Stochastic Processes and Financial Mathematics
(part one)

1.3 Arbitrage

We now introduce a key concept in mathematical finance, known as arbitrage. We say that arbitrage occurs in a market if it possible to make money, for free, without risk of losing money.

There is a subtle distinction to be made here. We might sometimes expect to make money, on average. But an arbitrage possibility only occurs when it is possible to make money without any chance of losing it.

Example 1.3.1 is obviously artificial. It does illustrates an important point: no one should sell anything at a price that makes an arbitrage possible. However, if nothing is sold at a price that would permit arbitrage then, equally, nothing can be bought for a price that would permit arbitrage. With this in mind:

Let us step back and ask a natural question, about our market. Suppose we wish to have a single unit of stock delivered to us at time \(T=1\), but we want to agree in advance, at time \(0\), what price \(K\) we will pay for it. To do so, we would enter into a contract. A contract is an agreement between two (or more) parties (i.e. people, companies, institutions, etc) that they will do something together.

Consider a contract that refers to one party as the buyer and another party as the seller. The contract specifies that:

At time \(1\), the seller will be paid \(K\) cash and will deliver one unit of stock to the buyer.

A contract of this form is known as a forward contract. Note that no money changes hands at time \(0\). The price \(K\) that is paid at time \(1\) is known as the strike price. The question is: what should be the value of \(K\)?

In fact, there is only one possible value for \(K\). This value is

\begin{equation} \label {eq:forward_strike} K=s(1+r). \end{equation}

Let us now explain why. We argue by contradiction.

• • Suppose that a price \(K>s(1+r)\) was agreed. Then we could do the following:

  • 1. At time \(0\), enter into a forward contract as the seller.

  • 2. Borrow \(s\) from the bank, and use it buy a single unit of stock.

  • 3. Wait until time \(1\).

  • 4. Sell the stock (as agreed in our contract) in return for \(K\) cash.

  • 5. We owe the bank \(s(1+r)\) to pay back our loan, so we pay this amount to the bank.

  • 6. We are left with \(K-s(1+r)>0\) profit, in cash.

  With this strategy we are certain to make a profit. This is arbitrage!

• • Suppose, instead, that a price \(K<s(1+r)\) was agreed. Then we could:

  • 1. At time \(0\), enter into a forward contract as the buyer.

  • 2. Borrow a single unit of stock from the stockbroker.

  • 3. Sell this stock, in return for \(s\) cash.

  • 4. Wait until time \(1\).

  • 5. We now have \(s(1+r)\) in cash. Since \(K<s(1+r)\) we can use \(K\) of this cash to buy a single unit of stock (as agreed in our contract).

  • 6. Use the stock we just bought to pay back the stockbroker.

  • 7. We are left with \(s(1+r)-K>0\) profit, in cash.

  Once again, with this strategy we are certain to make a profit. Arbitrage!

Therefore, we reach the surprising conclusion that the only possible choice is \(K=s(1+r)\). We refer to \(s(1+r)\) as the arbitrage free value for \(K\). This is our first example of an important principle:

Expectation versus arbitrage

What of pricing by expectation? Let us, temporarily, forget about arbitrage and try to use pricing by expectation to find \(K\).

The value of our forward contract at time \(1\), from the point of view of the buyer, is \(S_1-K\). It costs nothing to enter into the forward contract, so if we believed that we should price the contract by its expectation, we would want it to cost nothing! This would mean that \(\E [S_1-K]=0\), which means we’d choose

\begin{equation} \label {eq:naive_K} K=\E [S_1]=sup_u+sdp_d. \end{equation}

This is not the same as the formula \(K=s(1+r)\), which we deduced in the previous section.

It is possible that our two candidate formulas for \(K\) ‘accidentally’ agree, that is \(up_u+dp_d=s(1+r)\), but they are only equal for very specific values of \(u,p_u,d,p_d,s\) and \(r\). Observations of real markets show that this doesn’t happen.

It may feel very surprising that (1.2) is different to (1.1). The reality is, that financial markets are arbitrage free, and the correct strike price for our forward contract is \(K=s(1+r)\). However intuitively appealing it might seem to price by expected value, it is not what happens in reality.

Does this mean that, with the correct strike price \(K=s(1+r)\), on average we either make or lose money by entering into a forward contract? Yes, it does. But investors are often not concerned with average payoffs – the world changes too quickly to make use of them. Investors are concerned with what happens to them personally. Having realized this, we can give a short explanation, in economic terms, of why markets are arbitrage free.

If it is possible to carry out arbitrage within a market, traders will¹ discover how and immediately do so. This creates high demand to buy undervalued commodities. It also creates high demand to borrow overvalued commodities. In turn, this demand causes the price of commodities to adjust, until it is no longer possible to carry out arbitrage. The result is that the market constantly adjusts, and stays in an equilibrium in which no arbitrage is possible.

Of course, in many respects our market is an imperfect model. We will discuss its shortcomings, as well as produce better models, as part of the course.