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The Arbitrage Theorem

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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  1. The Arbitrage Theorem Henrik Jönsson Mälardalen University Sweden Gurzuf, Crimea, June 2001

  2. Contents • Necessary conditions • European Call Option • Arbitrage • Arbitrage Pricing • Risk-neutral valuation • The Arbitrage Theorem Gurzuf, Crimea, June 2001

  3. 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

  4. C - Option Price K - Strike price T - Expiration day Exercise only at T Payoff function, e.g. European Call Option Gurzuf, Crimea, June 2001

  5. 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

  6. 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

  7. 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

  8. q prob. 1-q Arbitrage Pricing The Binomial price model 0du 0q1 Gurzuf, Crimea, June 2001

  9. 1+r < d: 1+r > u: Arbitrage Pricing r = risk-free interest rate d < (1+r) < u Gurzuf, Crimea, June 2001

  10. Equivalence portfolio Call option (t=0) (t=T) Arbitrage Pricing (t=0) (t=T) r = risk-free interest rate Gurzuf, Crimea, June 2001

  11. No Arbitrage Opportunity Arbitrage Pricing Choose  and B such that Gurzuf, Crimea, June 2001

  12. Arbitrage Pricing Gurzuf, Crimea, June 2001

  13. 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

  14. 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

  15. 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

  16. 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

  17. The Arbitrage Theorem • If the outcome X=j then Gurzuf, Crimea, June 2001

  18. 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

  19. 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

  20. The Arbitrage Theorem Primal problem Dual problem Proof (cont.): Gurzuf, Crimea, June 2001

  21. 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

  22. 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

  23. The Arbitrage Theorem Return functions: • i=1: • i=2: Gurzuf, Crimea, June 2001

  24. The Arbitrage Theorem Expected return • i=1: • i=2: Gurzuf, Crimea, June 2001

  25. 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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