Derivative Securities – Fall 2007 – Section 1
Notes by Robert V. Kohn, extended and improved by Steve Allen.
Courant Institute of Mathematical Sciences.
Forwards, puts, calls, and other contingent claims.
This section discusses the most
basic examples of contingent claims, and explains how considerations of arbitrage determine
or restrict their prices. This material is in Chapters 1 and 3, Sections 8.1 and 8.2, and
Chapter 10 of Hull (6th edition). For a more concise, less distributed treatment see also
Chapters 2 and 3 of Jarrow and Turnbull. We concentrate for simplicity on European
options rather than American ones, on forwards rather than futures, and on deterministic
rather than stochastic interest rates. We close with a detailed discussion about the pricing
of forwards, corresponding to Chapter 5 of Hull.
***********************
The most basic instruments:
Forward contract
with maturity T and delivery price K.
buy a forward
↔
ho
ldalongforward
↔
holder is obliged to buy the
underlying asset at price K on date T
.
European call option
with maturity T and strike price K.
buy a call
↔
ldalongca
l
l
↔
holder is entitled to buy the
underlying asset at price K on date T
.
European put option
with maturity T and strike price K.
buy a put
↔
hold a long put
↔
holder is entitled to sell the
underlying asset at price K on date T
.
These are
contingent claims
, i.e. their value at maturity is not known in advance. Payoﬀ
formulas and diagrams (value at maturity, as a function of
S
T
=value of the underlying) are
shownintheF
igure
.
Any long position has a corresponding (opposite)
short
position:
Buyer of a claim has a long position
↔
seller has a short position.
Payoﬀ diagram of short position = negative of payoﬀ diagram of long position.
1
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S
 K
(S
K)
(KS
)
Put
Call
S
KK
S
K
S
T
T
T+
+
T
T
T
Figure 1: Payoﬀs of forward, call, and put options.
An
American
option diﬀers from its European sibling by allowing early exercise. For exam
ple: the holder of an American call with strike
K
and maturity
T
has the right to purchase
the underlying for price
K
at any time 0
≤
t
≤
T
. A discussion of American options must
deal with two moreorless independent issues: the unknown future value of the underlying,
and the optimal choice of the exercise time. By focusing initially on European options we’ll
develop an understanding of the ﬁrst issue before addressing the second.
************************
Why do people buy and sell contingent claims? Brieﬂy, to
hedge
or to
speculate
. Examples
of hedging:
•
A US airline has a contract to buy a French airplane for a price ﬁxed in Euros, payable
one year from now. By going long on a forward contract for Euros (payable in dollars)
it can eliminate its foreign currency risk.
•
The holder of a forward contract has unlimited downside risk. Holding a call limits
the downside risk (but buying a call with strike K costs more than buying the forward
with delivery price K). Holding one long call and one short call costs less, but gives
up some of the upside beneﬁt:
(
S
T

K
1
)
+

(
S
T

K
2
)
+
K
1
<K
2
This is known as a “bull spread”. (See the ﬁgure.)
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 Fall '11
 Bayou
 Finance, Interest Rates

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