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International Assets Allocation with Risk Management via Multi-Stage Stochastic Programming

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Abstract

In this paper, we develop a multi-stage stochastic programming model for dynamic international portfolio risk management with options in an integrated view. Upon scenario trees, the model can automatically compute the optimal hedging strategies, which provides rolling and dynamic decisions for how much option positions should be established and how much should be liquidated, while simultaneously allocating the corresponding underlying assets. Extensive numerical analyses strongly verify the effectiveness of the model, especially in market downturns, and support the computational feasibility and performance of the model.

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Notes

  1. Rasmussen and Clausen (2007) point out that arbitrage opportunity may arise when using scenario reduction in portfolio optimization problems. Moreover, Geyer et al. (2011) also show that even if the scenario trees generated by scenario reduction are arbitrage-free, the solutions to the approximate optimization problem represented by the reduced tree are highly variable. By contrast, they both find that when using a simple problem with a known analytic solution, moment matching method accompanied by arbitrage-free check can exactly replicates this solution.

  2. The tests aim to verify that the scenario does not contain a feasible costless, self-financing decision that can generate a riskless profit (i.e., a non-positive initial endowment yield non-negative payoffs for all scenarios, or a strictly positive profit at least for one scenario).

  3. It should be noted that other option pricing methods which rely on specific distribution assumptions for the stochastic process of the underlying asset are unrealistic and are not met in the empirical tests in our paper. The scenario sets generated by moment-matching method capture skewness and excess kurtosis characteristics of the underlying asset that do not conform to the assumptions on which popular option pricing methods are based.

  4. Source: Closing price for S&P500 options are obtained from http://www.cmegroup.com, closing price for Nikke225 options are obtained from http://www.sgx.com, closing price for FTSE100 options are obtained from https://globalderivatives.nyx.com/stock-indices/nyse-liffe, closing price for DAX30 options are obtained from http://www.eurexchange.com/exchange-en/trading/.

  5. We introduce the upside potential and downside risk ratio (UP ratio) proposed by Sortino and Van der Meer (1991), which is a more appropriate measure for risk-adjusted performance. This ratio contracts the average excess return over some target with a measure of shortfall from the same benchmark, as suggested by Sortina et al. (1999). Here we use the log return of MSCI Stock Index as the benchmark. Let \(r_t \) be the realized return of a portfolio in month \(t=1,2,\ldots ,T\) of the simulation period 10/2007–12/2009. Let \(\rho _t \) be the return of the benchmark (risk-free asset) at the same period. Then the UP ratio can be measured as follows:

    $$\begin{aligned} \text{ UP } \;\text{ ratio }=\frac{\text{1 }}{\text{ T }}\sum _{t=1}^T {\frac{\max \left( {0,r_t -\rho _t } \right) }{\sqrt{\frac{\text{1 }}{\text{ T }}\sum _{t=1}^T {\left[ {\max \left( {0,\rho _t -r_t } \right) } \right] ^{2}}}}.} \end{aligned}$$
    Fig. 3
    figure 3

    Ex-post realized cumulative wealth of optimal portfolio of alternative hedging strategies with target return \(=\) 2 %

    Full size image

  6. The last two measures serve as assessment metrics for dynamic investment process based on information about the compositions of the evaluated portfolios, which are discussed extensively in Grinblatt and Titman (1993). The ESM reflects the realized excess returns of the actively managed portfolio in comparison to its unconditional expected return. The PCM quantifies the return resulting from the portfolio revision at the each decision stage. The intuition is that an effective strategy adjusts the portfolio structure in favor of assets that achieve higher returns and away from assets with lower returns. The \(ESM_t \) and \(PCM_t \) measures at each month \(t=1,2,\ldots ,T\) of the simulation period are computed as follows:

    $$\begin{aligned} ESM_t =\sum _{\kappa \in K} {\varpi _{\kappa ,t} \left( {\vartheta _{\kappa ,t} -\overline{\vartheta _t } } \right) } ,\quad PCM_t =\sum _{\kappa \in K} {\vartheta _{\kappa ,t} \left( {\varpi _{\kappa ,t} -\varpi _{\kappa ,t-1} } \right) } , \end{aligned}$$

    where, \(K\) is the set of all available investment instruments (stock indices and options). \(\varpi _{\kappa ,t} \) denotes the proportion of funds in instrument \(\kappa \in K\)of the portfolio at time \(t\), with respect to the total value of the portfolio, \(\vartheta _{\kappa ,t} \) represents the market return of instrument \(\kappa \in K\) of the portfolio at time \(t, \overline{\vartheta _t } =\frac{1}{T}\sum _{t=1}^T {\vartheta _{\kappa ,t} } \), is an estimate of the unconditional expected return of the investment instrument.

References

  • Abdelaziz, F. B., Aouni, B., & Fayedh, R. E. (2007). Multi-objective stochastic programming for portfolio selection. European Journal of Operational Research, 177(3), 1811–1823.

    Article  Google Scholar 

  • Aliprantis, C., Brown, D., & Werner, J. (2000). Minimum-cost portfolio insurance. Journal of Economic Dynamics and Control, 24(11–12), 1703–1719.

    Article  Google Scholar 

  • Blomvall, J., & Lindberg, P. (2003). Back-testing the performance of an actively managed option portfolio at the Swedish stock market, 1990–1999. Journal of Economic Dynamics and Control, 27(6), 1099–1112.

    Article  Google Scholar 

  • Brennan, M., & Cao, H. (1996). Information, trade, and derivative securities. Review of Financial Studies, 9(1), 163–208.

    Article  Google Scholar 

  • Dupacová, J., Gröwe-Kuska, N., & Römisch, W. (2003). Scenario reduction in stochastic programming: An approach using probability metrics. Mathematical Programming, Series A, 95, 493–511.

    Article  Google Scholar 

  • Ferstl, R., & Weissensteiner, A. (2010). Cash management using multi-stage stochastic programming. Quantitative Finance, 10(2), 209–219.

    Article  Google Scholar 

  • Ferstl, R., & Weissensteiner, A. (2011). Asset-liability management under time-varying investment opportunities. Journal of Banking and Finance, 35(1), 182–192.

    Article  Google Scholar 

  • Follmer, H., & Sondermann, D. (1986). Hedging of non-redundant contingent claims. In W. Hildenbrand & A. Mas-Colell (Eds.), Contributions to mathematical economics (pp. 205–223). Amsterdam: North-Holland.

    Google Scholar 

  • Gaivoronski, A. A., Krylov, S., & Van der Wijst, N. (2005). Optimal portfolio selection and dynamic benchmark traking. European Journal of Operational Research, 163(1), 115–131.

    Article  Google Scholar 

  • Gao, P. W. (2009). Options strategies with the risk adjustment. European Journal of Operation Research, 192, 975–980.

    Article  Google Scholar 

  • Geyer, A., Hanke, M., & Weissensteiner, A. (2011). Scenario tree generation and multi-asset financial optimization problems. Working paper. Available at SSRN. http://ssrn.com/abstract=1857411 or http://dx.doi.org/10.2139/ssrn.1857411.

  • Gondzio, J., Kouwenberg, R., & Vorst, T. (2003). Hedging options under transaction costs and stochastic volatility. Journal of Economic Dynamics and Control, 27(6), 1045–1068.

    Article  Google Scholar 

  • Grinold, R. C. (1999). Mean-variance and scenario-based approaches to portfolio selection. The Journal of Portfolio Management, 25(2), 10–22.

    Article  Google Scholar 

  • Grinblatt, M., & Titman, S. (1993). Performance measurement without benchmarks: An examination of mutual fund returns. Journal of Business, 66(1), 47–68.

    Article  Google Scholar 

  • Harrison, J. M., & Pliska, S. R. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications, 11, 215–260.

    Article  Google Scholar 

  • Haugh, M., & Lo, A. (2001). Asset allocation and derivatives. Quantitative Finance, 1(1), 45–72.

    Article  Google Scholar 

  • Heitsch, H., & Römisch, W. (2003). Scenario reduction algorithms in stochastic programming. Computational Optimization and Applications, 24, 187–206.

    Article  Google Scholar 

  • Hochreiter, R., & Pflug, G. Ch. (2007). Financial scenario generation for stochastic multi-stage decision processes as facility location problems. Annals of Operations Research, 152(1), 257–272.

  • Horasanh, M. (2008). Hedging strategy for a portfolio of options and stocks with linear programming. Applied Mathematics and Computation, 199, 804–810.

    Article  Google Scholar 

  • Høyland, K., Kaut, M., & Stein, W. (2003). A heuristic for moment-matching scenario generation. Computational Optimization and Applications, 24(2–3), 169–185.

    Article  Google Scholar 

  • Høyland, K., & Wallance, S. W. (2001). Generating scenario trees for multistage decision problems. Management Science, 47(2), 295–307.

    Article  Google Scholar 

  • Huang, D. S., Zhu, S. S., Fabozzi, F. J., & Fukushima, M. (2008). Portfolio selection with uncertain exit time: A robust CVaR approach. Journal of Economic Dynamics and Control, 32(2), 594–623.

    Article  Google Scholar 

  • Huang, D. S., Zhu, S. S., Fabozzi, F. J., & Fukushima, M. (2010). Portfolio selection under distributional uncertainty: A relative robust CVaR approach. European Journal of Operational Research, 203(1), 185–194.

    Article  Google Scholar 

  • Klaassen, P. (2002). Comment on “Generating scenario trees for multistage decision problems”. Management Science, 48(11), 1512–1516.

    Article  Google Scholar 

  • Korn, R., & Trautmann, S. (1999). Optimal control of option portfolios and applications. OR Spektrum, 21, 123–146.

    Article  Google Scholar 

  • Liu, J., & Pan, J. (2003). Dynamics derivative strategies. Journal of Financial Economics, 69(3), 401–430.

    Article  Google Scholar 

  • Merton, R., Scholes, M., & Gladstone, M. (1978). The return and risk of alternative call option portfolio investment strategies. Journal of Business, 51, 183–242.

    Article  Google Scholar 

  • Muck, M. (2010). Trading strategies with partial access to the derivatives market. Journal of Banking and Finance, 34(6), 1288–1298.

    Article  Google Scholar 

  • Neuberger, A., & Hodges, S. (2002). How large are the benefits from using options? Journal of Financial and Quantitative Analysis, 37(2), 202–220.

    Article  Google Scholar 

  • Papahristodoulou, C. (2004). Option strategies with linear programming. European Journal of Operation Research, 157, 246–256.

    Article  Google Scholar 

  • Pflug, G. C. (2001). Optimal scenario tree generation for multiperiod financial planning. Mathematical Programming, 89(2), 251–271.

    Article  Google Scholar 

  • Rasmussen, K. M., & Clausen, J. (2007). Mortgage loan portfolio optimization using multi-stage stochastic programming. Journal of Economic Dynamics and Control, 31(3), 742–766.

    Article  Google Scholar 

  • Scheuenstuhl, G., & Zagst, R. (2008). Integrated portfolio management with options. European Journal of Operational Research, 185(3), 1477–1500.

    Article  Google Scholar 

  • Sortino, F. A., & Van der Meer, R. (1991). Downside risk: Capturing what’s at stake in investment situations. Journal of Portfolio Management, 17(4), 27–31.

    Article  Google Scholar 

  • Sinha, P., & Johar, A. (2010). Hedging Greeks for a portfolio of options using linear and quadratic programming. The Journal of Prediction Markets, 4(1), 17–26.

  • Sortina, F., Van der Meer, R., & Plantinga, A. (1999). The Dutch triangle—A framework to measure upside potential relative to downside risk. Journal of Portfolio Management, 26(1), 50–58.

    Article  Google Scholar 

  • Topaloglou, N., Vladimirou, H., & Zenios, S. A. (2002). CVaR models with selective hedging for international asset allocation. Journal of Banking and Finance, 26(7), 1535–1561.

    Article  Google Scholar 

  • Topaloglou, N., Vladimirou, H., & Zenios, S. A. (2008a). A dynamic stochastic programming model for international portfolio management. European Journal of Operational Research, 185, 1501–1524.

    Article  Google Scholar 

  • Topaloglou, N., Vladimirou, H., & Zenios, S. A. (2008b). Pricing options on scenario trees. Journal of Banking and Finance, 32(2), 283–298.

    Article  Google Scholar 

  • Topaloglou, N., Vladimirou, H., & Zenios, S. A. (2011). Optimizing international portfolios with options and forwards. Journal of Banking and Finance, 35(12), 3188–3201.

    Article  Google Scholar 

  • Zhao, Y., & Ziemba, W. (2008). Calculating risk neutral probabilities and optimal portfolio policies in a dynamic investment model with downside risk control. European Journal of Operational Research, 185(3), 1525–1540.

    Article  Google Scholar 

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Acknowledgments

The authors would like to thank the National Natural Science Foundation of China for financially supporting under Contract No. 70831001 and 71173008.

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  1. School of Economics and Management, Beihang University, A1124, New Main Building, 37 Xueyuan Road, Haidian District, Beijing, 100191, China

    Libo Yin

  2. School of Economics and Management, Beihang University, A1101, New Main Building, 37 Xueyuan Road, Haidian District, Beijing, 100191, China

    Liyan Han

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Yin, L., Han, L. International Assets Allocation with Risk Management via Multi-Stage Stochastic Programming. Comput Econ 55, 383–405 (2020). https://doi.org/10.1007/s10614-013-9365-z

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