Chapter 2: FAQs 199
models tend to be of more academic than practical
interest.
No-arbitrage, or arbitrage-free, models represent the
point at which there aren’t any arbitrage profits to be
made. If the same future payoffs and probabilities can
be made with two different portfolios then the two port-
folios must both have the same value today, otherwise
there would be an arbitrage. In quantitative finance
the obvious example of the two portfolios is that of an
option on the one hand and a cash and dynamically
rebalanced stock position on the other. The end result
being the pricing of the option relative to the price of
the underlying asset. The probabilities associated with
future stock prices falls out of the calculation and pref-
erences are never needed. When no-arbitrage pricing is
possible it tends to be used in practice. The price out-
put by a no-arbitrage model is supposedly correct in a
relative sense.
For no-arbitrage pricing to work we need to have mar-
kets that arecomplete, so that we can price one con-
tract in terms of others. If markets are not complete and
we have sources of risk that are unhedgeable then we
need to be able to quantify the relevantmarket price
of risk. This is a way of consistently relating prices of
derivatives with the same source of unhedgeable risk, a
stochastic volatility for example.
Both the equilibrium and no-arbitrage models suffer
from problems concerning parameter stability.
In the fixed-income world, examples of equilibrium
models are Vasicek, CIR, Fong & Vasicek. These have
parameters which are constant, and which can be esti-
mated from time series data. The problem with these
is that they permit very simple arbitrage because the
prices that they output for bonds will rarely match