1
NBA 6730: Derivative Securities
Lecture 10: Introduction to Option Pricing
09/28/2010
George Gao
NBA6730Derivative Securities I
2
Agenda
Pricing an option needs assumptions about the behavior of
the underlying security’s prices. The basic idea is to
construct a replicating portfolio of stock and bond that
dynamically
replicates the payoff of an option. In this
lecture, we learn Binomial option pricing that is used most
on the Street.
Bounds on options
Early exercise of American options
Binomial model (plainvanilla European call and put)
RiskNeutral pricing
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
NBA6730Derivative Securities I
3
(1) Bounds on Options
Recall our notation: small letters for European options,
big letters for American options
Strike, maturity, underlying security price
ܥ
ݐ
≜ ܥ
ܵ
ݐ , ܭ, ݐ, ܶ , ܲ
ݐ
≜ ܲ
ܵ
ݐ , ܭ, ݐ, ܶ , ܿ
ݐ , ሺݐሻ
Options never have negative value
ܥሺݐሻ 0, ܿሺݐሻ 0, ܲሺݐሻ 0, ሺݐሻ 0
A call option is never worth more than the stock
ܥ
ݐ
ܵ
ݐ , ܿ
ݐ
ܵ
ݐ
Otherwise, you make a riskless profit by buying a stock and
selling a call option immediately
NBA6730Derivative Securities I
4
(1) Bounds on Options
A put option is never worth more than the strike
ܲ
ݐ
ܭ,
ݐ
ܭ
Otherwise, you make a riskless profit by selling a put option
(you pay the strike price
ܭ
at most
)
A European put option is never worth more than the
present value of the strike price
ݐ
ܸܲ
ܭ
ൌ ܭ݁
ିሺ்ି௧ሻ
A European put can only be exercised at maturity
ܶ
. Thus, you
sell a put at
ݐ
and get
, and meantime you can invest
ܸܲሺܭሻ
in
Tbill at
ݐ
and get
ܭ
at maturity
ܶ
.
This relation doesn’t always hold for an American put option
since an American put can be exercised at any time before
ܶ
.
3
NBA6730Derivative Securities I
5
(1) Bounds on Options
American options with more time to maturity are at
least as valuable as the same options with less time to
maturity. For
ܶ
ଶ
ܶ
ଵ
:
ܥ
ݐ, ܶ
ଶ
ܥ
ݐ, ܶ
ଵ
, ܲ
ݐ, ܶ
ଶ
ܲ
ݐ, ܶ
ଵ
An American option is at least as valuable as its
intrinsic
value
ܥ
ݐ
max
0, ܵ
ݐ
െ ܭ , ܲሺݐሻ maxሾ0, ܭ െ ܵ
ݐ ሿ
Otherwise, buy a call and execute the option immediately. You
make riskless profit by getting
SK
and paying
C
. Similar
arguments for a put.
The two types of bounds above don’t always hold for
European options
NBA6730Derivative Securities I
6
(1) Bounds on Options
Options on
non dividend
paying stocks
Call option:
ܥ
ݐ
ܿ
ݐ
max
0, ܵ
ݐ
െ ܭ ∙ ܤ
ݐ, ܶ
Put option:
ܲሺݐሻ ሺݐሻ maxሾ0, ܭ ∙ ܤ
ݐ, ܶ
െ ܵ
ݐ ሿ
Options on
dividend
paying stocks
Call option:
ܥ
ݐ
ܿ
ݐ
ܵ
ݐ
െ ܭ ∙ ܤ
ݐ, ܶ
െ ܸܲሺܦሻ
Put option:
ܲ
ݐ
ݐ
ܭ ∙ ܤ
ݐ, ܶ
െ ܵ
ݐ
ܸܲሺܦሻ
Intuitions
Recall the payoff diagrams in deriving putcall parity
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 GeorgeGao
 Intel, NBA6730Derivative Securities

Click to edit the document details