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On Optimal Arbitrage Strategy

On Optimal Arbitrage Strategy. Shuhong Fang shfang@fudan.edu.cn Department of Finance, School of Management Fudan University July 6, 2007. 1 Introduction.

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On Optimal Arbitrage Strategy

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  1. On Optimal Arbitrage Strategy Shuhong Fang shfang@fudan.edu.cn Department of Finance, School of Management Fudan University July 6, 2007 www.swingtum.com/institute/IWIF

  2. 1 Introduction • Currently, the financial theory mainly based on the standard portfolio theory originated from Markowitz (1952), and followed by Sharpe (1964), Merton (1972) and so on. Basic assumption: no-arbitrage opportunity! www.swingtum.com/institute/IWIF

  3. 1 Introduction • However, arbitrage opportunities do exist. Banz (1981), Litzenberger & Ramaswamy (1979), DeBondt & Thaler (1985), Shefrin & Statman (1985), Fama & French (1992), Jegadeesh & Titman (1993) and so on. More recent evidences: Deviations from put-call parity in options markets (Ofek，Richardson and Whitelaw, JFE, 2004) Paradoxical behavior of prices in some equity carve-outs (Lamont and Thaler, JPE, 2003) www.swingtum.com/institute/IWIF

  4. 1 Introduction • Trade-off of the risk and return of arbitrages Korkie and Turtle (2002) : Mean-variance analysis of arbitrage portfolios based on the current definition of arbitrage portfolios. Fang (2007) : strictly define the arbitrage portfolio and its return, and then mean-variance analysis is presented. • What about risk-free arbitrages? www.swingtum.com/institute/IWIF

  5. 2Optimal Arbitrage Portfolio • States s = 1, 2, …, S Financial assets j = 1, 2, …, N current value vj, end-period value xsj rate of return on asset j in state s: rsjxsj/vj-1. The economy is characterized by current value vector v  (v1, v2, …, vN)T and state space tableauofprices www.swingtum.com/institute/IWIF

  6. 2Optimal Arbitrage Portfolio • Investor’s trading strategy n  (n1, n2, …, nN)T, nj = number of units held of asset j. investor’s commitment in asset j Wjnj vj. • An arbitrage is a non-zero vector of commitments summing to zero, that is nTv = 0 and (W1, W2, …, WN)T0. (2) www.swingtum.com/institute/IWIF

  7. 2Optimal Arbitrage Portfolio • Arbitrage size: • Arbitrage portfolio : w =(W1/W0, W2/W0, …, WN/W0), wjWj/W0, weight of the arbitrage in asset j . • Return of arbitrage portfolio: www.swingtum.com/institute/IWIF

  8. 2Optimal Arbitrage Portfolio • An arbitrage opportunity implies an arbitrage that enjoys a sure profit, that is • X n 0, and X n 0. • An arbitrage opportunity implies there exists an arbitrage portfolio such that www.swingtum.com/institute/IWIF

  9. 2Optimal Arbitrage Portfolio • Assume that the probability p of state s is ps, s = 1, 2, …, S. An arbitrager will face the following optimization problem (OAP) such that www.swingtum.com/institute/IWIF

  10. 2Optimal Arbitrage Portfolio • Claim: The optimal arbitrage strategy problem (OAS) enjoys a solution if and only if there is an arbitrage opportunity. That is • F {w RN| Rw 0, Rw0, wT1＝0}. www.swingtum.com/institute/IWIF

  11. 3A Simple Approach • The original optimal problem (OAS) may be transformed to the following linear programming problem eT(w+-w-) (LPP) such that w+ , w-  0, R(w + - w -)  0, R(w + - w -)  0, (w+)T 1= (w-)T 1 = 1. where R is thestate space tableau of return . www.swingtum.com/institute/IWIF

  12. 4Further Considerations • Classification of arbitrage opportunities • Risk-free arbitrage opportunity • Surely-profitable arbitrage opportunity • Constantly-profitable arbitrage opportunity • Risky arbitrage opportunity www.swingtum.com/institute/IWIF

  13. Comments Welcome！ www.swingtum.com/institute/IWIF

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