale.
15
Thus a martingale measure can be viewed
as a representation of the market’s current
opinion on the evolution of values of underlying assets and the prices of all derivatives
contingent to them.
Consequently, the knowledge of the martingale measure is all that is needed, in principle,
to value whatever derivative securities by the
formula of the form (15).
16
Then given a stock price process St with
probability measure P , the goal is to find the
martingale measure P̃ .
This can be accomplished if there is an invertible function (one-to-one) ξ(z) such that
(16)
dP̃ (z) = ξ(z) dP (z).
Now recalling from standard calculus; if G(z) =
R
g(z) dz then g(z) = dG(z)/dz = G0(z), i.e.,
g(z) is the (mathematical) derivative of G(z).
Alternatively, we can write dG(z) = g(z)dz,
and hence
Z
(17)
G(z) =
Z
dG(z) =
g(z)dz.
17
In the same manner, because
(18) P̃ (z) =
Z z
−∞
dP̃ (t) =
Z z
−∞
ξ(t) dP (t),
so that we can adopt notation
(19)
dP̃ (z)
= ξ(z),
dP (z)
and call ξ as a derivative of P̃ with respect
to P .
In mathematical measure theory this is known
as the Radon-Nikodym derivative.
18
The existence of ξ is guaranteed if P and P̃
satisfy
(20) P̃ (dz) > 0 if and only if P (dz) > 0.
Actually this condition guarantees besides the
existence of ξ (known as Radon-Nikodym Theorem), also the equivalence of P̃ and P in the
sense
(21)
dP̃ (z) = ξ(z) dP (z)
and
(22)
dP (z) = ξ(z)−1dP̃ (z).
In this sense P̃ and P are equivalent probability measures.
19
Example 7.2: (GBM) Consider the geometric Brownian motion
(23)
dSt = µSt dt + σSt dWt ,
so that
(24)
µ
Zt = log (St/S0 ) =
t≥0
¶
1 2
µ − σ t + σ Wt
2
and
(25)
Zt ∼ N (µ∗ t, σ 2 t),
where µ∗ = µ − σ 2 /2.
Thus
(26)
dP (z) = √
1
2π σ 2 t
e
∗ 2
t)
− (z−µ
2σ 2 t
dz.
20
Let
(27)
µ
Z̃t =
¶
rf −
1 2
σ t + σ W̃t,
2
where rf is a riskfree rate, and W̃t is another Wiener
process (the next example shows the relationship between W and W̃ ).
Then
(28)
dP̃ (z̃) = √
1
2π σ 2t
∗ 2
t)
− (z−r
2σ 2 t
e
dz̃,
where r∗ = rf − 21 σ 2.
Because the density function of the normal distribution is always positive, trivially
(29)
P (dz) > 0 ⇐⇒ P̃ (dz) > 0.
21
The transformation between these two probability measures is
(30)
−
ξ(z) = e
2
2
(µ∗ −r∗ )z− 1 (µ∗ −r∗ )t
2
2
σ
,
so that
(31)
dP̃ (z) = ξ(z) dP (z).
22
The Girsanov Theorem
The Radon-Nikodym theorem gives the conditions under which the derivative ξ exist (i.e.,
that we can move to another probability measure without losing any information, which
means that those events that have positive
probability have also positive probability under the other measure).
The Girsanov Theorem provides the conditions under which the Radon-Nikodym derivative exists for cases where Zt is a continuous
stochastic process.
23
For the purpose, let {It}, t ∈ [0, T ] be a family
of information sets (T < ∞). Define
Rt
(32) ξt = e
Rt 2
1
0 Xu dWu − 2 0 Xu du ,
t ∈ [0, T ],
where Xt is an It-measurable process (that is
once It is given, the value of Xt is known),
and Wt is a Wiener process with probability
measure P .
24
It is assumed that Xt does not increase too
fast, so that
(33)
· Rt
E e
0 Xu du
¸
< ∞, t ∈ [0, T ],
called Novikov condition (after a Russian mathematician).
Using Itô
(34)
dξt = ξtXt dWt,
from which we immediately see that ξt is a
martingale, because Wt is a Wiener process
and there is no drift component in (34).
25
This is also easy to see formally.
Obviously, from (32)
(35)
ξ0 = 1.
Thus,
ξt = ξ0 +
(36)
= 1+
·Z
(37)
E
t
Z t
0
Z t
0
¸
ξsXs dWs
ξsXs dWs.
Z
u
ξs Xs dWs | Iu =
0
ξs Xs dWs , u < t,
0
Rt
i.e., 0 ξsXs dWs is a martingale, and hence
Z u
(38) E[ξt|Iu] = 1 +
ξsXs dWs = ξu,
0
implying that ξt is a martingale.
26
Theorem. (Girsanov) Let Wt be a Wiener
process w.r.t probability measure P and w.r.t
information sets It, and let Xt be as defined
above. Then if the process ξt is a martingale
w.r.t information sets It, then W̃t defined by
(39)
W̃t = Wt −
Z t
0
Xu du,
t ∈ [0, T ]
is a Wiener process w.r.t information sets It
and w.r.t probability measure
(40)
P̃ (A) = EP [1AξT ],
where A ∈ IT and 1A is the indicator function
of the event A.
27
In heuristic terms: If Wt is a Wiener process
with probability measure P , then
(41)
dW̃t = dWt − Xt dt
is a Wiener process with probability measure
P̃ , such that dP̃ = ξT dP .
Remark 7.1: Generally the theorem gives us a method
to find the (equivalent) probability measures with respect to which a drifting process can be turned to a
martingale.
Remark 7.2: We only change the drift and live the
volatility intact.
28
Example 7.3: Consider the general diffusion process
(42)
dS = a(S, t)dt + b(S, t)dW ,
where W is the Wiener process w.r.t the probability
measure P ,
(43)
dP (w) = √
1
w2
2πt
e− 2t dw,
the normal distribution N (0, t).
Define
a(S, t)
b(S, t)
and assume that the drift a(S, t) and diffusion b(S, t)
(44)
Xt =
are such that the Novikov condition (33) holds for Xt.
Then defining
Z
(45)
W̃ = W −
0
t
a(S, u)
du
b(S, u)
is a Wiener process with respect to the probability P̃
given by (40) and
(46)
dS = b(S, t)dW̃
is a martingale w.r.t the probability measure P̃ .
29
Example 7.4: As in Example 7.2, consider again the
geometric Brownian motion
(47)
dS = µSdt + σSdW ,
where µ and σ are constants. The probability measure
for W is again (43).
Let r be the risk-free rate and consider the discounted
price series
S̃t = e−rt St.
(48)
Using Ito,
(49)
dS̃ = (µ − r)S̃dt + σSdW .
Now (44) becomes simply
(50)
Xt =
(µ − r)S̃
µ−r
=
,
σ
σ S̃
µ−r
t
σ
is Wiener process w..r.t the probability measure P̃ ,
and (w.r.t. this measure)
(51)
(52)
W̃ = W −
dS̃ = σ S̃dW̃
is martingale.
P̃ ?
30
In the Girsanov theorem
Rt
Rt 2
Xu dWu − 12
Xu du
0
ξt = e 0
(53)
1
1
2
= e σ (µ−r)Wt − 2σ2 (µ−r) t.
Furthermore, we can consider only the events A =
{Wt ≤ w}, w ∈ IR (because here the information sets
on the real line are Borel sets that are essentially open
intervals).
Because Wt ∼ N (0, t) the associated probability measure is (43). Then
Z w
Z w
u2
1
P̃ (A) = EP [1A ξt ] =
ξt(u) √
e− 2t du =
ξ(u)dP (u),
2π t
−∞
−∞
(54)
i.e.,
dP̃ (w) = ξt (w)dP (w)
1
1
2
1
1
2
= e σ (r−µ)w− 2σ2 (r−µ) tdP (w)
(55)
w2
1
− 2t
e
= e σ (r−µ)w− 2σ2 (r−µ) t √2π
dw
t
=
1
√ 1 e− 2t (w−
2π t
r−µ
σ
t)2
dw.
31
Denoting
w̃ = w −
µ−r
t,
σ
we have finally
(56)
dP̃ (w̃) = √
1
1
e− 2t w̃ dw̃,
2
2π t
which is again the density of the N (0, t) distribution.
The end result is that discounted price process (48)
is martingale with respect to the probability measure
P̃ .
The solution of (48) is
(57)
1
S̃t = S̃0e− 2 σ
2
t+σdW̃t
.
In terms of the original process from (48) St = ert S̃,
we would have
1
(58)
St = S0 e(r− 2 σ
or
(59)
dS = rS dt + σSdW̃ ,
2
)t+σ W̃t
,
i.e., we have essentially replaced the original drift µ
with the risk free rate r, and the end result is a process whose discounted price process, S̃t = e−rt St is a
martingale. Process (58) is usally called the risk neutral process.
32
Remark 7.3: When pricing options, we calculate the
expected values with respect to the distribution of
risk neutral process St given in (58). Its distribution
is log-normal with density
(60)
fSt (y) = √
1
2πt σy
e
θ̃t )
− (log(y)−
2σ 2 t
2
,
y > 0,
where
(61)
θ̃t = log S0 + (r −
1 2
σ )t.
2
Remark 7.4: The original distribution of St is lognormal with density
(62)
fSt (y) = √
1
2πt σy
e
t)
− (log(y)−θ
2σ 2 t
2
,
y > 0,
where
(63)
θt = log S0 + (µ −
1 2
σ )t.
2
33