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Derivative Securities – Fall 2007– Section 2
Notes by Robert V. Kohn, extended and improved by Steve Allen.
Courant Institute of Mathematical Sciences.
Binomial and trinomial oneperiod models.
This section explores the implications of
arbitrage for the pricing of contingent claims in a oneperiod setting.
We do not claim, of course, that any market can realistically be modelled using a one
period binomial or trinomial framework. But we will argue soon that many markets can be
modelled using
multiperiod
trees, in much the same way that diﬀusion can be modelled by
random walk on a lattice. A good understanding of the singleperiod setting will lead, with
just a little extra work, to an understanding of the multiperiod models.
Many books discuss only the binomial case. I discuss also the trinomial setting, because
it provides the simplest example of a market that’s not complete. Moreover it permits me
to introduce (ever so brieﬂy) the connection between riskneutral pricing and the duality
theory of linear programming. However this is “enrichment” material: my discussion of
the trinomial model, and the related discussion of linear programming duality, will not be
tested by the HW’s or exams.
We’ll do two passes. In the ﬁrst, we’ll focus (like most texts, including Hull and Baxter
Rennie) on hedging by a combination of a bond and the “underlying.” In the second, we’ll
discuss how hedging can alternatively be achieved by buying or selling just a forward or
futures contract. The second viewpoint (hedging by forwards or futures) is in some ways
better than the ﬁrst, as we’ll explain when we begin that discussion. (For this reason, Steve
Allen’s version of Section 2 focuses entirely on hedging by forwards or futures.)
****************
The binomial model.
We consider a oneperiod market which has
•
just two securities: a stock (paying no dividend, initial unit price per share
s
1
dollars)
and a bond (interest rate
r
, one bond pays one dollar at maturity).
•
just one maturity date
T
•
just two possible states for the stock price at time
T
:
s
2
and
s
3
,w
ith
s
2
<s
3
(see the ﬁgure). We could suppose we know the probability
p
that the stock will be worth
s
3
at time
T
. This would allow us to calculate the expected value of any contingent claim.
However we will make no use of such knowledge. Pricing by arbitrage considerations makes
no use of information about probabilities – it uses just the list of possible events.
The reasonable values of
s
1
,s
2
3
are not arbitrary: the economy should permit no arbitrage.
This requires that
s
2
1
e
rT
3
.
1
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View Full Document S
1
S
3
S
2
Figure 1: Prices in the oneperiod binomial market model.
It’s easy to see that if this condition is violated then an arbitrage is possible. The converse
is extremely plausible; a simple proof will be easy to give a little later.
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This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.
 Fall '11
 Bayou
 Finance, Arbitrage

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