The binomial asset pricing model provides a powerful tool to understand arbitrage pricing theory
and probability theory. In this course, we shall use it for both these purposes.
In the binomial asset pricing model, we model stock prices in discrete time, assuming that at each
step, the stock price will change to one of two possible values. Let us begin with an initial positive
stock price S0. There are two positive numbers, d and u, with
0 < d < u; (1.1)
such that at the next period, the stock price will be either dS0 or uS0. Typically, we take d and u
to satisfy 0 < d < 1 < u, so change of the stock price from S0 to dS0 represents a downward
movement, and change of the stock price from S0 to uS0 represents an upward movement. It is
common to also have d = 1
u, and this will be the case in many of our examples. However, strictly
speaking, for what we are about to do we need to assume only (1.1) and (1.2) below.
Of course, stock price movements are much more complicated than indicated by the binomial asset
pricing model. We consider this simple model for three reasons. First of all, within this model the
concept of arbitrage pricing and its relation to risk-neutral pricing is clearly illuminated. Secondly,
the model is used in practice because with a sufficient number of steps, it provides a good, computationally
tractable approximation to continuous-time models. Thirdly, within the binomial model
we can develop the theory of conditional expectations and martingales which lies at the heart of
continuous-timemodels.
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