Derivative Securities – Fall 2007– Section 3
Notes by Robert V. Kohn, extended and improved by Steve Allen.
Courant Institute of Mathematical Sciences.
Multiperiod Binomial Trees.
We turn to the valuation of derivative securities in a
timedependent setting. We focus for now on multiperiod binomial models, i.e. binomial
trees. This setting is simple enough to let us do everything explicitly, yet rich enough to
approximate many realistic problems.
The material covered in this section is very standard (and very important). The treatment
here tracks closely with Baxter and Rennie (Chapter 2). Hull addresses this topic in Chapter
11 (6th edition).
Another good treatment is that of Jarrow and Turnbull (Chapter 5,
2nd edition), which includes many examples.
In the Section 4 notes we’ll discuss how
the parameters should be chosen to mimic the conventional (BlackScholes) hypothesis of
lognormal stock prices, and we’ll pass to the continuoustime limit.
Binomial trees are widely used in practice, in part because they are easy to implement
numerically. (Also because the scheme can easily be adjusted to price American options.)
For a nice discussion of alternative numerical implementations, see the article “Nine ways
to implement the binomial method for option valuation in Matlab,” by D.J. Higham, SIAM
Review 44 (2002) 661677.
As we saw in Section 2, an option can be replicated by trading the underlying, or alterna
tively by trading forwards or futures on the underlying. We’ll focus ﬁrst on replication by
trading the underlying (since this is the ﬁrst thing you’ll see in most books). Then we’ll
make a second pass, replicating with futures (the viewpoint emphasized by Steve Allen’s
version of these notes).
The practical question is not how to replicate an option but rather how to hedge it. But
replication and hedging are intimately connected. Simple example: when you sell a call
with strike
K
and maturity
T
you receive cash for it now, but you owe (
s
T

K
)
+
to the
holder when the option matures. We’ll explain (in the binomial setting) how the apparent
uncertainty in your ﬁnaltime obligation can be entirely eliminated (hedged) by pursuing
a suitable trading strategy. It is, of course, the strategy that replicates your ﬁnaltime
obligation (
s
T

K
)
+
.
*******************************
First pass: trading the underlying.
A multiperiod binomial model generalizes the
singleperiod binomial model we considered in Section 2. It has
•
just two securities: a risky asset (a “stock,” paying no dividend) and a riskless asset
(“bond”);
•
a series of times 0
,δt,
2
δt,.
..,Nδt
=
T
at which trades can occur;
•
interest rate
r
i
during time interval
i
for the bond;
•
a binomial tree of possible states for the stock prices.
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Figure 1: States of a nonrecombining binomial tree.
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 Fall '11
 Bayou
 Finance, Valuation, binomial tree

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