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7. Equivalent Martingale Measures

So far we have considered derivative asset

pricing exploiting PDEs implied by arbitrage-

free portfolios.

Another approach is to change the probabil-ity 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 Euro-

pean call option.

The aim is to find the fair price for the option

given the available information.

To price the option (C t), we use the bestprediction of the end value in the light of

available information, such that

C t = Et [ρ max(S T − K, 0)],(1)

where Et

is the conditional expectation given

information up to time t, and ρ is a discount

factor.

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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 expec-

tation must be calculated is such that the

discounted price process

S t = e−r tS t,(2)

where r is the riskfree return, is martingale.

3

To illustrate the situation, consider the fol-

lowing 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 S 1 > K > S 2 of the stock.

So

S 1 = u S 0, with u > 1

S 0

S 2 = d S 0, with d < 1,

p

1 − p

where p is the probability that the price goes up to

S = u S 0, and S 0 is the current price of the stock.

Then the option with initial cost C has the end value

max

{S 1

−K, 0

}. To replicate this with the stock and

bank account with (riskfree) interest rate, r, we may

construct the following strategy:

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Buy one share of the stock by financing it with cash

and S 2/(1+r) borrowed at rate r from the bank (after

one period the repayment is accordinly S 2). The value

of the initial position is then

S 0 − 1

1 + rS 2.(3)

The end value of the position is according to the stockprice as

S = S 1 S = S 2

Stock value S 1 S 2Loan repayment −S 2 −S 2Total payoff S 1 − S 2 0

5

We observe that in the case of S = S 2 the payoff is

0, the same as with the call options, and in the case

S = S 1 the total payoff is S 1 − S 2 = a(S 1 − K ), where

a = (S 1 − S 2)/(S 1 −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 = S 0 − 1

1 + rS 2

or

C = (S 0−

1

1 + r

S 2)/a = 1

1 + r

p∗(S 1−

K ),

where

p∗ = 1 + r − d

u − d .

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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 proba-

bility given the initial price S 0 of the stock (the prob-

ability depends on u and d which are dependent on

S 0, i.e., how much the price should go up to reach

the given value S 1, or to decrease to go down to the

other possible given value S 2). These probabilities are

called risk neutral, hedging or martingale probabilities

or probability measures. The last name is because

they make the discounted price process S a martin-

gale.

7

This is seen as follows: We easily find that

(4)

S 0 = 1

1 + r ( p∗S 1 + (1 − p∗)S 2) = p∗S 1 + (1− p∗)S 2,

where S = S/(1 + r) is the discounted price pro-

cess. That is, the conditional expectations of the

discounted price process S given the information I 0 =

{S 0} is

E∗[S |I 0] = p∗S 1 + (1− p∗)S 2 = S 0 = S 0,(5)

so that S is martingale with respect to the risk neutral

probability measure, as stated above.

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We observe that the option value does not at all de-

pend on the true probabilities p and q = 1− p!

However, if we write the above martingale equaton as

(q∗ = 1− p∗)

E∗[S |I 0] = p∗S 1 + q∗S 2 = p∗ p

pS 1 + q∗

q qS 2,(6)

where p∗/p and q∗/q can be considered kinds of like-

lihood ratios or odds ratios, judged by the markets

for the events that the stock price will be S 1 and S 2,

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 distribu-

tion,

f (z) = 1√ 2π

e−12z2

.(7)

Then probability of that the random variable

Z is near a specific value z is

P

z − 1

2∆ < Z < z +

1

2∆

=

z+

12 ∆

z−12

∆

1√ 2π

e−12

z2

dz,(8)

which is a real number (between zero and

one).

Thus, the probability associates a real num-ber (in this case between zero and one) to

intervals on real line, or more generally to

(Borel) sets.

Such functions are called measures in math-

ematics or measure functions. 10

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Because ∆ is small

z+12∆

z−12∆

1√ 2π

e−12z2

dz ≈ 1√ 2π

e−12z2

z+12∆

z−12∆

dz

= 1√

2πe−

12z2

∆.

(9)

For infinitesimal ∆, denoted as dz, we desig-

nate the associated measure by symbol dP (z),

or simply by dP .

Thus, in the above case we have

dP (z) = 1√

2πe−

12z2

dz.(10)

11

Generally, if P is a probability measure, we

have ∞−∞

dP = 1.(11)

With these notations, e.g.,

E[X ] = ∞

−∞x dP (x).(12)

So the expected value is mathematically an

integral with respect to probability measure.

dP is called sometimes the density of the

probability measure P .

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Changing Probability Measure

Martingale model is a central tool for mod-

eling fair prices of derivative securities.

However, generally, if S t is a risky asset, then

given information up to time point t, we have

Et[S t+h] > (1 + rf )S t, (h > 0),(13)

because investors want some compensation

for the risk, where Et is the conditional ex-

pectation, and rf

is the risk-free rate.

13

However, we observed in the PDE approach

that under the arbitrage-free pricing the risk-

free rate should be a proper discounting fac-

tor in pricing risky derivative assets.

More importantly, the fundamental theorem

of asset pricing establishes the equivalence of the absence of arbitrage opportunity and ex-

istence of martingale measure in (the stochas-

tic model of) financial markets.

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A probability measure, P , is a martingale mea-

sure for the discounted price process S t =

e−rtS t, if S t is martingale under P , i.e.

EP t [S s] = S t, whenever s ≥ t,(14)

where r is the riskfree rate, and EP t indicates

that the conditional expectation is taken withrespect to probability measure P .

This theory implies that the no-arbitrage price

of a contingent claim with underlying secu-

rity S t and (random) payoff X at maturity T

is obtained by

C t = EP t

e−rτ X

,(15)

where τ = T − t, and P is the martingale

measure for the discounted price process S t.

We say that S t is P -martingale.

15

Thus a martingale measure can be viewed

as a representation of the market’s current

opinion on the evolution of values of under-

lying assets and the prices of all derivatives

contingent to them.

Consequently, the knowledge of the martin-gale measure is all that is needed, in principle,

to value whatever derivative securities by the

formula of the form (15).

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Then given a stock price process S t with

probability measure P , the goal is to find the

martingale measure P .

This can be accomplished if there is an in-

vertible function (one-to-one) ξ(z) such that

dP (z) = ξ(z) dP (z).(16)

Now recalling from standard calculus; if G(z) = g(z) dz then g(z) = dG(z)/dz = G(z), i.e.,

g(z) is the (mathematical) derivative of G(z).

Alternatively, we can write dG(z) = g(z)dz,

and hence

G(z) =

dG(z) =

g(z)dz.(17)

17

In the same manner, because

P (z) = z

−∞dP (t) =

z

−∞ξ(t) dP (t),(18)

so that we can adopt notation

dP (z)

dP (z) = ξ(z),(19)

and call ξ as a derivative of P with respect

to P .

In mathematical measure theory this is known

as the Radon-Nikodym derivative .

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The existence of ξ is guaranteed if P and P

satisfy

P (dz) > 0 if and only if P (dz) > 0.(20)

Actually this condition guarantees besides the

existence of ξ (known as Radon-Nikodym The-

orem), also the equivalence of P and P in thesense

dP (z) = ξ(z) dP (z)(21)

and

dP (z) = ξ(z)−1dP (z).(22)

In this sense P and P are equivalent proba-

bility measures .

19

Example 7.2: (GBM) Consider the geometric Brown-

ian motion

dS t = µS t dt + σS t dW t, t ≥ 0(23)

so that

Z t = log (S t/S 0) =

µ− 1

2σ2

t + σ W t(24)

and

Z t ∼ N (µ∗t, σ2t),(25)

where µ∗ = µ − σ2/2.

Thus

dP (z) = 1√ 2π σ2t

e−(z−µ∗t)2

2σ2t dz.(26)

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Let

Z t =

rf − 12

σ2

t + σ W t,(27)

where rf is a riskfree rate, and W t is another Wiener

process (the next example shows the relationship be-

tween W and W ).

Then

dP (z) = 1√ 2π σ2t

e−(z−r∗t)2

2σ2t dz,(28)

where r∗ = rf − 12

σ2.

Because the density function of the normal distribu-

tion is always positive, trivially

P (dz) > 0 ⇐⇒ P (dz) > 0.(29)

21

The transformation between these two probability mea-

sures is

ξ(z) = e−(µ∗−r∗)z−1

2(µ∗2−r∗2

)t

σ2 ,(30)

so that

dP (z) = ξ(z) dP (z).(31)

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The Girsanov Theorem

The Radon-Nikodym theorem gives the con-

ditions under which the derivative ξ exist (i.e.,

that we can move to another probability mea-

sure without losing any information, which

means that those events that have positiveprobability have also positive probability un-

der the other measure).

The Girsanov Theorem provides the condi-

tions under which the Radon-Nikodym deriva-

tive exists for cases where Z t is a continuous

stochastic process.

23

For the purpose, let

{I t

}, t

∈[0, T ] be a family

of information sets (T < ∞). Define

ξt = e t

0 X u dW u−12

t0 X 2u du, t ∈ [0, T ],(32)

where X t is an I t-measurable process (that is

once I t is given, the value of X t is known),

and W t is a Wiener process with probabilitymeasure P .

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It is assumed that X t does not increase too

fast, so that

E

e t

0 X u du

< ∞, t ∈ [0, T ],(33)

called Novikov condition (after a Russian math-

ematician).

Using Ito

dξt = ξtX t dW t,(34)

from which we immediately see that ξt is a

martingale, because W t is a Wiener process

and there is no drift component in (34).

25

This is also easy to see formally.

Obviously, from (32)

ξ0 = 1.(35)

Thus,

ξt = ξ0 +

t

0ξsX s dW s

= 1 + t

0ξsX s dW s.

(36)

E

t

0

ξsX s dW s | I u

=

u

0

ξsX s dW s, u < t,(37)

i.e., t

0 ξsX s dW s is a martingale, and hence

E[ξt|I u] = 1 + u

0ξsX s dW s = ξu,(38)

implying that ξt is a martingale.

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Theorem. (Girsanov) Let W t be a Wiener

process w.r.t probability measure P and w.r.t

information sets I t, and let X t be as defined

above. Then if the process ξt is a martingale

w.r.t information sets I t, then W t defined by

W t = W

t − t

0X

udu, t

∈[0, T ](39)

is a Wiener process w.r.t information sets I tand w.r.t probability measure

P (A) = EP [1AξT ],(40)

where A∈ I T

and 1A

is the indicator function

of the event A.

27

In heuristic terms: If W t is a Wiener process

with probability measure P , then

d W t = dW t − X t dt(41)

is a Wiener process with probability measure

P , such that dP = ξT dP .

Remark 7.1: Generally the theorem gives us a method

to find the (equivalent) probability measures with re-

spect to which a drifting process can be turned to a

martingale.

Remark 7.2: We only change the drift and live the

volatility intact.

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Example 7.3: Consider the general diffusion process

dS = a(S, t)dt + b(S, t)dW ,(42)

where W is the Wiener process w.r.t the probabilitymeasure P ,

dP (w) = 1√

2πte−

w2

2t dw,(43)

the normal distribution N (0, t).

Define

X t = a(S, t)

b(S, t)(44)

and assume that the drift a(S, t) and diffusion b(S, t)

are such that the Novikov condition (33) holds for X t.

Then defining

W = W − t

0

a(S, u)

b(S, u)du(45)

is a Wiener process with respect to the probability P

given by (40) and

dS = b(S, t)d W (46)

is a martingale w.r.t the probability measure P .

29

Example 7.4: As in Example 7.2, consider again the

geometric Brownian motion

dS = µSdt + σSdW,(47)

where µ and σ are constants. The probability measurefor W is again (43).

Let r be the risk-free rate and consider the discountedprice series

S t = e−rtS t.(48)

Using Ito,

dS = (µ − r)Sdt + σSdW.(49)

Now (44) becomes simply

X t = (µ− r)S

σS

= µ− r

σ

,(50)

W = W − µ− r

σ t(51)

is Wiener process w..r.t the probability measure P ,and (w.r.t. this measure)

dS = σSd W (52)

is martingale.

P ?

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In the Girsanov theorem

ξt = e

t

0X u dW u−1

2

t

0X 2u du

= e1

σ(µ−r)W t− 1

2σ2 (µ−r)2t.(53)

Furthermore, we can consider only the events A =

{W t ≤ w}, w ∈ IR (because here the information sets

on the real line are Borel sets that are essentially open

intervals).

Because W t ∼ N (0, t) the associated probability mea-sure is (43). Then

P (A) = EP [1Aξt] =

w

−∞ξt(u)

1√ 2π t

e−u2

2t du =

w

−∞ξ(u)dP (u),

(54)i.e.,

dP (w) = ξt(w)dP (w)

= e1

σ(r−µ)w− 1

2σ2 (r−µ)2tdP (w)

= e1

σ(r−µ)w− 1

2σ2 (r−µ)2t 1√ 2π t

e−w2

2t dw

=

1

√ 2π te−1

2t(w

−r−µ

σ t)2

dw.

(55)

31

Denoting

w = w − µ− rσ

t,

we have finally

dP (w) = 1√

2π te−

1

2t w2

dw,(56)

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

P .

The solution of (48) is

S t = S 0e−1

2σ2t+σd W t.(57)

In terms of the original process from (48) S t = ertS ,we would have

S t = S 0e(r−1

2σ2)t+σ W t,(58)

or

dS = rS dt + σSd W ,(59)

i.e., we have essentially replaced the original drift µ

with the risk free rate r, and the end result is a pro-

cess whose discounted price process, S t = e−rtS t is a

martingale. Process (58) is usally called the risk neu-

tral process.

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Remark 7.3: When pricing options, we calculate the

expected values with respect to the distribution of

risk neutral process S t given in (58). Its distribution

is log-normal with density

f S t(

y) =

1

√ 2πt σy e−(log(y)−θt)2

2σ2t

, y > 0

,(60)

where

θt = log S 0 + (r − 1

2σ2)t.(61)

Remark 7.4: The original distribution of S t is log-normal with density

f S t(y) = 1√ 2πt σy

e−(log(y)−θt)2

2σ2t , y > 0,(62)

where

θt = log S 0 + (µ − 1

2σ2)t.(63)

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