1. Gurzuf, Crimea, June 20011 The Arbitrage Theorem Henrik Jönsson Mälardalen University Sweden

2. Gurzuf, Crimea, June 20012 Contents Necessary conditions European Call Option Arbitrage Arbitrage Pricing Risk-neutral valuation The Arbitrage Theorem

3. Gurzuf, Crimea, June 20013 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

4. Gurzuf, Crimea, June 20014 European Call Option C - Option Price K - Strike price T - Expiration day Exercise only at T Payoff function, e.g.

5. Gurzuf, Crimea, June 20015 Arbitrage The Law of One Price: In a competitive market, if two assets are equivalent, they will tend to have the same market price.

6. Gurzuf, Crimea, June 20016 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.

7. Gurzuf, Crimea, June 20017 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

8. Gurzuf, Crimea, June 20018 Arbitrage Pricing The Binomial price model prob. q 1-q 0du0du0q10q1

9. Gurzuf, Crimea, June 20019 Arbitrage Pricing 1+r < d: 1+r > u: Action at time 0 t=0t=T BorrowS-(1+r)S Buy stock-SdS Return0>0 Action at time 0 t=0t=T Lend-S(1+r)S Sell stockS-uS Return0>0 r = risk-free interest rate d < (1+r) < u

10. Gurzuf, Crimea, June 200110 Arbitrage Pricing Equivalence portfolioCall option r = risk-free interest rate (t=T)(t=0) (t=T)(t=0)

11. Gurzuf, Crimea, June 200111 Arbitrage Pricing Choose  and B such that No Arbitrage Opportunity

12. Gurzuf, Crimea, June 200112 Arbitrage Pricing

13. Gurzuf, Crimea, June 200113 Risk-neutral valuation p = risk-neutral probability prob. q 1-q ( p = equivalent martingale probability ) Expected rate of return = (1+r)

14. Gurzuf, Crimea, June 200114 Risk-neutral valuation Expected present value of the return = 0 C = (1+r) -1 [pC u + (1-p)C d ] ( p = equivalent martingale probability ) Price of option today = Expected present value of option at time T Risk-neutral probability p

15. Gurzuf, Crimea, June 200115 The Arbitrage Theorem Let X  {1,2,…,m} be the outcome of an experiment Let p = (p 1,…,p m ), p j = P{X=j}, for all j=1,…,m Let there be n different investment opportunities Let  = (  1,…,  n ) be an investment strategy (  i pos., neg. or zero for all i)

16. Gurzuf, Crimea, June 200116 The Arbitrage Theorem Let r i (j) be the return function for a unit investment on investment opportunity i 11  1 r 1 (1)  1 r 1 (2)  1 r 1 (m) prob. p1p1 p2p2 pmpm Example: i=1

17. Gurzuf, Crimea, June 200117 The Arbitrage Theorem If the outcome X=j then

18. Gurzuf, Crimea, June 200118 The Arbitrage Theorem Exactly one of the following is true: Either a)there exists a probability vector p = (p 1,…,p m ) for which or b) there is an investment strategy  = (  1,…,  m ) for which

19. Gurzuf, Crimea, June 200119 The Arbitrage Theorem Proof: Use the Duality Theorem of Linear Programming Primal problemDual problem If x * primal feasible & y * dual feasible then c T x * =b T y * x * primal optimum & y * dual optimum If either problem is infeasible, then the other does not have an optimal solution.

20. Gurzuf, Crimea, June 200120 The Arbitrage Theorem Proof (cont.): Primal problemDual problem

21. Gurzuf, Crimea, June 200121 The Arbitrage Theorem Dual feasible iff y probability vector under which all investments have the expected return 0 Primal feasible when  i = 0, i=1,…, n, c T  * = b T y * = 0  Optimum! No sure win is possible! Proof (cont.):

22. Gurzuf, Crimea, June 200122 The Arbitrage Theorem Example: Stock (S 0 ) with two outcomes Two investment opportunities: i=1: Buy or sell the stock i=2: Buy or sell a call option (C)

23. Gurzuf, Crimea, June 200123 The Arbitrage Theorem Return functions: i=1: i=2:

24. Gurzuf, Crimea, June 200124 The Arbitrage Theorem Expected return i=1: i=2:

25. Gurzuf, Crimea, June 200125 The Arbitrage Theorem (1) and the Arbitrage theorem gives: (2), (3) & the Arbitrage theorem gives the non-arbitrage option price: (3)