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Options strategies for international portfolios with overall risk management via multi-stage stochastic programming

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Abstract

This paper proposes a multi-stage stochastic programming model to explore optimal options strategies for international portfolios with overall risk management on Greek letters, extending existing Greek-based analysis to dynamic and nondeterministic programming under uncertainty. The contribution to the existing literature are overall control on the time-varying Greek letters, state-contingent decision dynamics in consistent with the projected outcomes of the changing information, and a holistic view for optimizing the portfolio of assets and options. Empirical results show the model possesses considerable benefits in terms of larger profit margins, greater stability of returns and higher hedging efficiency compared to traditional methods.

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Notes

  1. Readers can refer to Chap. 17 of “Options, futures and other derivatives” (7th ed.) written by John C. Hull.

  2. There are three reasons for us to choose this period for simulation. Firstly, it contains both downturns and upturns in underlying market. Secondly, the subprime crisis swept across the United States, the European Union and Japan along with other major financial markets since August 2007, providing a good platform to test whether our model can escape or endure such severe market shocks. Thirdly, since the problem is considered from the view of a Chinese investor, the effect of series reforms of deepening the international investment and exchange rate mechanism appears gradually since October 2007.

  3. Hedging efficiency indicator is designed to measure the effect of hedging. We give the following two indicators in terms of minimum risk case and maximum utility case, respectively. In the framework of minimum risk case, we apply the formula proposed by Ederington (1979), defined as follows:

    where Var(R s), Var(R f) present the variance of the proportional change rate of exchange rates and the variance of the log returns of currency options, h represents the optimal risk-minimizing hedge ratio. The higher the indicator HE1, the better the hedging effect.

    We also calculate a second indicator for the hedging efficiency, defined as follows:

    $$\mathit{HE}_2=\bigl[ 1-E\bigl(R^p\bigr)/E\bigl(R^s\bigr)\bigr]\big/ \big[1-\mathit{Var}\bigl(R^p\bigr)\bigr]\big/\mathit{Var}\bigl(R^s\bigr), \qquad E\bigl(R^p\bigr)=E\bigl(R^s\bigr)-hE\bigl(R^f\bigr)$$

    where E(R s), E(R f), present the expectation of the proportional change rate of exchange rates and the expectation of the log returns of currency options, h represents the optimal mean-risk hedge ratio. Lower value of the indicator HE 2 indicates better hedging efficiency.

  4. Traders usually ensure that their portfolios are Delta-neutral firstly. Whenever the opportunity arises, they improve Gamma and Vega. Gamma is the rate of change of Delta with respect to the price of the underlying asset, which states the stability of Delta. If Gamma is large, Delta is highly sensitive to the stock price, and then it will be quite risky to leave a Delta-neutral portfolio unchanged. Vega is the rate of change of the value of a derivatives portfolio with respect to volatility. As to Theta and Rho, since they are less important compared to Delta, Gamma and Vega, we take Theta as an example to show the effect of risk adjustment.

  5. The upside potential and downside risk ratio (UP_ratio) is 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 Sortino et al. (1999).

    We use the risk-free rate of three-month US T-bills as the benchmark. Let r t be the realized return of a portfolio in month t=1,2,…,T of the simulation period 10/2007–12/2009. Let ρ t be the return of the benchmark (risk-free asset) at the same period. Then the UP ratio can be measured as follows:

    $$\mathit{UP}\_\mathit{ratio}=\frac{1}{T}\sum^T_{t=1}\max[0,r_t-\rho_t]\Big/ \sqrt{\frac{1}{T}\sum^T_{t=1}\bigl( \max[0,\rho_t-r_t] \bigr)^2 } . $$

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Acknowledgements

The authors would like to thank the National Natural Science Foundation of China for financial support with projects No. 70831001 and 71173008.

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  1. School of Economics and Management, Beihang University, Beijing, 100191, China

    Libo Yin & Liyan Han

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  1. Libo Yin
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Correspondence to Libo Yin.

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Yin, L., Han, L. Options strategies for international portfolios with overall risk management via multi-stage stochastic programming. Ann Oper Res 206, 557–576 (2013). https://doi.org/10.1007/s10479-013-1375-7

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