# A Framework for Dynamic Hedging under Convex Risk Measures

## Abstract

We consider the problem of minimizing the risk of a financial position (hedging) in an incomplete market. It is well known that the industry standard for risk measure, the Value-at-Risk, does not take into account the natural idea that risk should be minimized through diversification. This observation led to the recent theory of coherent and convex risk measures. But, as a theory on bounded financial positions, it is not ideally suited for the problem of hedging because simple strategies such as buy-hold strategies may not be bounded. Therefore, we propose as an alternative to use convex risk measures defined as functionals on *L* ^{2} (or by simple extension *L* ^{ p }, *p* > 1). This framework is more suitable for optimal hedging with *L* ^{2}-valued financial markets. A dual representation is given for this minimum risk or market adjusted risk when the risk measure is real valued. In the general case, we introduce constrained hedging and prove that the market adjusted risk is still a *L* ^{2} convex risk measure and the existence of the optimal hedge. We illustrate the practical advantage in the shortfall risk measure by showing how minimizing risk in this framework can lead to a HJB equation and we give an example of computation in a stochastic volatility model with the shortfall risk measure

## Keywords

Hedging convex risk measures shortfall risk.## Preview

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## References

- 1.P. Artzner, F. Delbaen, J.M. Eber, and D Heath,
*Coherent measures of risk*, Math. Finance, 9 (3) (1999), 203**–**228.MathSciNetzbMATHCrossRefGoogle Scholar - 2.P. Barrieu and N. El Karoui,
*Optimal derivatives design under dynamic risk measures*, Article in Mathematics of Finance, Contemporary Mathematics (A.M.S. Proceedings), (2004), 13**–**26.Google Scholar - 3.P. Barrieu and N. El Karoui,
*Inf-convolution of risk measures and optimal risk transfer*, Finance and Stochastics, 9 (2005), 269**–**298.MathSciNetzbMATHCrossRefGoogle Scholar - 4.S. Biagini and M. Fritelli,
*Utility maximization in incomplete markets for unbounded processes*, Finance and Stochastics, 9 (2005), 493**–**517.MathSciNetzbMATHCrossRefGoogle Scholar - 5.S. Biagini and M. Frittelli,
*On the extension of the Namioka-Klee theorem and on the Fatou property for risk measures*, in:*Optimality and Risk***–***Modern Trends in Mathematical Finance*, Springer Berlin, (2009), 1**–**28.Google Scholar - 6.P. Cheridito and T. Li,
*Risk measures on Orlicz hearts*, Mathematical Finance, 19 (2) (2009), 189**–**214.MathSciNetzbMATHCrossRefGoogle Scholar - 7.Delbaen andW. Schachermayer,
*The Mathematics of Arbitrage*, Springer Finance, 2006.Google Scholar - 8.F. Delbaen, P. Monat, W. Schachermayer, M. Schweizer, and C. Stricker,
*Weighted norm inequalities and closedness of a space of stochastic integrals*, Finance and Stochastics, 1 (1997), 181**–**227.zbMATHCrossRefGoogle Scholar - 9.D. Filipovic and G. Svindland,
*Convex risk measures on Lp*, Working Paper, 2005.Google Scholar - 10.D. Filipovic and G. Svindland,
*Convex risk measures beyond bounded risks, or the canonical model space for law-invariant convex risk measures is L*1, short version to appear in Mathematical Finance, 2008.Google Scholar - 11.H. F
**ö**llmer and A. Schied,*Stochastic Finance, An Introduction in Discrete Time*, Walter de Gruyter, 2002.Google Scholar - 12.K. Giesecke and S. Weber,
*Measuring the risk of large losses*, Journal of Investment Management, 6 (4) (2008), 1**–**15.Google Scholar - 13.M. Harrison and S. Pliska,
*Martingales and stochastic integrals in the theory of continuous trading*, Stochastic Processes and Their Applications, 11 (1981), 215**–**260.MathSciNetzbMATHCrossRefGoogle Scholar - 14.A. Ilhan, M. Jonsson, and R. Sircar,
*Optimal static-dynamic hedge for exotic options under convex risk measures*, Stochastic Processes and their Applications, (2009), in press.Google Scholar - 15.E. Jouini, W. Schachermayer, and N. Touzi,
*Law invariant risk measures have the Fatou property*, Advances in Mathematical Economics, 9 (2006), 49**–**72.MathSciNetCrossRefGoogle Scholar - 16.S. Kl
**ö**ppel and M. Schweizer,*Dynamic indifference valuation via convex risk measures*, Mathematical Finance, 17 (4) (2007), 599**–**627.MathSciNetzbMATHCrossRefGoogle Scholar - 17.M. Musiela and P. Rutkowski,
*Martingale Methods in Financial Modelling*, Springer, 1998.Google Scholar - 18.P. Protter,
*Stochastic Integration and Differential Equations*, second edition, Springer-Verlag, 2005.Google Scholar - 19.B. Rudloff,
*Hedging in incomplete markets and testing compound hypotheses via convex duality*, Thesis Dissertation, 2006.Google Scholar - 20.A. Ruszczynski and A. Shapiro,
*Optimization of convex risk functions*, Mathematics of Operations Research, 31 (3) (2006), 433**–**452.MathSciNetzbMATHCrossRefGoogle Scholar - 21.M. Schweizer,
*A guided tour through quadratic hedging approaches*, In:*Option Pricing, Interest Rates and Risk Management*, E. Jouini, J. Cvitanic, and M. Musiela, Eds., Cambridge University Press, (2001), 538**–**574.Google Scholar - 22.A. Toussaint,
*Hedging with L*2*convex risk measures*, Dissertation, Princeton University, 2007.Google Scholar