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On Optimal Arbitrage Strategy

This document provides an in-depth discussion on the optimal arbitrage strategy, building upon traditional portfolio theory originally formulated by Markowitz and later expanded by other financial theorists. It explores the existence of arbitrage opportunities that deviate from the no-arbitrage assumption of standard models, addressing risk-return trade-offs and optimal asset allocation in arbitrage portfolios. The text clarifies definitions of arbitrage opportunities and examines various classifications, ultimately leading to a linear programming approach to solve the optimal arbitrage strategy problem.

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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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