The arbitrage theorem
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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

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The Arbitrage Theorem

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The arbitrage theorem

The Arbitrage Theorem

Henrik Jönsson

Mälardalen University

Sweden

Gurzuf, Crimea, June 2001


Contents

Contents

  • Necessary conditions

  • European Call Option

  • Arbitrage

  • Arbitrage Pricing

  • Risk-neutral valuation

  • The Arbitrage Theorem

Gurzuf, Crimea, June 2001


Necessary conditions

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


European call option

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

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


Arbitrage1

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


Arbitrage2

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


Arbitrage pricing

q

prob.

1-q

Arbitrage Pricing

The Binomial price model

0du

0q1

Gurzuf, Crimea, June 2001


Arbitrage pricing1

1+r < d:

1+r > u:

Arbitrage Pricing

r = risk-free interest rate

d < (1+r) < u

Gurzuf, Crimea, June 2001


Arbitrage pricing2

Equivalence portfolio

Call option

(t=0)

(t=T)

Arbitrage Pricing

(t=0)

(t=T)

r = risk-free interest rate

Gurzuf, Crimea, June 2001


Arbitrage pricing3

No Arbitrage Opportunity

Arbitrage Pricing

Choose  and B such that

Gurzuf, Crimea, June 2001


Arbitrage pricing4

Arbitrage Pricing

Gurzuf, Crimea, June 2001


Risk neutral valuation

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 valuation1

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 theorem1

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


The arbitrage theorem2

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 theorem3

The Arbitrage Theorem

  • If the outcome X=j then

Gurzuf, Crimea, June 2001


The arbitrage theorem4

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 theorem5

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 theorem6

The Arbitrage Theorem

Primal problem

Dual problem

Proof (cont.):

Gurzuf, Crimea, June 2001


The arbitrage theorem7

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 theorem8

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 theorem9

The Arbitrage Theorem

Return functions:

  • i=1:

  • i=2:

Gurzuf, Crimea, June 2001


The arbitrage theorem10

The Arbitrage Theorem

Expected return

  • i=1:

  • i=2:

Gurzuf, Crimea, June 2001


The arbitrage theorem11

The Arbitrage Theorem

  • (1)and the Arbitrage theorem gives:

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

(3)

Gurzuf, Crimea, June 2001


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