Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

BR: Motivating No-arbitrage Pricing

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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”.


Although the warning against “expectation pricing” is well-intended, there are a couple of problems with their exposition:

  1. The choice of the term “expectation pricing” can only, to quote BR, “muddy the waters”. Neither the term is standard in the literature (for the purpose they use it), nor is it a good choice per se, because it can only potentially confuse the student when the time comes to introduce risk-neutral pricing.
  2. The idea/model of  lognormal distribution for the stock price is hurriedly introduced, if not clumsily. While it helps point out the problem/unenforceability of the “expectation pricing” based value, given its centrality in the Black-Scholes-Merton world, it warrants more time.
  3. It’s a matter of personal choice really, but use of example of call option to motivate “fair strike” could have been avoided. The entire discussion appearing in the text could have been done using only the forward contract.

Written by Vineet

August 19, 2010 at 12:42 pm

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