Partial Hedging Using Malliavin Calculus

Affiliation(s)

Department of Management Sciences, Rider University, Lawrenceville, USA.

Department of Statistics and Operations Research, New York University, New York, USA.

Department of Management Sciences, Rider University, Lawrenceville, USA.

Department of Statistics and Operations Research, New York University, New York, USA.

ABSTRACT

Under the constraint that the initial capital is not enough for a perfect hedge, the problem of deriving an optimal partial hedging portfolio so as to minimize the shortfall risk is worked out by solving two connected subproblems sequentially. One subproblem is to find the optimal terminal wealth that minimizes the shortfall risk. The shortfall risk is quantified by a general convex risk measure to accommodate different levels of risk tolerance. A convex duality approach is used to obtain an explicit formula for the optimal terminal wealth. The second subproblem is to derive the explicit expression for the admissible replicating portfolio that generates the optimal terminal wealth. We show by examples that to solve the second subproblem, the Malliavin calculus approach outperforms the traditional delta-hedging approach even for the simplest claim. Explicit worked-out examples include a European call option and a standard lookback put option.

Under the constraint that the initial capital is not enough for a perfect hedge, the problem of deriving an optimal partial hedging portfolio so as to minimize the shortfall risk is worked out by solving two connected subproblems sequentially. One subproblem is to find the optimal terminal wealth that minimizes the shortfall risk. The shortfall risk is quantified by a general convex risk measure to accommodate different levels of risk tolerance. A convex duality approach is used to obtain an explicit formula for the optimal terminal wealth. The second subproblem is to derive the explicit expression for the admissible replicating portfolio that generates the optimal terminal wealth. We show by examples that to solve the second subproblem, the Malliavin calculus approach outperforms the traditional delta-hedging approach even for the simplest claim. Explicit worked-out examples include a European call option and a standard lookback put option.

Cite this paper

L. Nygren and P. Lakner, "Partial Hedging Using Malliavin Calculus,"*Journal of Mathematical Finance*, Vol. 2 No. 3, 2012, pp. 203-213. doi: 10.4236/jmf.2012.23023.

L. Nygren and P. Lakner, "Partial Hedging Using Malliavin Calculus,"

References

[1] J. Cvitanic and I. Karatzas, “On Dynamic Measures of Risk,” Finance and Stochastics, Vol. 3, No. 4, 1999, pp. 451-482. doi:10.1007/s007800050071

[2] J. Cvitanic, “Minimizing Expected Loss of Hedging in Incomplete and Constrained Markets,” SIAM Journal on Control and Optimization, Vol. 38 No. 4, 2000, pp. 1050- 1066. doi:10.1137/S036301299834185X

[3] H. F?llmer and P. Leukert, “Efficient Hedging: Cost Versus Shortfall Risk,” Finance and Stochastics, Vol. 4, No. 2, 2000, pp. 117-146. doi:10.1007/s007800050008

[4] H.-P. Bermin, “Hedging Options: The Malliavin Calculus Approach Versus the -Hedging Approach,” Mathematical Finance, Vol. 13, No. 1, 2003, pp. 73-84. doi:10.1111/1467-9965.t01-1-00006

[5] D. Ocone and I. Karatzas, “A Generalized Clark Repre- sentation Formula with Applications to Optimal Portfolios,” Stochastics and Stochastic Reports, Vol. 34, No. 3- 4, 1991, pp. 187-220.

[6] P. Lakner, “Optimal Trading Strategy for an Investor: The Case of Partial Information,” Stochastic Processes and Their Applications, Vol. 76, No. 1, 1998, pp. 77-97. doi:10.1016/S0304-4149(98)00032-5

[7] H.-P. Bermin, “Hedging Lookback and Partial Lookback Options Using Malliavin Calculus,” Applied Mathematical Finance, Vol. 7, No. 2, 2000, pp. 75-100. doi:10.1080/13504860010014052

[8] H.-P. Bermin, “A General Approach to Hedging Options: Applications to Barrier and Partial Barrier Options,” Mathematical Finance, Vol. 12, No. 3, 2002, pp. 199-218. doi:10.1111/1467-9965.02007

[9] P. Lakner and L. M. Nygren, “Portfolio Optimization with Downside Constraints,” Mathematical Finance, Vol. 16, No. 2, 2006, pp. 283-299. doi:10.1111/j.1467-9965.2006.00272.x

[10] F. Black and M. Scholes, “The Pricing of Options on Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-659. doi:10.1086/260062

[11] V. Bawa, “Optimal Rules for Ordering Uncertain Prospects,” Journal of Financial Economics, Vol. 2, No. 1, 1975, pp. 95-121. doi:10.1016/0304-405X(75)90025-2

[12] I. Karatzas and S. E. Steven, “Methods of Mathematical Finance,” Springer-Verlag, New York, 1998.

[13] D. Nualart, “The Malliavin Calculus and Related Topics,” Springer-Verlag, Berlin Heidelberg, New York, 1995.

[14] B. K. Oksendal, “Stochastic Differential Equations: An Introduction with Applications,” 5th Edition, Springer- Verlag, Berlin Heidelberg, New York, 1998.

[15] M. Musiela and M. Rutkowski, “Martingale Methods in Financial Modelling,” Springer-Verlag, Berlin Heidelberg, 1997.

[16] R. M. Dudley, “Wiener Functionals as It? Integrals,” Annals of Probability, Vol. 5, No. 1, 1977, pp. 140-141. doi:10.1214/aop/1176995898

[17] I. Karatzas and S. E. Shreve, “Brownian Motion and Stochastic Calculus,” Springer-Verlag, New York, 1991. doi:10.1007/978-1-4612-0949-2

[1] J. Cvitanic and I. Karatzas, “On Dynamic Measures of Risk,” Finance and Stochastics, Vol. 3, No. 4, 1999, pp. 451-482. doi:10.1007/s007800050071

[2] J. Cvitanic, “Minimizing Expected Loss of Hedging in Incomplete and Constrained Markets,” SIAM Journal on Control and Optimization, Vol. 38 No. 4, 2000, pp. 1050- 1066. doi:10.1137/S036301299834185X

[3] H. F?llmer and P. Leukert, “Efficient Hedging: Cost Versus Shortfall Risk,” Finance and Stochastics, Vol. 4, No. 2, 2000, pp. 117-146. doi:10.1007/s007800050008

[4] H.-P. Bermin, “Hedging Options: The Malliavin Calculus Approach Versus the -Hedging Approach,” Mathematical Finance, Vol. 13, No. 1, 2003, pp. 73-84. doi:10.1111/1467-9965.t01-1-00006

[5] D. Ocone and I. Karatzas, “A Generalized Clark Repre- sentation Formula with Applications to Optimal Portfolios,” Stochastics and Stochastic Reports, Vol. 34, No. 3- 4, 1991, pp. 187-220.

[6] P. Lakner, “Optimal Trading Strategy for an Investor: The Case of Partial Information,” Stochastic Processes and Their Applications, Vol. 76, No. 1, 1998, pp. 77-97. doi:10.1016/S0304-4149(98)00032-5

[7] H.-P. Bermin, “Hedging Lookback and Partial Lookback Options Using Malliavin Calculus,” Applied Mathematical Finance, Vol. 7, No. 2, 2000, pp. 75-100. doi:10.1080/13504860010014052

[8] H.-P. Bermin, “A General Approach to Hedging Options: Applications to Barrier and Partial Barrier Options,” Mathematical Finance, Vol. 12, No. 3, 2002, pp. 199-218. doi:10.1111/1467-9965.02007

[9] P. Lakner and L. M. Nygren, “Portfolio Optimization with Downside Constraints,” Mathematical Finance, Vol. 16, No. 2, 2006, pp. 283-299. doi:10.1111/j.1467-9965.2006.00272.x

[10] F. Black and M. Scholes, “The Pricing of Options on Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-659. doi:10.1086/260062

[11] V. Bawa, “Optimal Rules for Ordering Uncertain Prospects,” Journal of Financial Economics, Vol. 2, No. 1, 1975, pp. 95-121. doi:10.1016/0304-405X(75)90025-2

[12] I. Karatzas and S. E. Steven, “Methods of Mathematical Finance,” Springer-Verlag, New York, 1998.

[13] D. Nualart, “The Malliavin Calculus and Related Topics,” Springer-Verlag, Berlin Heidelberg, New York, 1995.

[14] B. K. Oksendal, “Stochastic Differential Equations: An Introduction with Applications,” 5th Edition, Springer- Verlag, Berlin Heidelberg, New York, 1998.

[15] M. Musiela and M. Rutkowski, “Martingale Methods in Financial Modelling,” Springer-Verlag, Berlin Heidelberg, 1997.

[16] R. M. Dudley, “Wiener Functionals as It? Integrals,” Annals of Probability, Vol. 5, No. 1, 1977, pp. 140-141. doi:10.1214/aop/1176995898

[17] I. Karatzas and S. E. Shreve, “Brownian Motion and Stochastic Calculus,” Springer-Verlag, New York, 1991. doi:10.1007/978-1-4612-0949-2