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Min Dai Dept of Math National University of Singapore

Optimal Investment and Consumption Strategies with Finite Horizon and Transaction Costs: Theory and Computation. Min Dai Dept of Math National University of Singapore Joint works with Yi (2009), with Xu and Zhou (2007), with Jiang, Li and Yi (2009), with Yang (2009), with Zhong (2008).

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Min Dai Dept of Math National University of Singapore

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  1. Optimal Investment and Consumption Strategies with Finite Horizon and Transaction Costs: Theory and Computation Min Dai Dept of Math National University of Singapore Joint works with Yi (2009), with Xu and Zhou (2007), with Jiang, Li and Yi (2009), with Yang (2009), with Zhong (2008)

  2. Contents • Literature review • Problem formulation • An equivalent variational inequality • No consumption case • Consumption case • A numerical method: penalty method • Extensions and Future Work

  3. Literature review • Merton’s model (1969, 1971): without transaction costs to keep a constant fraction of total wealth in the risky asset. • Shortage: incessant trading • Transaction costs caused are unacceptable in practice. • It violates the conventional buy-and-hold strategy.

  4. Literature review • Introduction of transaction costs to Merton’s model • Magill and Constantinides (1976), J. of Economic Theory • Infinite time horizon (one stock) • Davis and Norman (1990), Mathematics of Operation Research • Shreve and Soner (1994), Annals of Applied Probability • Finite horizon optimal investment (one stock) • Gennotte and Jung (1994), Management Sciences • Liu and Loewenstein (2002), Review of Financial Studies • The case of multiple stocks • Akian, Menaldi, and Sulem (1996), SIAM J. Control and Optim. • Muthuraman and Kumar (2006), Mathematical Finance

  5. Asset market

  6. y x Investor’s problem

  7. With transaction costs: HJB equation • A formal derivation (Davis and Norman (1990))

  8. HJB equations: continued

  9. Change of variables (1)

  10. A further change of variables (2)

  11. Key idea: equivalence to a double obstacle problem

  12. The case of • Key proof (Dai and Yi (2009), J. Differential Eqn.) • SR is on the left hand side, and BR on the right hand side

  13. Characterization of buy and sell regions

  14. The case of

  15. The case of • Dai, Jiang, Li and Yi (2009), SIAM J. Control and Optimization • use a fixed point theorem to show the equivalence; • smoothness of free boundaries is indispensable and has been proved in Dai, Xu and Zhou (2008) in terms of cone property, where an auxiliary function 1/v is considered; • a technical assumption >0 is imposed to prove a critical inequality

  16. The case of

  17. Optimal investment strategy

  18. Continued: non-monotonicity

  19. Numerical method: penalty method Dai and Zhong, Journal of Computational Finance, to appear

  20. Penalty method (continued)

  21. Convergence analysis

  22. Numerical results

  23. Numerical results (continued)

  24. Trading strategy in the case of two stocks

  25. Extension and future work • Extension: • Dai, Jin and Liu (2008): with portfolio constraints • Dai, Li and Liu (2009): with market closure • Future work: • Theoretical analysis on multi-risky assets

  26. Conclusion • An equivalent variational inequality • Characterization of the optimal investment and consumption strategy • Efficient numerical algorithm: penalty method

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