Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
Stochastic Processes and
Advanced Mathematical Finance
Student: contains scenes of mild algebra or calculus that may require guidance.
It’s the day of the big game. You know that your rich neighbor really wants to buy tickets, in fact you know he’s willing to pay $50 a ticket. While on campus, you see a hand lettered sign oﬀering “two general-admission tickets at $25 each, inquire immediately at the mathematics department”. You have your phone with you, what should you do? Discuss whether this is a frequent event, and why or why not? Is this market eﬃcient? Is there any risk in this market?
The notion of arbitrage is crucial in the modern theory of ﬁnance. It is a cornerstone of the Black, Scholes and Merton option pricing theory, developed in 1973, for which Scholes and Merton received the Nobel Prize in 1997 (Fisher Black died in 1995).
An arbitrage opportunity is a circumstance where the simultaneous purchase and sale of related securities is guaranteed to produce a riskless proﬁt. Arbitrage opportunities should be rare, but in a world-wide basis some do occur.
This section illustrates the concept of arbitrage with simple examples.
Consider a stock that is traded in both New York and London. Suppose that the stock price is $172 in New York and £100 in London at a time when the exchange rate is $1.7500 per pound. An arbitrageur in New York could simultaneously buy 100 shares of the stock in New York and sell them in London to obtain a risk-free proﬁt of
in the absence of transaction costs. Transaction costs would probably eliminate the proﬁt on a small transaction like this. However, large investment houses face low transaction costs in both the stock market and the foreign exchange market. Trading ﬁrms would ﬁnd this arbitrage opportunity very attractive and would try to take advantage of it in quantities of many thousands of shares.
The shares in New York are underpriced relative to the shares in London with the exchange rate taken into consideration. However, note that the demand for the purchase of many shares in New York would soon drive the price up. The sale of many shares in London would soon drive the price down. The market would soon reach a point where the arbitrage opportunity disappears.
Suppose that the current market price (called the spot price) of an ounce of gold is $398 and that an agreement to buy gold in three months time would set the price at $390 per ounce (called a forward contract). Suppose that the price for borrowing gold (actually the annualized 3-month interest rate for borrowing gold, called the convenience price) is 10%. Additionally assume that the annualized interest rate on 3-month deposits (such as a certiﬁcate of deposit at a bank) is 4%. This set of economic circumstances creates an arbitrage opportunity. The arbitrageur can borrow one ounce of gold, immediately sell the borrowed gold at its current price of $398 (this is called shorting the gold), lend this money out for three months and simultaneously enter into the forward contract to buy one ounce of gold at $390 in 3 months. The cost of borrowing the ounce of gold is
and the interest on the 3-month deposit amounts to
The investor will therefore have in the bank account after 3 months. Purchasing an ounce of gold in 3 months, at the forward price of $390 and immediately returning the borrowed gold, he will make a proﬁt of $2.03. This example ignores transaction costs and assumes interests are paid at the end of the lending period. Transaction costs would probably consume the proﬁts in this one-ounce example. However, large-volume gold-trading arbitrageurs with low transaction costs would take advantage of this opportunity by purchasing many ounces of gold.
Figure 1 schematically diagrams this transaction. Time is on the horizontal axis, and cash ﬂow is vertical, with the arrow up if cash comes in to the investor, and the arrow down if cash ﬂows out from the investor.
Arbitrage opportunities as just described cannot last for long. In the ﬁrst example, as arbitrageurs buy the stock in New York, the forces of supply and demand will cause the New York dollar price to rise. Similarly as the arbitrageurs sell the stock in London, they drive down the London sterling price. The two stock prices will quickly become equal at the current exchange rate. Indeed the existence of proﬁt-hungry arbitrageurs (usually pictured as frenzied traders carrying on several conversations at once!) makes it unlikely that a major disparity between the sterling price and the dollar price could ever exist in the ﬁrst place. In the second example, once arbitrageurs start to sell gold at the current price of $398, the price will drop. The demand for the 3-month forward contracts at $390 will cause the price to rise. Although arbitrage opportunities can arise in ﬁnancial markets, they cannot last long.
Generalizing, the existence of arbitrageurs means that in practice, only tiny arbitrage opportunities are observed only for short times in most ﬁnancial markets. As soon as suﬃciently many observant investors ﬁnd the arbitrage, the prices quickly change as the investors buy and sell to take advantage of such an opportunity. As a consequence, the arbitrage opportunity disappears. The principle can stated as follows: in an eﬃcient market there are no arbitrage opportunities. In this text many arguments depend on the assumption that arbitrage opportunities do not exist, or equivalently, that we are operating in an eﬃcient market.
A joke illustrates this principle well: A mathematical economist and a ﬁnancial analyst are walking down the street together. Suddenly each spots a $100 bill lying in the street at the curb! The ﬁnancial analyst yells “Wow, a $100 bill, grab it quick!”. The mathematical economist says “Don’t bother, if it were a real $100 bill, somebody would have picked it up already.” Arbitrage opportunities are like $100 bills on the ground, they do exist in real life, but one has to be quick and observant. For purposes of mathematical modeling, we can treat arbitrage opportunities as non-existent as $100 bills lying in the street. It might happen, but we don’t base our ﬁnancial models on the expectation of ﬁnding them.
The principle of arbitrage pricing is that any two investments with identical payout streams must have the same price. If this were not so, we could simultaneously sell the more the expensive instrument and buy the cheaper one; the payment from our sale exceeds the payment for our purchase. We can make an immediate proﬁt.
Before the 1970s most economists approached the valuation of a security by considering the probability of the stock going up or down. Economists now ﬁnd the price of a security by arbitrage without the consideration of probabilities. We will use the principle of arbitrage pricing extensively in this text.
The ideas in this section are adapted from Options, Futures and other Derivative Securities by J. C. Hull, Prentice-Hall, Englewood Cliﬀs, New Jersey, 1993, Stochastic Calculus and Financial Applications, by J. Michael Steele, Springer, New York, 2001, pages 153–156, the article “What is a …Free Lunch” by F. Delbaen and W. Schachermayer, Notices of the American Mathematical Society, Vol. 51, Number 5, pages 526–528, and Quantitative Modeling of Derivative Securities, by M. Avellaneda and P. Laurence, Chapman and Hall, Boca Raton, 2000.
for at least one year. Suppose also that the Elbonian government has issued 1-year notes denominated in the EBB that pay a continuously compounded interest rate of 30%. Assuming that the corresponding continuously compounded interest rate for US deposits is 6%, show that an arbitrage opportunity exists. (Adapted from Quantitative Modeling of Derivative Securities, by M. Avellaneda and P. Laurence, Chapman and Hall, Boca Raton, 2000, Exercises 1.7.1, page 18).
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