2.1. SINGLE PERIOD BINOMIAL MODELS 79
combination of the risky stock and risk-free bond replicates the derivative.
As such the movement probabilities alone do not completely assess the risk
associated with the transaction.
Nevertheless, we are left with a nagging feeling that pricing by arbitrage
as done above ignores the probability associated with security price changes.
One could legitimately ask if there is a way to value derivatives by taking
some kind of expected value. The answer is yes, there isanotherprobability
distribution associated with the binomial model that correctly takes into
account the rest of the market. In fact, the quantitiesπand 1−πdefine this
probability distribution. This is called therisk-neutral measureor more
completely therisk-neutral martingale measureand we will talk more
about it later. Economically speaking, the market assigns a “fairer” set of
probabilitiesπand 1−πthat give a value for the option compatible with the
no arbitrage principle. Another way to say this is that the market changes
the odds to make option pricing fairer. The risk-neutral measure approach
is the very modern, sophisticated, and general way to approach derivative
pricing. However it is too advanced for us to approach just yet.
Summary
From R. C. Merton in “Influence of mathematical models in finance on prac-
tice: past, present and future”, inMathematical Models in Finance, edited
by S.D. Howison, F. P. Kelly, and P. Wilmott, Chapman and Hall, London,
1995, pages 1-15.
“The basic insight underlying the Black-Scholes model is that
a dynamic portfolio trading strategy in the stock can be found
which will replicate the returns from an option on that stock.
Hence, to avoid arbitrage opportunities, the option price must
always equal the value of this replicating portfolio.”Sources
This section is adapted from: “Chapter 2, Discrete Processes” inFinancial
Calculusby M. Baxter, A. Rennie, Cambridge University Press, Cambridge,
1996, [5].