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Unformatted text preview: Special Note on Introductory Financial Mathematics Mary R Hardy 1 The No Arbitrage Assumption 1.1 Arbitrage The no arbitrage assumption is the foundation of modern valuation methods in financial mathematics. The assumption is more colloquially known as the no free lunch assump- tion, and states quite simply that you cant get something for nothing. More formally, arbitrage in financial mathematics is a risk-free trading profit. An ar- bitrage opportunity exists if an investor can make a trading profit (or a possibility of a profit) with no possibility of a loss. In other words, an arbitrage exists if either (a) an investor can make a deal that would give her or him an immediate profit, with no risk of future loss; or (b) an investor can make a deal that has zero initial cost, no risk of future loss, and a non-zero probability of a future profit. If we assume that there are no arbitrage opportunities in a market, then it follows that any two securities or combinations of securities that give exactly the same payments must have the same price . Example 1 Consider a very simple securities market, consisting of two securities, A and B . At time t = 0 the prices of the assets are S A and S B respectively. The term of both the assets is 1 year. At time 1 there are two possible outcomes. Either the market goes up, in which case security A pays S A 1 ( u ) and B pays S B 1 ( u ), or it goes down, with payments S A 1 ( d ) and S B 1 ( d ) respectively. Investors can buy assets, in which case they pay the time 0 price and receive the time 1 income, or they can sell them, in which case they receive the time 0 price and must pay the time 1 outgo. At the outset, we dont know whether prices will go up or down. In 1 this simplified market, we assume there are no trading expenses or other frictions, and that all investors can buy or sell the assets freely. Now, assume first that we have the following payment table: Time 0 price Market goes up Market goes down Security: S S 1 ( u ) S 1 ( d ) $ $ $ A 5 8 4 B 9 16 8 There is an arbitrage opportunity here. An investor could buy one unit of security B and sell two units of security A. This would give income at time 0 of $10 from the sale of security A and an outgo of $9 from the purchase of security B which gives a net income at time 0 of $1. At time 1 the outgo due on the portfolio of 2 units of security A exactly matches the income due from security B, whether the market moves up or down. Thus, the investor makes a profit at time 0, with no risk of a future loss. It is clear that investment A is unattractive compared with investment B. This will cause pressure to reduce the price of A and to increase the price of B, as there will be no demand for A and a excessive demand for B. Ultimately we would achieve balance, when S A = S B / 2, when the arbitrage opportunity is eliminated, and the prices are consistent....
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This note was uploaded on 02/07/2010 for the course ACTSC 231 taught by Professor Chisholm during the Spring '09 term at Waterloo.
- Spring '09