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Recap on BlackScholes
Volatility
American options
Lecture 9  American options
BUSI 588, Fall 2011
Diego Garc´
ıa
KenanFlagler Business School
UNC at Chapel Hill
September 26th, 2011
c
±
Diego Garc´
ıa, BUSI 588, KenanFlagler, Fall 2011
Lecture 9  American options
1 / 19
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Volatility
American options
Outline
1
Recap on BlackScholes
Expected returns on options
Hedging
2
Volatility
Basic estimates
ARCH/GARCH
3
American options
A simple example
Early exercise in Galitzin
Handouts today:
Class slides.
Case 9 solutions.
c
±
Diego Garc´
ıa, BUSI 588, KenanFlagler, Fall 2011
Lecture 9  American options
2 / 19
Recap on BlackScholes
Volatility
American options
Risk and return for call options
What is the expected return of a call option on an underlying asset
with a
β
S
= 1? Assume
r
f
= 0
.
05 and
E
[
r
m
]

r
f
= 5%, so
E
[
r
s
] = 10%.
Further let
σ
= 0
.
5,
T
= 1,
δ
= 0,
S
= 1300,
K
= 1350. From
BlackScholes:
C
= 262
.
4 and Δ = 0
.
6072
Key: a call is just a long position in the underlying asset (Δ shares)
ﬁnanced with borrowing. As a portfolio, its beta must satisfy:
β
C
=
Δ
S
Δ
S
+
B
β
S
+
B
Δ
S
+
B
β
B
= Δ
S
C
β
S
= Ω
β
S
with Ω
≥
1.
Using
S
= 1300,
C
= 262
.
4, Δ = 0
.
6072, we have Ω = 3 and
E
[
r
c
] = 20%.
c
±
Diego Garc´
ıa, BUSI 588, KenanFlagler, Fall 2011
Lecture 9  American options
3 / 19
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Volatility
American options
Risk and return for put options
A put is also simply a short position in the underlying asset
coupled with lending.
As a portfolio, its beta must satisfy:
β
P
=
Δ
S
Δ
S
+
B
β
S
+
B
Δ
S
+
B
β
B
= Δ
S
P
β
S
= Ω
β
S
but now Ω
≤
0, since Δ
≤
0.
With the parameters as in the previous example we have Ω
≈ 
2,
so that
E
[
r
p
] =

5%.
c
±
Diego Garc´
ıa, BUSI 588, KenanFlagler, Fall 2011
Lecture 9  American options
4 / 19
Volatility
American options
One last hedging example
Suppose we just sold a straddle with a oneyear maturity on an
underlying asset trading at $100 (
σ
= 0
.
4 and
r
f
= 0
.
05), with a strike of
$100. We have the following information on the straddle (ﬁrst column)
and one set of liquid traded options (shorter maturity).
Sold options
Hedging instruments
Strike
100
100
Time
1
0.5
Call price
17.97
12.36
Put price
13.21
9.95
Delta (call)
0.6263
0.5900
Gamma
0.0095
0.0137
(a) How could one hedge the straddle risk by trading in the shortterm
calls? (b) Can one create a delta and gammaneutral position using the
options with
K
= 100 (a call and a put)?
c
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This note was uploaded on 11/25/2011 for the course BUSI 588 taught by Professor Staff during the Fall '10 term at UNC.
 Fall '10
 Staff
 Derivatives, Options, Volatility

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