Stochastic Systems

Portfolio rebalancing error with jumps and mean reversion in asset prices

Paul Glasserman and Xingbo Xu

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Abstract

We analyze the error between a discretely rebalanced portfolio and its continuously rebalanced counterpart in the presence of jumps or mean-reversion in the underlying asset dynamics. With discrete rebalancing, the portfolio’s composition is restored to a set of fixed target weights at discrete intervals; with continuous rebalancing, the target weights are maintained at all times. We examine the difference between the two portfolios as the number of discrete rebalancing dates increases. With either mean reversion or jumps, we derive the limiting variance of the relative error between the two portfolios. With mean reversion and no jumps, we show that the scaled limiting error is asymptotically normal and independent of the level of the continuously rebalanced portfolio. With jumps, the scaled relative error converges in distribution to the sum of a normal random variable and a compound Poisson random variable. For both the mean-reverting and jump-diffusion cases, we derive “volatility adjustments” to improve the approximation of the discretely rebalanced portfolio by the continuously rebalanced portfolio, based on on the limiting covariance between the relative rebalancing error and the level of the continuously rebalanced portfolio. These results are based on strong approximation results for jump-diffusion processes.

Article information

Source
Stoch. Syst., Volume 1, Number 1 (2011), 109-145.

Dates
First available in Project Euclid: 24 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.ssy/1393252125

Digital Object Identifier
doi:10.1214/10-SSY015

Mathematical Reviews number (MathSciNet)
MR2948919

Zentralblatt MATH identifier
1291.91193

Subjects
Primary: 91G10: Portfolio theory
Secondary: 60F05: Central limit and other weak theorems 60H35: Computational methods for stochastic equations [See also 65C30]

Keywords
Jump-diffusion processes portfolio analysis strong approximation Ito-Taylor expansion

Citation

Glasserman, Paul; Xu, Xingbo. Portfolio rebalancing error with jumps and mean reversion in asset prices. Stoch. Syst. 1 (2011), no. 1, 109--145. doi:10.1214/10-SSY015. https://projecteuclid.org/euclid.ssy/1393252125


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References

  • [1] Avellaneda, M. and Zhang, S. (2010). Path-dependence of Leveraged ETF returns. SIAM Journal on Financial Mathematics 1 586–603.
  • [2] Basel Committee on Banking Supervision (2007). Guidelines for computing capital for incremental default risk in the trading book (BCBS 134). Bank for International Settlements, Basel, Switzerland. Available at www.bis.org.
  • [3] Basel Committee on Banking Supervision (2009). Guidelines for computing capital for incremental risk in the trading book (BCBS 149). Bank for International Settlements, Basel, Switzerland. Available at www.bis.org.
  • [4] Bertsimas, D., Kogan, L. and Lo, A. W. (2000). When is time continuous? Journal of Financial Economics 55 173–204.
  • [5] Boyle, P. P. and Emanuel, D. (1980). Discretely adjusted option hedges. Journal of Financial Economics 8 259–282.
  • [6] Bruti-Liberati, N. and Platen, E. (2005). On the Strong Approximation of Jump-Diffusion Processes. Technical Report, Quantitative Finance Research Papers 157, University of Technology, Sydney, Australia.
  • [7] Bruti-Liberati, N. and Platen, E. (2007). Approximation of jump diffusions in finance and economics. Computational Economics 29 283–312.
  • [8] Cont, R. and Tankov, P. (2003). Financial Modelling with Jump Processes. Chapman & Hall/CRC.
  • [9] Das, S. R. and Uppal, R. (2004). Systemic risk and international portfolio choice. The Journal of Finance 59 2809–2834.
  • [10] Dempster, M. A. H., Pflug, G. C. and Mitra, G. (2009). Quantitative Fund Management. Chapman & Hall/CRC Press, Boca Raton, Florida.
  • [11] Duffie, D. and Sun, T.-S. (1990). Transactions costs and portfolio choice in a discrete-continuous-time setting. Journal of Economic Dynamics and Control 14 35–51.
  • [12] Glasserman, P. (2010). Risk Horizon and Rebalancing Horizon in Portfolio Risk Measurement. Mathematical Finance To appear.
  • [13] Glasserman, P. and Xu, X. (2010). Importance Sampling for Tail Risk in Discretely Rebalanced Portfolio. Proceedings of the Winter Simulation Conference 2655–2665.
  • [14] Guasoni, P., Huberman, G. and Wang, Z. (2010). Performance maximization of actively managed funds. Journal of Financial Economics to appear.
  • [15] Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Annals of Probability 26 267–307.
  • [16] Jessen, C. (2009). Constant Proportion Portfolio Insurance: Discrete-time Trading and Gap Risk Coverage. Working Paper Series, University of Copenhagen.
  • [17] Kallsen, J. (2000). Optimal portfolios for exponential Lévy processes. Mathematical methods of operations research 51 357–374.
  • [18] Kim, T. S. and Omberg, E. (1996). Dynamic nonmyopic portfolio behavior. Review of Financial Studies 9 141.
  • [19] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin.
  • [20] Kurtz, T. and Protter, P. (1991). Wong-Zakai corrections, random evolutions and numerical schemes for SDEs. Stochastic Analysis, Academic Press, Boston,MA 331–346.
  • [21] Leland, H. E. (1985). Option pricing and replication with transactions costs. Journal of Finance 40 1283–1301.
  • [22] Morton, A. J. and Pliska, S. R. (1995). Optimal portfolio management with fixed transaction costs. Mathematical Finance 5 337–356.
  • [23] Platen, E. (1982). A Generalized Taylor Formula for Solutions of Stochastic Equations. Sankhya: The Indian Journal of Statistics 44, No. 2 163–172.
  • [24] Sepp, A. (2009). An Approximate Distribution of Delta-Hedging Errors in a Jump-Diffusion Model with Discrete Trading and Transaction Costs. Working Paper Series, Merrill Lynch & Co.
  • [25] Studer, M. (2001). Stochastic Taylor Expansions and Saddlepoint Approximations for Risk Management. Dissertation No. 14242, Department of Mathematics, ETH Zurich, Available at e-collection.ethbib.ethz.ch.
  • [26] Tankov, P. and Voltchkova, E. (2009). Asymptotic analysis of hedging errors in models with jumps. Stochastic Processes and their Applications 119 2004–2027.