# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## BR: Motivating No-arbitrage Pricing

BR begin with a warning to beginners that a statistical mind-set in pricing derivative securities can be misleading.

Taking the coin tossing experiment (win 1 for H, 0 for T), they start out with saying that a tempting way to price a fair game is to use expectations – they call it “Expectation Pricing” – and for the coin tossing game they argue that the fair price is 50 paise. They remind, however, that in the “short run”/market, such a price may not exist/be enforceable.

They digress then to introducing time value of money into the game, and within one paragraph go from coins to the world of stocks. Assuming (without motivating) that stocks follow a lognormal distribution, they show that “expectation pricing” implies that the fair strike for a forward contract should be $S_0 \displaystyle e^{(\mu + \frac{1}{2} \sigma^2)T}$.

This, of course, is not the price enforced by the market. Because one can replicate the payoff from a forward contract exactly by borrowing and holding the stock till maturity, arbitrage implies that the fair strike should be $S_0 \displaystyle e^{rT}$ and not $S_0 \displaystyle e^{(\mu + \frac{1}{2} \sigma^2)T}$. This is not to say that the expectation is “wrong” – but that the expectation is conditional on the buyer’s/seller’s beliefs about the world (needless to say, whatever the assumed distribution for the stock price, the buyer expects the forward to be in-the-money and vice-versa), but is just not enforceable in the market. Any other strike different from $S_0 \displaystyle e^{rT}$ would lead to arbitrage.

They conclude by pointing out that “all derivatives can be built from the underlying – arbitrage lurks everywhere”.