Arbitrage in Option Pricing

71 Pages·2016·0.21 MB·English

Arbitrage in Option Pricing
(cid:2)c 2016 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 196
All general laws are attended with inconveniences,
when applied to particular cases.
— David Hume (1711–1776)
(cid:2)c 2016 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 197
Arbitrage
•
The no-arbitrage principle says there is no free lunch.
•
It supplies the argument for option pricing.
•
A riskless arbitrage opportunity is one that, without any
initial investment, generates nonnegative returns under
all circumstances and positive returns under some.
•
In an efficient market, such opportunities do not exist
(for long).
(cid:2)c 2016 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 198
Portfolio Dominance Principle
•
Consider two portfolios A and B.
•
A should be more valuable than B if A’s payoff is at
least as good as B’s under all circumstances and better
under some.
(cid:2)c 2016 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 199
Two Simple Corollaries
•
A portfolio yielding a zero return in every possible
scenario must have a zero PV.
–
Short the portfolio if its PV is positive.
–
Buy it if its PV is negative.
–
In both cases, a free lunch is created.
•
Two portfolios that yield the same return in every
a
possible scenario must have the same price.
a
Aristotle, “those who are equal should have everything alike.”
(cid:2)c 2016 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 200
The PV Formula (p. 32) Justified
Theorem 1 C , C , . . . , C
For a certain cash flow ,
1 2 n
(cid:2)n
P C d i .
= ( )
i
i=1
• P∗ < P
Suppose the price .
• n n
Short the zeros that match the security’s cash flows.
• P
The proceeds are dollars.
(cid:2)c 2016 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 201
P
C
C 2 C (cid:5)
(cid:3) 3 C security
1
n
(cid:3) (cid:3) · · · (cid:2)
(cid:3)(cid:4) (cid:4) · · · (cid:3)(cid:4)
(cid:4) (cid:5)
C zeros
(cid:4) C C n
1 C
3
2
P∗
(cid:2)c 2016 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 202
The Proof (concluded)
• P∗
Then use of the proceeds to buy the security.
•
The cash inflows of the security will offset exactly the
obligations of the zeros.
• P − P∗
A riskless profit of dollars has been realized now.
• P∗ > P
If , just reverse the trades.
(cid:2)c 2016 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 203
Two More Examples
•
A put or a call must have a nonnegative value.
–
Suppose otherwise and the option has a negative
price.
–
Buy the option for a positive cash flow now.
–
It will end up with a nonnegative amount at
expiration.
–
So an arbitrage profit is realized now.
(cid:2)c 2016 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 204
Two More Examples (continued)
•
An American option cannot be worth less than the
a
intrinsic value.
–
This is true if the intrinsic value is zero (p. 204).
–
Suppose the intrinsic value is positive but the claim
is false.
–
So the American option is cheaper than its intrinsic
value.
– X
For the call: Short the stock and lend dollars.
– X
For the put: Borrow dollars and buy the stock.
a ,S − X ,X − S
max(0 t ) for the call and max(0 t) for the put.
(cid:2)c 2016 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 205