# The Arbitrage Theorem - PowerPoint PPT Presentation

1 / 25

The Arbitrage Theorem. Henrik Jönsson Mälardalen University Sweden. Contents . Necessary conditions European Call Option Arbitrage Arbitrage Pricing Risk-neutral valuation The Arbitrage Theorem. Necessary conditions. No transaction costs

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

The Arbitrage Theorem

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

## The Arbitrage Theorem

Henrik Jönsson

Mälardalen University

Sweden

Gurzuf, Crimea, June 2001

### Contents

• Necessary conditions

• European Call Option

• Arbitrage

• Arbitrage Pricing

• Risk-neutral valuation

• The Arbitrage Theorem

Gurzuf, Crimea, June 2001

### Necessary conditions

• 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.

### European Call Option

Gurzuf, Crimea, June 2001

### Arbitrage

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

### Arbitrage

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

### Arbitrage

• Two portfolios A & B have the same value at t=T

• No risk-less arbitrage opportunity

• They have the same value at any time tT

Gurzuf, Crimea, June 2001

q

prob.

1-q

### Arbitrage Pricing

The Binomial price model

0du

0q1

Gurzuf, Crimea, June 2001

1+r < d:

1+r > u:

### Arbitrage Pricing

r = risk-free interest rate

d < (1+r) < u

Gurzuf, Crimea, June 2001

Equivalence portfolio

Call option

(t=0)

(t=T)

### Arbitrage Pricing

(t=0)

(t=T)

r = risk-free interest rate

Gurzuf, Crimea, June 2001

No Arbitrage Opportunity

### Arbitrage Pricing

Choose  and B such that

Gurzuf, Crimea, June 2001

### Arbitrage Pricing

Gurzuf, Crimea, June 2001

q

prob.

1-q

### Risk-neutral valuation

p = risk-neutral probability

Expected rate of return = (1+r)

( p = equivalent martingale probability )

Gurzuf, Crimea, June 2001

### Risk-neutral valuation

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

### The Arbitrage Theorem

• 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

### The Arbitrage Theorem

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

Gurzuf, Crimea, June 2001

### The Arbitrage Theorem

• If the outcome X=j then

Gurzuf, Crimea, June 2001

### The Arbitrage Theorem

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

### The Arbitrage Theorem

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

### The Arbitrage Theorem

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,

### The Arbitrage Theorem

Proof (cont.):

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

Gurzuf, Crimea, June 2001

### The Arbitrage Theorem

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

### The Arbitrage Theorem

Return functions:

• i=1:

• i=2:

Gurzuf, Crimea, June 2001

### The Arbitrage Theorem

Expected return

• i=1:

• i=2:

Gurzuf, Crimea, June 2001

### The Arbitrage Theorem

• (1)and the Arbitrage theorem gives:

• (2), (3) & the Arbitrage theoremgives the non-arbitrage option price:

(3)

Gurzuf, Crimea, June 2001