2.1 The single period binomial model.
- A single stock of initial valueS, in the time interval [0,T] it can either
increase by a factorUto valueSUwith probabilityp, or it can decrease
in value by factorDto valueSDwith probabilityq= 1−p. - A single bond with a continuously compounded interest raterover the
interval [0,T]. If the initial value of the bond isB, then the final value
of the bond will beBexp(rT). - A market for derivatives (such as options) dependent on the value of
the stock at the end of the period. The payoff of the derivative to
some investor would be the rewards (or penalties)f(SU) andf(SD).
For example, a futures contract with strike priceKwould have value
f(ST) =ST−K. A call option with strike priceK, would have value
f(ST) = max(ST−K,0).
A realistic financial assumption would be thatD <exp(rT)< U. Then
investment in the risky security may pay better than investment in a risk free
bond, but it may also pay less! The mathematics only requires thatU 6 =D,
see below.
We can attempt to find the value of the derivative by creating a portfolio
of the stock and the bond which will have the same value as the derivative
itself in any circumstance, called areplicating portfolio. Consider a port-
folio consisting ofφunits of the stock worthφS andψunits of the bond
worthψB. (Note we are making the assumption that the stock and bond
are divisible, we can buy them in any amounts including negative amounts
which are short positions.) If we were to buy the this portfolio at time zero,
it would cost
φS+ψB.One time period of lengthT on the trading clock later, the portfolio would
be worth
φSD+ψBexp(rT)after a down move and
φSU+ψBexp(rT)after an up move. You should find this mathematically meaningful: there
are two unknown quantitiesφandψto buy for the portfolio, and we have
two expressions to match with the two values of the derivative! That is, the