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The Arbitrage Theorem

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The Arbitrage Theorem

Henrik Jönsson

Mälardalen University

Sweden

Gurzuf, Crimea, June 2001

- Necessary conditions
- European Call Option
- Arbitrage
- Arbitrage Pricing
- Risk-neutral valuation
- The Arbitrage Theorem

Gurzuf, Crimea, June 2001

- No transaction costs
- Same risk-free interest rate r for borrowing & lending
- Short positions possible in all instruments
- Same taxes
- Momentary transactions between different assets possible

Gurzuf, Crimea, June 2001

C - Option Price

K - Strike price

T - Expiration day

Exercise only at T

Payoff function, e.g.

Gurzuf, Crimea, June 2001

The Law of One Price:

In a competitive market, if two assets are equivalent, they will tend to have the same market price.

Gurzuf, Crimea, June 2001

Definition:

- A trading strategy that takes advantage of two or more securities being mispriced relative to each other.
- The purchase and immediate sale of equivalent assets in order to earn a sure profit from a difference in their prices.

Gurzuf, Crimea, June 2001

- Two portfolios A & B have the same value at t=T
- No risk-less arbitrage opportunity
- They have the same value at any time tT

Gurzuf, Crimea, June 2001

q

prob.

1-q

The Binomial price model

0du

0q1

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1+r < d:

1+r > u:

r = risk-free interest rate

d < (1+r) < u

Gurzuf, Crimea, June 2001

Equivalence portfolio

Call option

(t=0)

(t=T)

(t=0)

(t=T)

r = risk-free interest rate

Gurzuf, Crimea, June 2001

No Arbitrage Opportunity

Choose and B such that

Gurzuf, Crimea, June 2001

Gurzuf, Crimea, June 2001

q

prob.

1-q

p = risk-neutral probability

Expected rate of return = (1+r)

( p = equivalent martingale probability )

Gurzuf, Crimea, June 2001

Expected present value of the return = 0

Price of option today = Expected present value of option at time T

C = (1+r)-1[pCu + (1-p)Cd]

- Risk-neutral probability p

( p = equivalentmartingale probability )

Gurzuf, Crimea, June 2001

- Let X{1,2,…,m} be the outcome of an experiment
- Let p = (p1,…,pm), pj = P{X=j}, for all j=1,…,m
- Let there be n different investment opportunities
- Let = (1,…, n) be an investmentstrategy (i pos., neg. or zero for all i)

Gurzuf, Crimea, June 2001

1r1(1)

p1

1r1(2)

p2

Example: i=1

1

prob.

1r1(m)

pm

- Let ri(j) be the return function for a unit investment on investment opportunity i

Gurzuf, Crimea, June 2001

- If the outcome X=j then

Gurzuf, Crimea, June 2001

Exactly one of the following is true: Either

- there exists a probability vector p=(p1,…,pm) for which
or

b) there is an investment strategy =(1,…, m) for which

Gurzuf, Crimea, June 2001

Primal problem

Dual problem

Proof: Use the Duality Theorem of Linear Programming

- If x* primal feasible & y* dual feasible then
- cTx* =bTy*
- x* primal optimum & y* dual optimum
- If either problem is infeasible, then the other does not have an optimal solution.

Gurzuf, Crimea, June 2001

Primal problem

Dual problem

Proof (cont.):

Gurzuf, Crimea, June 2001

Dual feasible iff y probability vector under which all investments have the expected return 0

Primal feasible when i = 0, i=1,…, n,

Proof (cont.):

cT* = bTy* = 0 Optimum! No sure win is possible!

Gurzuf, Crimea, June 2001

Example:

- Stock (S0) with two outcomes
- Two investment opportunities:
- i=1: Buy or sell the stock
- i=2: Buy or sell a call option (C)

Gurzuf, Crimea, June 2001

Return functions:

- i=1:
- i=2:

Gurzuf, Crimea, June 2001

Expected return

- i=1:
- i=2:

Gurzuf, Crimea, June 2001

- (1)and the Arbitrage theorem gives:
- (2), (3) & the Arbitrage theoremgives the non-arbitrage option price:

(3)

Gurzuf, Crimea, June 2001