ECON 337901 FINANCIAL ECONOMICSirelandp.com/econ3379/notes/ch11slides.pdf · The Setting and the Intuition p0 j = 1 1 + r f XN i=1 ˇRN i p i j That is, the price at t = 0 of each

ECON 337901

FINANCIAL ECONOMICS

Peter Ireland

Boston College

Spring 2020

These lecture notes by Peter Ireland are licensed under a Creative Commons Attribution-NonCommerical-ShareAlike4.0 International (CC BY-NC-SA 4.0) License. http://creativecommons.org/licenses/by-nc-sa/4.0/.

11 Martingale Pricing

A Approaches to Valuation

B The Setting and the Intuition

C Definitions and Basic Results

D Relation to Arrow-Debreu

E Market Incompleteness and Arbitrage Bounds

Approaches to Valuation

One approach to pricing a risky asset payoff or cash flow C̃ isby taking its expected value E (C̃ ) and then “penalizing” theriskiness by either discounting at a rate that is higher than therisk-free rate

PA =E (C̃ )

1 + rf + ψ

or by reducing its value more directly as

PA =E (C̃ ) − Ψ

1 + rf


PA =E (C̃ )

1 + rf + ψor PA =

E (C̃ ) − Ψ

1 + rf

In this context, the CAPM and CCAPM are both models ofthe risk premia ψ and Ψ.

The CAPM associates the risk premia with the correlationbetween returns on the risky asset and the market portfolio.

The CCAPM associates the risk premia with the correlationbetween returns on the risk asset and the IMRS.


An alternative approach is to break the risky payoff or cashflow C̃ down “state-by-state” C 1,C 2, . . . ,CN and then pricingit as a bundle of contingent claims:

PA =N∑i=1

qiC i

In this Arrow-Debreu approach, the contingent claims pricescan either be inferred from other asset price throughno-arbitrage arguments or linked to investors’ IMRS throughequilibrium analysis.


Now we will consider a third approach, the martingale orrisk-neutral approach to pricing, developed in the late 1970s byMichael Harrison and David Kreps, “Martingales and Arbitragein Multiperiod Securities Markets,” Journal of EconomicTheory Vol.20 (June 1979): pp.381-408.

This approach is used extensively at the frontiers of assetpricing theory, as in Darrell Duffie, Dynamic Asset PricingTheory, Princeton: Princeton University Press, 2001.


Although the terminology associated with the two approachesdiffers, the connections between Arrow-Debreu and martingalepricing methods are very strong.

To an extent, therefore, we can learn about martingalemethods simply by taking what we already know aboutArrow-Debreu pricing and “translating” our previous insightsinto the new language.

The Setting and the Intuition

Consider, once again, a setting in which there are two dates,t = 0 and t = 1, and N possible states i = 1, 2, . . . ,N att = 1.

Let πi , i = 1, 2, . . . ,N , denote the probability of each state iat t = 1.


Here is what we know about the objective, physical, or trueprobability measure:

πi > 0 for all i = 1, 2, . . . ,N

A probability cannot be negative, and if πi = 0 you can alwaysdelete it by shortening the list of possible states.

N∑i=0

πi = 1

If you have accounted for all the possible states, theirprobabilities must sum to one.


In this environment, let there be a risk-free security that sellsfor p0b = 1 at t = 0 and pays off pib = 1 + rf in all statesi = 1, 2, . . . ,N at t = 1. Then rf is the risk-free rate.

Let there also be j = 1, 2, . . . ,M fundamental risky securities:

p0j = price of security j at t = 0

pij = payoff from (price of) security j in state i at t = 1

As in A-D no-arbitrage theory, we will make no assumptionsabout preferences or the distribution of asset returns.


Martingale pricing methods attempt to find a risk-neutralprobability measure, summarized in this environment by a setof risk-neutral probabilities πRN

i for i = 1, 2, . . . ,N , which canand usually will differ from the true or objective probabilities,but are such that

p0j =1

1 + rf

N∑i=1

πRNi pij

for each fundamental asset j = 1, 2, . . . ,M .


p0j =1

1 + rf

N∑i=1

πRNi pij

That is, the price at t = 0 of each fundamental security mustequal

1. The expected value of the price or payoff at t = 1, butcomputed using the risk-neutral instead of the trueprobability measure.

2. Discounted back to t = 0 using the risk-free rate.

Hence, martingale methods correct for risk by adjustingprobabilities instead of adjusting the risk-free rate.


In probability theory, a martingale is a stochastic process, thatis, a sequence of random variables Xt , t = 0, 1, 2, . . ., thatsatisfies

Xt = Et(Xt+1)

so that the expected value at t of Xt+1 equals Xt .

Equivalently, a martingale satisfies

0 = Et(Xt+1 − Xt)

implying that it is not expected to change between t and t + 1.


The classic example of a martingale occurs when you keeptrack of Xt as your accumulated stock of winnings or losings ina fair coin-flip game, where you gain 1 if the coin comes upheads and lose 1 if the coin comes up tails.

Since the coin flip is “fair,” the probability of either outcomeis 0.50 and

Et(Xt+1) = Xt + 0.50 × 1 + 0.50 × (−1) = Xt


Since

p0j =1

1 + rf

N∑i=1

πRNi pij

can be rewritten as

p0j(1 + rf )0

=N∑i=1

πRNi

[pij

(1 + rf )1

]= ERN

0

[pij

(1 + rf )1

]

the “discounted security price process is a martingale underthe risk-neutral probability measure.” Hence the name“martingale pricing.”

The Setting and the IntuitionMathematically, finding the risk-neutral probabilities amountsto collecting the equations

p0j =1

1 + rf

N∑i=1

πRNi pij for all j = 1, 2, . . . ,M

andN∑i=1

πRNi = 1

and trying to solve this system of equations subject to the“side conditions”

πRNi > 0 for all i = 1, 2, . . . ,N .


p0j =1

1 + rf

N∑i=1


N∑i=1

πRNi = 1

πRNi > 0 for all i = 1, 2, . . . ,N .

Hence, the risk-neutral probabilities must price all Mfundamental assets . . .


p0j =1

1 + rf

N∑i=1


N∑i=1

πRNi = 1

πRNi > 0 for all i = 1, 2, . . . ,N

. . . and the risk-neutral probabilities must sum to one andassign positive probability to the same N states identified bythe true probability measure.


In probability theory, the two sets of requirements

πi > 0 for all i = 1, 2, . . . ,N

andπRNi > 0 for all i = 1, 2, . . . ,N

make the objective and risk-neutral probability measuresequivalent.

In this context, the notion of equivalence is something moreakin to continuity than to equality.


p0j =1

1 + rf

N∑i=1


N∑i=1

πRNi = 1

πRNi > 0 for all i = 1, 2, . . . ,N .

Let’s start by considering a case in which this system ofequations does not have a solution.


Suppose that two of the fundamental securities have the sameprice at t = 0; for example:

p01 = p02

But suppose that security j = 1 pays off at least as much asj = 2 in every state and more than j = 1 in at least one stateat t = 1; for example:

p11 > p12

andpi1 = pi1 for all i = 2, 3, . . . ,N


With

p01 = p02 and p11 > p12 and pi1 = pi2 for all i = 2, 3, . . . ,N

the two martingale pricing equations

p01 =1

1 + rf

N∑i=1

πRNi pi1

p02 =1

1 + rf

N∑i=1

πRNi pi2

must have the same left-hand sides, but there is no way tochoose πRN

1 > 0 to make this happen.


On the other hand, with

p01 = p02 and p11 > p12 and pi1 = pi2 for all i = 2, 3, . . . ,N

there is an arbitrage opportunity.

The portfolio constructed by buying one share of asset 1 andselling short one share of asset 2 costs nothing at t = 0 butgenerates a positive payoff in state 1 at t = 1!


This is, in fact, one of the main messages of martingale pricingtheory.

There is a tight link between the absence of arbitrageopportunities and the existence of a risk-neutral probabilitymeasure for pricing securities.

This result is referred to as the Fundamental Theorem ofAsset Pricing by Philip Dybvig and Stephen Ross, “Arbitrage,”in The New Palgrave Dictionary of Economics, 2008.

Definitions and Basic Results

Still working in the two-date, N-state framework, consider aportfolio W consisting of wb bonds (units of the risk-freeasset) and wj shares (units) of each fundamental risky assetj = 1, 2, . . . ,M .

As usual, negative and/or fractional values for wb and the wj ’sare allowed. Assets are “perfectly divisible” and short selling ispermitted.


The value (cost) of the portfolio at t = 0 is

V 0w = wb +

M∑j=1

wjp0j

and the value (payoff) in each state i = 1, 2, . . . ,N at t = 1 is

V iw = wb(1 + rf ) +

M∑j=1

wjpij


The portfolio W constitutes an arbitrage opportunity if all ofthe following conditions hold:

1. V 0w = 0

2. V iw ≥ 0 for all i = 1, 2, . . . ,N

3. V iw > 0 for at least one i = 1, 2, . . . ,N

This definition of an arbitrage opportunity is slightly morespecific than the one we’ve been using up until now. Itrequires no money down today but allows for only thepossibility of profit – with no possibility of loss – in the future.


The portfolio W constitutes an arbitrage opportunity if all ofthe following conditions hold:

1. V 0w = 0

2. V iw ≥ 0 for all i = 1, 2, . . . ,N

3. V iw > 0 for at least one i = 1, 2, . . . ,N

The “no money down” requirement (1) is often described bysaying that the portfolio must be self-financing. Shortpositions in the portfolio must offset long positions so that, onnet, the entire portfolio can be assembled at zero cost.


Next, we’ll consider three examples:

1. With complete markets and no arbitrage opportunities . . .

2. With incomplete markets and no arbitrage opportunities. . .

3. With arbitrage opportunities . . .

And see that

1. . . . a unique risk-neutral probability measure exists.

2. . . . multiple risk-neutral probability measures exist.

3. . . . no risk-neutral probability measure exists.


Each example has two periods, t = 0 and t = 1, and threestates i = 1, 2, 3, at t = 1.

Example 1, with complete markets and no arbitrageopportunities, features the bond and two risky stocks asfundamental securities.


Example 1 p0j p1j p2j p3jBond j = b 1 1.1 1.1 1.1Stock j = 1 2 3 2 1Stock j = 2 3 1 4 6

With three assets having linearly independent payoffs, marketsare complete; and there are no arbitrage opportunities.

In this setting, a risk-neutral probability measure, if it exists,amounts to choice of πRN

1 , πRN2 , and πRN

3 .

Definitions and Basic ResultsExample 1 p0j p1j p2j p3jBond j = b 1 1.1 1.1 1.1Stock j = 1 2 3 2 1Stock j = 2 3 1 4 6

The risk-neutral probabilities must price the two risky assets

2 =1

1.1

(3πRN

1 + 2πRN2 + πRN

3

)3 =

1

1.1

(πRN1 + 4πRN

2 + 6πRN3

)sum to one

1 = πRN1 + πRN

2 + πRN3

and satisfy πRN1 > 0, πRN

2 > 0, and πRN3 > 0.


2 =1

1.1

(3πRN

1 + 2πRN2 + πRN

3

)3 =

1

1.1

(πRN1 + 4πRN

2 + 6πRN3

)1 = πRN

1 + πRN2 + πRN

3

We have a system of 3 linear equations in 3 unknowns. Theunique solution

πRN1 = 0.3 and πRN

2 = 0.6 and πRN2 = 0.1

also satisfies the side conditions πRN1 > 0, πRN

2 > 0, andπRN3 > 0.


Our first example confirms that when markets are completeand there is no arbitrage, there exists a risk-neutral probabilitymeasure and, moreover, that risk-neutral probability measureis unique.

To see what happens when markets are incomplete, let’s dropthe second risky stock from our first example. Since there willthen be fewer fundamental assets M than states N at t = 1,markets will be incomplete, although there will still be noarbitrage opportunities.


Example 2 p0j p1j p2j p3jBond j = b 1 1.1 1.1 1.1Stock j = 1 2 3 2 1

The risk-neutral probabilities must price the one risky asset

2 =1

1.1

(3πRN

1 + 2πRN2 + πRN

3

)sum to one

1 = πRN1 + πRN

2 + πRN3


2 > 0, and πRN3 > 0.


2 =1

1.1

(3πRN

1 + 2πRN2 + πRN

3

)1 = πRN

1 + πRN2 + πRN

3

We now have a system of 2 equations in 3 unknows, albeit onethat also imposes the side conditions πRN

1 > 0, πRN2 > 0, and

πRN3 > 0.


2 =1

1.1

(3πRN

1 + 2πRN2 + πRN

3

)1 = πRN

1 + πRN2 + πRN

3

To see what possibilities are allowed for, let’s temporarily holdπRN1 fixed and use the two equations to solve for

πRN2 = 1.2 − 2πRN

1

πRN3 = πRN

1 − 0.2


πRN2 = 1.2 − 2πRN

1

πRN3 = πRN

1 − 0.2

Now see what the side conditions require:

πRN2 = 1.2 − 2πRN

1 > 0 requires 0.6 > πRN1

πRN3 = πRN

1 − 0.2 > 0 requires πRN1 > 0.2


2 =1

1.1

(3πRN

1 + 2πRN2 + πRN

3

)1 = πRN

1 + πRN2 + πRN

3

Evidently, any risk-neutral probability measure with

0.6 > πRN1 > 0.2

πRN2 = 1.2 − 2πRN

1

πRN3 = πRN

1 − 0.2

will work.


2 =1

1.1

(3πRN

1 + 2πRN2 + πRN

3

)1 = πRN

1 + πRN2 + πRN

3

0.6 > πRN1 > 0.2

πRN2 = 1.2 − 2πRN

1

πRN3 = πRN

1 − 0.2

Notice that all of the possible risk-neutral probability measuresimply the same prices for the fundamental securities, which arealready being traded. They may, however, imply differentprices for securities that are not yet traded in this setting withincomplete markets.


Hence, our first two examples confirm that it is the absence ofarbitrage opportunities that is crucial for the existence of arisk-neutral probability measure.

The completeness or incompleteness of markets thendetermines whether or not the risk-neutral probability measureis unique.

As a third example, let’s confirm that the presence of arbitrageopportunities, a risk-neutral probability measure fails to exist.


Example 3 p0j p1j p2j p3jBond j = b 1 1.1 1.1 1.1Stock j = 1 2 1 2 3Stock j = 2 3 3 4 5

Here, there is an arbitrage opportunity, since buying one shareof stock j = 2 and selling one bond and one share of stockj = 1 costs nothing, on net, at t = 0 but generates positivepayoffs in all three states at t = 1.

Definitions and Basic ResultsExample 3 p0j p1j p2j p3jBond j = b 1 1.1 1.1 1.1Stock j = 1 2 1 2 3Stock j = 2 3 3 4 5

The risk-neutral probabilities, if they exist, must price the tworisky assets

2 =1

1.1

(πRN1 + 2πRN

2 + 3πRN3

)3 =

1

1.1

(3πRN

1 + 4πRN2 + 5πRN

3

)sum to one

1 = πRN1 + πRN

2 + πRN3


2 > 0, and πRN3 > 0.


2 =1

1.1

(πRN1 + 2πRN

2 + 3πRN3

)3 =

1

1.1

(3πRN

1 + 4πRN2 + 5πRN

3

)1 = πRN

1 + πRN2 + πRN

3

We still have a system of three equations in the threeunknowns. If, however, we add the first and third of theseequations to get

3 =1

1.1

(2.1πRN

1 + 3.1πRN2 + 4.1πRN

3

)


But

3 =1

1.1

(3πRN

1 + 4πRN2 + 5πRN

3

)3 =

1

1.1

(2.1πRN

1 + 3.1πRN2 + 4.1πRN

3

)cannot both hold, if the probabilities must satisfy the sideconditions πRN

1 > 0, πRN2 > 0, and πRN

3 > 0.

Again, we see that a risk-neutral probability measure fails toexist in the presence of arbitrage opportunities.


The lessons from these three examples generalize to yield thefollowing propositions.

Proposition 1 (Fundamental Theorem of Asset Pricing) Thereexists a risk-neutral probability measure if and only if there areno arbitrage opportunities among the set of fundamentalsecurities.


Proposition 2 If there are no arbitrage opportunities amongthe set of fundamental securities, then the value at t = 0 ofany portfolio W of the fundamental securities must equal thediscounted expected value of the payoffs generated by thatportfolio at t = 1, when the expected value is computed usingany risk-neutral probability measure and the discounting usesthe risk-free rate:

V 0w =

1

1 + rf

N∑i=1

πRNi V i

w


Proposition 3 Suppose there are no arbitrage opportunitiesamong the set of fundamental securities. Then markets arecomplete if and only if there exists a unique risk-neutralprobability measure.

Relation to Arrow-Debreu

Obscured behind differences in terminology are very close linksbetween martingale pricing methods and Arrow-Debreu theory.

To see these links, let’s remain in the martingale pricingenvironment with two periods, t = 0 and t = 1, and statesi = 1, 2, . . . ,N at t = 1, but imagine as well that thefundamental securities are, in fact, Arrow-Debreu contingentclaims.


Equivalently, we can assume that markets are complete, sothat A-D contingent claims for each state can be constructedas portfolios of the the fundamental securities.

Proposition 2 then implies that these claims can be pricedusing the risk-neutral probability measure.


In either case, the risk-neutral probability measure allowscontingent claims to be priced as

qi =1

1 + rfπRNi

for all i = 1, 2, . . . ,N , since the contingent claim for state ipays off one in that state and zero otherwise.

Hence, if we know the risk-neutral probability measure, we alsoknow all contingent claims prices.


Next, sum

qi =1

1 + rfπRNi

over all i = 1, 2, . . . ,N to obtain

N∑i=1

qi =1

1 + rf

N∑i=1

πRNi =

1

1 + rf

By itself, this equation states a no-arbitrage argument: sincethe payoff from a bond can replicated by buying a portfolioconsisting of 1 + rf contingent claims for each statei = 1, 2, . . . ,N , the price of this portfolio must equal the bondprice p0b = 1.


ButN∑i=1

qi =1

1 + rfand qi =

1

1 + rfπRNi

can also be combined to yield

πRNi =

qi∑Ni=1 q

ifor all i = 1, 2, . . . ,N .

Hence, if we know the contingent claims prices, we also knowthe risk-neutral probability measure.


Thus far, we have only relied on the no-arbitrage version ofA-D theory. But we can deepen our intuition if we are willingto invoke the equilibrium conditions

qi =βπiu

′(c i)

u′(c0)for all = 1, 2, . . . ,N .

where u(c) is a representative investor’s Bernoulli utilityfunction, c0 and c i denote his or her consumption at t = 0and in state i at t = 1, β is the investor’s discount factor, andπi is what we are now calling the objective or true probabilityof state i .


Combine

qi =1

1 + rfπRNi

and

qi =βπiu

′(c i)

u′(c0)for all = 1, 2, . . . ,N .

to obtain

πRNi = β(1 + rf )

[u′(c i)

u′(c0)

]πi


πRNi = β(1 + rf )

[u′(c i)

u′(c0)

]πi

Thus, apart from the scaling factor β(1 + rf ) that does notdepend on the particular state i , the risk-neutral probabilityπRNi “twists” the true or objective probability πi by adding

weight if u′(c i) is large and down-weighting if u′(c i) is small.

But if the representative investor is risk averse, so that u(c) isconcave, when is u′(c) large?


πRNi = β(1 + rf )

[u′(c i)

u′(c0)

]πi

Thus, apart from the scaling factor β(1 + rf ) that does notdepend on the particular state i , the risk-neutral probabilityπRNi “twists” the true or objective probability πi by adding

weight if u′(c i) is large and down-weighting if u′(c i) is small.

u′(c) is large during a recession and small during a boom.


πRNi = β(1 + rf )

[u′(c i)

u′(c0)

]πi

Thus, the risk-neutral probabilities might more accurately becalled “risk-adjusted” probabilities, since they “correct” thetrue probabilities to overweight states in which aggregateoutcomes are particularly bad.

This is why martingale pricing reliably discounts risky payoffsand cash flows.


πRNi = β(1 + rf )

[u′(c i)

u′(c0)

]πi

But like the no-arbitrage version of A-D theory, martingalepricing infers the risk-neutral probabilities from observed assetprices, not based on assumptions about preferences andconsumption.


πRNi = β(1 + rf )

[u′(c i)

u′(c0)

]πi

Suppose, however, that the representative investor isrisk-neutral, with

u(c) = a + bc

for fixed values of a and b > 0, so that u′′(c) = 0 instead ofu′′(c) < 0.


With risk-neutral investors,

u(c) = a + bc

implies u′(c) = b, and

πRNi = β(1 + rf )

[u′(c i)

u′(c0)

]πi

collapses toπRNi = β(1 + rf )πi


With risk-neutral investors

πRNi = β(1 + rf )πi

the risk-neutral and objective probabilities differ only by aconstant scaling factor.

Hence, the term “risk-neutral” probabilities: the risk-neutralprobabilities are (up to a constant scaling factor) whatprobabilities would have to be in an economy where assetprices are the same as what we observe but investors were riskneutral.


With risk-neutral investors

πRNi = β(1 + rf )πi

the risk-neutral and objective probabilities differ only by aconstant scaling factor.

Or, put differently, using risk-neutral probabilities instead ofobjective probabilities allows us to price assets “as if” investorswere risk neutral.


Finally, consider any asset that yields payoffs X i in each statei = 1, 2, . . . ,N at t = 1.

The martingale approach will price this asset using therisk-neutral probabilities as

pA =1

1 + rf

N∑i=1

πRNi X i =

1

1 + rfERN(X i)

where the RN superscript indicates that the expectation iscomputed using the risk-neutral probabilities.


Since, in an Arrow-Debreu equilibrium, the risk-neutralprobabilities are linked to the representative investor’s IMRSand the objective probabilities as

πRNi = (1 + rf )

[βu′(c i)

u′(c0)

]πi

we can rewrite the martingale pricing equation

pA =1

1 + rf

N∑i=1

πRNi X i =

1

1 + rfERN(X i)

in terms of the IMRS and the objective probabilities.


pA =1

1 + rfERN(X i) =

1

1 + rf

N∑i=1

πRNi X i

=1

1 + rf

N∑i=1

{(1 + rf )

[βu′(c i)

u′(c0)

]πi

}X i

=N∑i=1

πi

[βu′(c i)

u′(c0)

]X i = E

{[βu′(c i)

u′(c0)

]X i

}where now the expectation is computed with the objectiveprobabilities.


Hence, the martingale pricing equation

pA =1

1 + rf

N∑i=1

πRNi X i =

1

1 + rfERN(X i)

is closely linked to the Euler equation from the equilibriumversion of Arrow-Debreu theory

pA =N∑i=1

πi

[βu′(c i)

u′(c0)

]X i = E

{[βu′(c i)

u′(c0)

]X i

}


Let

mi =βu′(c i)

u′(c0)

denote the representative investor’s IMRS, so that the A-DEuler equation can be written more compactly as

pA =N∑i=1

πi

[βu′(c i)

u′(c0)

]X i =

N∑i=1

miX i = E (miX i)


In this more compact form

pA = E (miX i)

compares more directly to the martingale pricing equation

pA =1

1 + rfERN(X i)

In analyses that use the martingale approach, the IMRS mi isreferred to synonymously as the stochastic discount factor orthe pricing kernel.


pA =1

1 + rfERN(X i) and pA = E (miX i)

highlight the similarities between two equivalent approachespricing.

1. Deflate by the risk-free rate after computing theexpectation of the random payoff using the risk-neutralprobabilities.

2. Deflate with the stochastic discount factor beforecomputing the expectation of the random payoff usingthe objective probabilities.


qi =1

1 + rfπRNi

πRNi =

qi∑Ni=1 q

ifor all i = 1, 2, . . . ,N .

Ultimately, the “one-to-one” correspondence betweenrisk-neutral probabilities and contingent claims prices impliesthat there’s nothing we can do with martingale pricing theorythat we cannot do with A-D theory instead – and vice-versa.


Still, there are certain complex problems in asset valuation –particularly in pricing options and other derivative securities –and portfolio allocation that are easier, computationally, tosolve with martingale methods.

Before moving on, therefore, let’s go back to one of ourprevious examples to see how the martingale approach canlead us to interesting and useful results more quickly than theA-D method does.

Market Incompleteness and Arbitrage Bounds

To illustrate the practical usefulness of martingale pricingmethods, let’s return to the second example, in which wecharacterized the multiplicity of risk-neutral probabilitymeasures that exist under incomplete markets.


Here, we have three states at t = 1 but only two assets, somarkets are incomplete.



What we already know is that the bond, the stock, and allportfolios of these existing fundamental securities will bepriced, accurately and uniquely, by any risk-neutral probabilitymeasure with

0.6 > πRN1 > 0.2

πRN2 = 1.2 − 2πRN

1

πRN3 = πRN

1 − 0.2



But suppose one of your clients asks if you will sell him or hera contingent claim for state 1. What price should you ask for,assuming you decide to issue this new security?

Since markets are incomplete, there’s no way to “synthesize”that contingent claim by constructing a portfolio of the twoexisting assets: the bond and stock j = 1. There is no “purearbitrage” argument that will give you the “right” price.


The martingale approach, however, tells you that a contingentclaim for state 1 ought to sell for

q1 =1

1 + rfπRN1

at t = 0, so your data, which tell you that 1 + rf = 1.1 andyour previous result that every risk-neutral probability measuremust have 0.6 > πRN

1 > 0.2 indicate right away that the pricewill have to satisfy

0.5455 =0.6

1.1> qi >

0.2

1.1= 0.1818


How can we verify that you are not going to get0.5455 = 0.6/1.1 or more if you decide to issue (sell) acontingent claim for state 1 to your client?

Let’s go back to the data for the existing assets, and see ifthere is a way that your client can construct a portfolio of thebond and stock that:

1. Costs 0.6/1.1 = 0.5455 at t = 0

2. Pays off one in state 1 at t = 1

3. Pays off at least zero in states 2 and 3 at t = 1.


Is there is a way that your client can construct a portfolio ofthe bond and stock that:

1. Costs 0.6/1.1 = 0.5455 at t = 0

2. Pays off one in state 1 at t = 1

3. Pays off at least zero in states 2 and 3 at t = 1

Such a portfolio will be better for your client than acontingent claim for state 1; if it exists, that will confirm thatyou will not get 0.5455 (or more) for selling the claim.

Market Incompleteness and Arbitrage BoundsExample 2 p0j p1j p2j p3jBond j = b 1 1.1 1.1 1.1Stock j = 1 2 3 2 1

A portfolio consisting of wb bonds and w1 shares of stock willcost

V 0w = wb + 2w1

at t = 0 and will have payoffs

V 1w = 1.1wb + 3w1

V 2w = 1.1wb + 2w1

V 3w = 1.1wb + w1

in the three possible states at t = 1.


V 0w = wb + 2w1

V 1w = 1.1wb + 3w1

V 2w = 1.1wb + 2w1

V 3w = 1.1wb + w1

Your client wants

V 0w = wb + 2w1 =

0.6

1.1

V 1w = 1.1wb + 3w1 = 1


Your client wants

V 0w = wb + 2w1 =

0.6

1.1

V 1w = 1.1wb + 3w1 = 1

This is a system of two linear equations in two unknowns, withsolution

wb = − 1

2.2and ws =

1

2


But the choices

wb = − 1

2.2and ws =

1

2

also implyV 2w = 1.1wb + 2w1 = 0.5 > 0

V 3w = 1.1wb + w1 = 0

So this portfolio is better for your client than a contingentclaim for state 1. If you try to sell the claim for 0.5455 ormore, he or she will just buy the portfolio of the bond andstock instead.


0.5455 =0.6

1.1> qi >

0.2

1.1= 0.1818

Now let’s ask: should you sell the claim for 0.1818 or less?

The answer is no, if there is a way for you to construct aportfolio of the bond and stock that

1. Provides you with 0.2/1.1 = 0.1818 at t = 0

2. Requires you to make a payment of one in state 1 att = 0

3. Provides you with at least zero in states 2 and 3 at t = 1

Market Incompleteness and Arbitrage BoundsExample 2 p0j p1j p2j p3jBond j = b 1 1.1 1.1 1.1Stock j = 1 2 3 2 1

A portfolio consisting of wb bonds and w1 shares of stock willcost

V 0w = wb + 2w1

at t = 0 and will have payoffs

V 1w = 1.1wb + 3w1

V 2w = 1.1wb + 2w1

V 3w = 1.1wb + w1

in the three possible states at t = 1.


V 0w = wb + 2w1

V 1w = 1.1wb + 3w1

V 2w = 1.1wb + 2w1

V 3w = 1.1wb + w1

You want

V 0w = wb + 2w1 = −0.2

1.1

V 1w = 1.1wb + 3w1 = −1

where the numbers are negative because you want to receivethe negative price at t = 0 and make the negative payoff att = 1.


You want

V 0w = wb + 2w1 = −0.2

1.1

V 1w = 1.1wb + 3w1 = −1

This is another system of two linear equations in twounknowns, with solution

wb =2

1.1and ws = −1


But the choices

wb =2

1.1and ws = −1

also implyV 2w = 1.1wb + 2w1 = 0

V 3w = 1.1wb + w1 = 1 > 0

So buying portfolio is better for you than selling the contingentclaim. If your client won’t pay more than 0.1818 for the claim,you should buy this portfolio (or do nothing) instead.


It took us awhile to confirm the result, but that underscoresthe fact that the martingale approach initially led us to answervery quickly.

Martingale pricing methods require you to learn a newlanguage, and the economic intuition is not as direct as it iswith Arrow-Debreu.

But these methods can be extremely useful in solving verydifficult problems in asset pricing.

ECON 337901 FINANCIAL ECONOMICSirelandp.com/econ3379/notes/ch11slides.pdf · The Setting and the Intuition p0 j = 1 1 + r f XN i=1 ˇRN i p i j That is, the price at t = 0 of each

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