financial economics

CITY UNIVERSITY LONDON

Financial Economics What is the significance for financial economics of the concept of

arbitrage?

Natalia Lopez

Reg: 120013401

November 2014

Abstract

The principle of arbitrage plays a fundamental role in financial economics as it bonds and

underpins the major subfields of neoclassical finance. The arbitrage principle implies that risk

free investments earn the risk free rate of return and there is not risk free opportunities

requiring zero outlay that yield positive returns. Agents engaging in arbitrage activity

unintendedly close arbitrage opportunities bringing the market price to equilibrium because of

the demand and supply laws. In equilibrium, assets are correctly priced and the Law of One

Price (LoOP) holds not allowing for further arbitrage opportunities.

Word count: 2,200

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Natalia Lopez November 2014

1. Introduction

The concept of arbitrage is of such importance to financial economics that the building blocks

of neoclassical theory are interconnected by its presence bringing together a set of theories

that will define the financial system. The role of arbitrage in the financial markets is comparable

to a force that holds the entire system together. The aim of this paper is to interpret and

demonstrate the implications of arbitrage activity upon financial economics. The Efficient

Markets Hypothesis (EMH) is a major subfield of neoclassical finance which establishes that

assets are correctly priced when information available is impounded. This paper also analyses

the binomial option pricing model and applies the arbitrage principle when building the

replicating portfolio. The LoOP is composed of state prices which are the discounted risk

neutral measure of probability also known as martingale probabilities. This measure is heavily

used in finance and so will this paper to tie together the EMH and the binomial option pricing

model demonstrating the introductory statement.

2. Interpreting the arbitrage principle

In neoclassical finance agents are assumed to be rational and they aim to maximise their utility

in an uncertain environment. It is a socially desirable behaviour as rational agents drive the

economy to equilibrium when attempting to exploit profit opportunities. This concept leads to

the underlying principle known as the “Law of One Price” (LoOP): assets which are close

substitutes must sell at the same price in every part of the market. In equilibrium, the LoOP

must hold for those assets having similar risk and return profiles, i.e. close substitutes. Here

is where arbitrage comes into play.

Arbitrage is the exploitation of price differences of the same asset in different markets making

an instantaneous, unlimited and risk free profit. When arbitrageurs take advantage of rate

discrepancies they bring the market price to balance through demand and supply forces. An

arbitrage portfolio returning unlimited, instantaneous and risk free profits is also known as a

money pump which agents will infinitely exploit assuming the non-satiation aspect of rational

preferences. The arbitrage portfolio is the collection of holdings that reunite these

characteristics:

𝑃1𝑋1 + 𝑃2𝑋2 + ⋯ + 𝑃𝑛𝑋𝑛 = 0 This is the initial outlay today where at least one of the

assets is short and another is long all liquidate at the end of the period.

𝑃𝑛𝑋𝑛 : Present asset n price times its number of units.

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𝑉𝐾1𝑋1 + 𝑉𝐾2𝑋2 + ⋯ + 𝑉𝐾𝑛𝑋𝑛 ≥ 0 By definition, there will not be negative payoffs in the

future period.

𝑉𝐾𝑛𝑋𝑛: Future asset price n in state K times its number of units.

An arbitrage opportunity is a risk free set of transactions that secures a non negative payoff in

all states of the world and a positive payoff in at least one state without requiring an initial

investment. It is agents closing arbitrage opportunities when engaging in arbitrage activities

the force that brings the market to balance. The arbitrage principle states that arbitrage

opportunities are absent. Dybvig & Ross (1998) p.100 provide a useful comparison of the

arbitrage principle and a firm with constant returns to scale and zero economic profit. If there

was a firm yielding positive payoffs the limit at which the firm would run the activity would be

inexistent so there would not be an optimum point of production. The same applies for the

arbitrage portfolio: if an arbitrage opportunity exists an optimal portfolio cannot be constructed

because there will not be limits as to how much profit the agent makes. Then an optimal

portfolio can be constructed iff the arbitrage principle holds. This leads to the concept of state

price.

3. The implications of the arbitrage principle

A security that pays one unit of wealth in a particular state of the world i and zero in all others

has a state price qi today. Suppose there is a future possible outcome with 0.4 probability and

one unit of payoff. The agent has 0.6 probability of receiving zero payoff so adds a risk

premium: 1 UNIT*0.4 + RISK PREMIUM (all discounted to present value). This is the

magnitude known as state price (represented by qi) which is multiplied by the sum of its

corresponding payoffs in each state (represented by vij) to obtain:

𝑃𝑗 = 𝑞1𝑣1𝑗 + 𝑞2𝑣2𝑗 + ⋯ + 𝑞𝑖𝑣𝑖𝑗 𝑗 = 1,2, … 𝑛 Pj represents the present asset price.

This is known as the linear pricing rule which is satisfied when the arbitrage principle holds.

As stated by Bailey (2005) p.174-75, the linear pricing rule is equivalent to the risk- neutral

valuation relationship (RNVR) with the addition of a discount factor:

𝛿 ≡ 1

(1+𝑟0) r0 is the risk-free rate of return.

And the agent’s beliefs are represented by the martingale probabilities π1, π2, … πl in each

state. The state price is the discounted martingale probability:

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𝑃𝑗 = 𝑞1𝑣1𝑗 + ⋯ 𝑤ℎ𝑒𝑟𝑒 𝑠𝑡𝑎𝑡𝑒 𝑝𝑟𝑖𝑐𝑒: 𝑞1 =𝜋1

1 + 𝑟0 𝑡ℎ𝑒𝑛: 𝑃𝑗 = (

1

1 + 𝑟0) 𝜋1𝑣𝑖𝑗 + ⋯

A fundamental valuation relationship can be constructed:

𝑃𝑗 = 𝛿𝐸∗[𝑉𝑗] 𝑗 = 1,2,3, … , 𝑛

𝐸∗[𝑉𝑗] Is the expectation operator of the known payoff of asset j in individual states weighted

by the martingale probability of that state occurring added up with all other states and δ is the

discount to the present value. The RNVR is crucial in option pricing theory as it eases the

calculation of equilibrium asset prices in constructions such as the binomial model being a

simplification of the Black-Scholes model. The concept of martingale probabilities is also

fundamental to the subfield of efficient markets.

4. The Efficient Markets Hypothesis (EMH)

In financial markets asset pricing rests on information that determines the value at which

assets are traded. Information available is impounded into prices through the actions of

informed agents and the positions they take in the market. If the EMH is satisfied, the markets

will reflect all the available information on prices preventing agents from trading on that

information to generate excess returns. Then, the arbitrage principle is at the heart of EMH’s

foundations implying that the expected return is the opportunity cost of the funds invested on

that asset. According to Ross (1987) p. 323, a formulation of the EMH equation can be

expressed following the martingale:

𝑝𝑡 = 1

(1 + 𝑟𝑡)𝐸(𝑝𝑡+1|𝐼𝑡)

This is an application of the arbitrage principle meaning that the future (t+1) martingale

expected value given a set of information today (t) discounted back to the present is equal to

today’s price. The same principle can be applied to option pricing models.

5. The binomial option pricing model

The binomial approach to option pricing involves three assets; the share (S), the call C(S)

which is a function of the share value and the bond with the risk free rate of return (1+r). The

value of the call will depend on the movement of the share which follows a binomial process:

𝐶(𝑆𝑡+1) = {𝐶(𝑎𝑆𝑡)𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑎𝑙𝑙 𝑖𝑛 𝑢𝑝𝑡𝑖𝑐𝑘 𝑤𝑖ℎ 𝑝𝑟𝑜𝑏 𝜋

𝐶(𝑏𝑆𝑡) 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑎𝑙𝑙 𝑖𝑛 𝑑𝑜𝑤𝑛𝑡𝑖𝑐𝑘 𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏 1 − 𝜋

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At time t + 1 (future) there are two possible states of the world; a and b which assume a normal

distribution of asset returns. That is, a represents the distribution of possible outcomes if the

share price goes up and b if the share price goes down. However, note that, bSt does not

necessarily have to be a downtick; it could also be an uptick but not as high as the risk free

rate of return (1+r) because:

a > b > (1+r) If the share returns more than the bond in both states the agent would

engage in arbitrage activity by selling the bond without limits in order to purchase the share.

Equally:

(1+r) > a > b If the bond returns more than the share in both states the agent would

engage in arbitrage activity by selling the share without limits in order to purchase the bond.

Then:

a > (1+r) > b Is the condition that must be satisfied in the absence of arbitrage

opportunities and the equilibrium of the model.

Ross (1987) p. 330 explains that the particularity of the binomial approach is that probabilities

π and (1 - π) will not play any further role from this point as the agent’s choice of a or b will be

based on their preferences about risk and their assessment of the probabilities of the two

states. A portfolio can be constructed so that it replicates the call’s position in one of the states

by adjusting α representing the holding of the share and (1- α) representing the holding of the

risk free bond.

In the uptick the rate of return of the call is:

𝟏. 𝐶(𝑎𝑆𝑡+1)

𝐶(𝑆𝑡)= ∝ 𝑎 + (1−∝)(1 + 𝑟)

The call value is a function of the increase in the share price as ratio of the original investment

which is equal to the payoff on the replicating portfolio. In the uptick, ∝ 𝑎 is the payoff of the

share and (1−∝)(1 + 𝑟) is the payoff of the bond.

In the downtick the rate of return of the call is:

𝟐. 𝐶(𝑏𝑆𝑡+1)

𝐶(𝑆𝑡)= ∝ 𝑏 + (1−∝)(1 + 𝑟)

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The call value is a function of the decrease in the share price as the ratio of the original

investment which is equal to the payoff on the replicating portfolio. In the downtick, ∝ 𝑏 is the

payoff of the share and (1−∝)(1 + 𝑟) is the payoff of the bond.

In both states, the return of the call is equal to the payoff of the portfolio in its respective state.

If the return on the portfolio is equal to the return of the call in state a it must also be equal to

the return of the call in state b. This is because, if that was not the case, either the portfolio or

the call would dominate each other creating an arbitrage opportunity. Then equation 2 must

hold in order to satisfy the arbitrage principle and equation 1 is true by construction. This is

the equilibrium condition that brings into existence neoclassical finance. If both equations are

brought together this result is obtained:

(1 + 𝑟)𝐶(𝑆𝑡) = 𝜋∗𝐶(𝑎𝑆𝑡) + (1 − 𝜋∗)𝐶(𝑏𝑆𝑡)

Meaning that the value of the call today multiplied by the risk free rate equals to the value of

the call in the two possible states multiplied by their respective martingale probability.

Where:

𝜋∗ =(1 + 𝑟) − 𝑏

𝑎 − 𝑏

1 − 𝜋∗ = 𝑎 − (1 + 𝑟)

𝑎 − 𝑏

However note that, on this occasion, π is the martingale risk neutral measure of probability;

the market’s perception about the subjective probability of an uptick or downtick occurring,

their preferences about risk and the predicted value.

Say a=1.08, b= 1.02 and r = 4%, then:

𝜋∗ =1.04 − 1.02

0.06=

1

3

1 − 𝜋∗ =1.08 − 1.04

0.06=

2

3

The agent behaves as if s/he was risk neutral and the probability of an uptick occurring is 1/3

and the probability of a downtick occurring is 2/3.

Suppose that:

𝐼𝑛 𝑠𝑡𝑎𝑡𝑒 𝑎 ∶ 𝐶(𝑎𝑆𝑡) = 0 𝑆𝑡+1 < 𝑋

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𝐼𝑛 𝑠𝑡𝑎𝑡𝑒 𝑏: 𝐶(𝑏𝑆𝑡) = 𝑆𝑡+1 − 𝑋 𝑆𝑡+1 ≥ 𝑋

In state b the agent is in the money as share price is greater than the strike price. The agent

exercises the call in state b and abandons in state a (its value is zero).

Given that the call value today is:

𝐶(𝑆𝑡) =𝜋∗

1 + 𝑟𝐶(𝑎𝑆𝑡) +

(1 − 𝜋∗)

1 + 𝑟𝐶(𝑏𝑆𝑡)

The final result is:

𝐶(𝑆𝑡) = 𝑎 − (1 + 𝑟)

𝑎 − 𝑏

1

1 + 𝑟(𝑆𝑡+1 − 𝑋)

Which is the share price minus the exercise price (the degree at which the option is in the

money) discounted and multiplied by the martingale probability. Here the same application of

the arbitrage principle that was introduced in the EMH section can be used. Furthermore,

recalling that the arbitrage principle implies the existence of a linear pricing rule and associated

martingale probabilities, it is easy to see the similarity between the linear pricing rule (1) and

the option pricing formula (2):

𝟏. 𝑃𝑗 = 𝑞1𝑣1𝑗 + 𝑞2𝑣2𝑗 + ⋯ + 𝑞𝑖𝑣𝑖𝑗 𝑗 = 1,2, … 𝑛

𝟐. 𝐶(𝑆𝑡) =𝜋∗

1 + 𝑟𝐶(𝑎𝑆𝑡) +

(1 − 𝜋∗)

1 + 𝑟𝐶(𝑏𝑆𝑡)

Where:

𝜋∗

1 + 𝑟 𝑎𝑛𝑑

(1 − 𝜋∗)

1 + 𝑟

Are the state prices for the different states of the world a and b with the numerators being the

martingale probabilities. Summarising, the binomial option pricing model depends upon the

arbitrage principle and it becomes evident when displaying its consistency with the linear

pricing rule which was introduced in the first section of the paper.

6. Conclusion

Arbitrageurs push prices to equilibrium when attempting to exploit price discrepancies

between same assets in different markets. Arbitrage activity returns unlimited, instantaneous

and risk free profits and an optimal portfolio cannot be constructed under these characteristics.

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The LoOP must hold for arbitrage opportunities to be inexistent. The subfields of EMH and

option pricing are simplified using the state price logic: the martingale expected value

discounted back to the present is equal to the present price of the asset. The arbitrage principle

is invoked to build the replicating portfolio in the binomial model; it behaves in the same way

in both states of the world because to do otherwise the principle would be violated. It is evident

the link between EMH and option pricing through the use of state prices and the significance

of arbitrage through the imposition of the arbitrage principle on portfolio construction.

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Bibliography

Bailey R.E., (2005). “The Economics of Financial Markets”. USA: Cambridge University Press.

Ch: 7

Dybvig, Philip H. & Stephen A. Ross., (1998). "Arbitrage". The New Palgrave Dictionary of

Economics. Second Edition. Eds. Steven N. Durlauf & Lawrence E. Blume. Palgrave

Macmillan, 2008. The New Palgrave Dictionary of Economics Online. [09 November 2014].

Pp. 100 - 104

Ross, Stephen A., (1987). "Finance”. The New Palgrave Dictionary of Economics. Second

Edition. Eds. Steven N. Durlauf & Lawrence E. Blume. Palgrave Macmillan, 2008. The New

Palgrave Dictionary of Economics Online. [09 November 2014]. Pp. 323 - 332

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