Stochastic Processes and Financial Mathematics
(part one)
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Chapter 1 Expectation and Arbitrage
In this chapter we look at our first example of a financial market. We introduce the idea of arbitrage free pricing, and discuss what tools we would need to build better models.
1.1 Betting on coin tosses
We begin by looking at a simple betting game. Someone tosses a fair coin. They offer to pay you \(\$1\) if the coin comes up heads and nothing if the coin comes up tails. How much are you prepared to pay to
play the game?
One way that you might answer this question is to look at the expected return of playing the game. If the (random) amount of money that you win is \(X\), then you’d expect to make
\[\E [X]=\tfrac {1}{2}\$1+\tfrac {1}{2}\$0=\$0.50.\]
So you might offer to pay \(\$0.50\) to play the game.
We can think of a single play as us paying some amount to buy a random quantity. That is, we pay \(\$0.50\) to buy the random quantity \(X\), then later on we discover if \(X\) is \(\$1\) or \(\$0\).
We can link this ‘pricing by expectation’ to the long term average of our winnings, if we played the game multiple times. Formally this uses the strong law of large numbers:
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Let \((X_i)_{i\in \N }\) be a sequence of random
variables that are independent and identically distributed. Suppose that \(\E [X_1]=\mu \) and \(\var (X_1)<\infty \), and set
\[S_n=\frac {X_1+X_2+\ldots +X_n}{n}.\]
Then, with probability one, \(S_n\to \mu \) as \(n\to \infty \).
In our case, if we played the game a large number \(n\) of times, and on play \(i\) our winnings were \(X_i\), then our average winnings would be \(S_n\approx \E [X_1]=\frac 12\). So we might regard
\(\$0.50\) as a ‘fair’ price to pay for a single play. If we paid less, in the long run we’d make money, and if we paid more, in the long run we’d lose money.
Often, though, you might not be willing to pay this price. Suppose your life savings were \(\$20,000\). You probably (hopefully) wouldn’t gamble it on the toss of a single coin, where you would get \(\$40,000\)
on heads and \(\$0\) on tails; it’s too risky.
It is tempting to hope that the fairest way to price anything is to calculate its expected value, and then charge that much. As we will explain in the rest of Chapter 1, this tempting idea turns out to be completely wrong.