Mean Square Optimal Hedges Using Higher Order Moments

Implied volatility0.160.1550.150.1450.140.135the first m cumulants are matched at each step. We thenanalyzed the effect of these higher order moments in the underlyingasset process on the price of derivative securities.The relationship between the term structure of the volatilitysmile and smirk and higher order cumulants was illustratedthrough numerical experiments.REFERENCESImplied volatilityImplied volatilityImplied volatility0.130.1250.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12K/S 0Figure 2: 10 days expiration0.160.1550.150.1450.140.1350.130.1250.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12K/S 0Figure 3: 20 days expiration0.160.1550.150.1450.140.1350.130.1250.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12K/S 0Figure 4: 40 days expiration0.160.1550.150.1450.140.1350.130.1250.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12K/S 0Figure 5: 80 days expiration[1] F. Black and M. Scholes, “The Pricing of Options andCorporate Liabilities,” Journal of Political Economy,81:637–654, 1973.[2] J.C. Cox and S.A. Ross, “The valuation of options foralternative stochastic processes,” Journal of FinancialEconomics, 3:145–166, 1976.[3] J.C. Cox, S.A. Ross, and M. Rubinstein, “Option pricing:A simplified approach,” Journal of Financial Economics,7:229–263, 1979.[4] S.R. Das and R.K. Sundaram, ”Of Smiles and Smirks:A Term-Structure Perspective,” Research report, 1998.[5] E. Derman and I. Kani, “Implied Trinomial Treesof the Volatility Smile,” Goldman Sachs QuantitativeStrategies Research Notes, February, 1994.[6] E. Derman and I. Kani, “Riding on a Smile,” Risk,7:18–20, 1994.[7] D. Duffie and J. Pan, “An Overview of Value at Risk,”Journal of Derivatives, Spring:7-49, 1997. “Riding ona Smile,” Risk, 7:18–20, 1994.[8] D. Duffie and H.R. Richardson, “Mean-variance hedgingin continuous time.” Annals Appl. Probability, 1,1-15, 1991.[9] S. Fedotov and S. Mikhailov, “Option Pricing for IncompleteMarkets via Stochastic Optimization: TransactionCosts, Adaptive Control, and Forecast,” Int. J. ofTheoretical and Applied Finance, 4(1):179–195, 2001.[10] H. Follmer and M. Schweizer, “Hedging of contingentclaims under incomplete information”, In AppliedStochastic Analysis (M.H.A. Davis and R.J. Elliott,eds.), Stochastics Monographs 5, 389-414. Gordonand Breach, New York, 1991.[11] D.T. Gillespie, Markov process: an introduction forphysical scientists, Academic Press, 1992.[12] O. Hammarlid, “On minimizing risk in incompletemarkets option pricing models,” Int. J. Theoretical andApplied Finance, 1(2):227–233, 1998.[13] J. Hull, Options, Futures, and Other Derivative Securities,4th edition. Englewood Cliffs: Prentice-Hall,1999.[14] R. Jarrow and A. Rudd, Option Pricing. McGraw-HillProfessional Book Group, 1983.