# Hedging Derivatives on Two Assets with Model Risk

• Koichi Matsumoto
• Keita Shimizu
Original Research

## Abstract

This paper studies a static hedging problem of derivatives when the model risk exists. When the payoff of derivative depends on one asset, Matsumoto (Int J Financ Eng 4(4):1750042, 2017b) solves the problem. We extend his result to derivatives on two assets. Though the optimal solution is more complicated, we show that the problem can be solved numerically in an algebraic way. Further we give some simple numerical examples to show our method works well.

## Keywords

Hedging Derivatives Model risk

91G20 91G60

G13 D81

## Notes

### Acknowledgements

We wish to thank the participants of the Japanese Association of Financial Econometrics and Engineering (JAFEE), Winter Conference 2017, for helpful discussions and comments. We also thank the anonymous referees for the valuable comments. This research was partially supported by JSPS KAKENHI Grant Number JP 15K03544.

## References

1. Avellaneda, M., Levy, A., & Paras, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Applied Mathematical Finance, 2, 73–88.
2. Bertsimas, D., Kogan, L., & Lo, A. W. (2001). Hedging derivative securities and incomplete markets: An $$\epsilon$$ arbitrage approach. Operations Research, 49, 372–397.
3. Černý, A. (2004). Dynamic programming and mean-variance hedging in discrete time. Applied Mathematical Finance, 11, 1–25.
4. Gugushvili, S. (2003). Dynamic programming and mean-variance hedging in discrete time. Georgian Mathematical Journal, 2, 237–246.Google Scholar
5. Lyons, T. J. (1995). Uncertain volatility and the risk free synthesis of derivatives. Applied Mathematical Finance, 2, 117–133.
6. Matsumoto, K. (2009). Mean-variance hedging with uncertain trade execution. Applied Mathematical Finance, 16(3), 219–252.
7. Matsumoto, K. (2017). Mean–variance hedging with model risk. International Journal of Financial Engineering, 4(4), 1750042.
8. Melnikov, A. V., & Nechaev, M. L. (1999). On the mean-variance hedging problem. Theory of Probability and Its Applications, 43, 588–603.
9. Pinar, M. C. (2008). On robust quadratic hedging contingent claims in incomplete markets under ambiguous uncertainty. Paper presented at the first conference on advanced mathematical methods in finance.Google Scholar
10. Schäl, M. (1994). On quadratic cost criteria for option hedging. Mathematics of Operations Research, 19, 121–131.
11. Schweizer, M. (1995). Variance-optimal hedging in discrete time. Mathematics of Operations Research, 20, 1–32.
12. Tevzadze, R., & Uzunashvili, T. (2012). Robust mean-variance hedging and pricing of contingent claims in a one period model. International Journal of Theoretical and Applied Finance, 15(3), 1250024.