78 CHAPTER 2. BINOMIAL OPTION PRICING MODELS
same quantity. At the end of the period, the value of the derivative would be
exactly the same as the portfolio. So selling each derivative would repay the
short with a profit ofV−P and the trade carries no risk! SoPwould not
have been a rational price for the trader to quote and the market would have
quickly mobilized to take advantage of the “free” money on offer in arbitrary
quantity. (This ignores transaction costs. For an individual, transaction
costs might eliminate the profit. However for large firms trading in large
quantities, transaction costs can be minimal.)
Similarly if a seller quoted the derivative at a price P greater thanV,
anyone could short sell the derivative and buy the (φ,ψ) portfolio to lock
in a risk-free profit ofP−V per unit trade. Again the market would take
advantage of the opportunity. Hence,V is the only rational price for the
derivative. We have determined the price of the derivative through arbitrage.
HowNOT to price the derivative and a hint of a better way.
Note that we did not determine the price of the derivative in terms of the
expected value of the stock or the derivative. A seemingly logical thing to do
would be to say that the derivative will have valuef(SU) with probability
pand will have valuef(SD) with probability 1−p. Therefore the expected
value of the derivative at timeT is
E[f] =pf(SU) + (1−p)f(SD).The present value of the expectation of the derivative value is
exp(−rT)E[f] = exp(−rT)[pf(SU) + (1−p)f(SD)].Except in the peculiar case that the expected value just happened to match
the valueV of the replicating portfolio, pricing by expectation would be
driven out of the market by arbitrage! The problem is that the probability
distribution (p, 1 −p) only takes into account the movements of the security
price. The expected value is the value of the derivative over many identical
iterations or replications of that distribution, but there will be only one trial
of this particular experiment, so expected value is not a reasonable way to
weight the outcomes. Also, the expected value does not take into account
the rest of the market. In particular, the expected value does not take into
account that an investor has the opportunity to simultaneously invest in
alternate combinations of the risky stock and the risk-free bond. A special