On quadratic hedging in continuous time

Abstract. We review the main results in the theory of quadratic hedging in a general incomplete model of continuous trading with semimartingale price process. The objective is to hedge contingent claims by using portfolio strategies. We describe two types of criteria: the so-called (local) risk-minimization and the mean-variance approaches. From a mathematical viewpoint, these optimization problems lead to new variants of decomposition theorems in stochastic analysis.

[1]  Walter Schachermayer,et al.  The variance-optimal martingale measure for continuous processes , 1996 .

[2]  M. Schweizer.  Approximation pricing and the variance-optimal martingale measure , 1996 .

[3]  Martin Schweizer,et al.  Hedging of options in a general semimartingale model , 1988 .

[4]  M. Schweizer.  A guided tour through quadratic hedging approaches , 1999 .

[5]  D. Lamberton,et al.  Hedging Index Options With Few Assets , 1993 .

[6]  Martin Schweizer,et al.  Approximating random variables by stochastic integrals , 1994 .

[7]  H. Kunita,et al.  On Square Integrable Martingales , 1967, Nagoya Mathematical Journal.

[8]  Thorsten Rheinländer,et al.  On L2-projections on a space of stochastic integrals , 1997 .

[9]  Saul D. Jacka,et al.  A Martingale Representation Result and an Application to Incomplete Financial Markets , 1992 .

[10]  Christophe Stricker,et al.  Couverture des actifs contingents et prix maximum , 1994 .

[11]  Martin Schweizer.  RISK‐MINIMIZING HEDGING STRATEGIES UNDER RESTRICTED INFORMATION , 1994 .

[12]  Walter Schachermayer,et al.  Weighted norm inequalities and hedging in incomplete markets , 1997, Finance Stochastics.

[13]  M. Schweizer.  Option hedging for semimartingales , 1991 .

[14]  J. Jacod.  Calcul stochastique et problèmes de martingales , 1979 .

[15]  Wolfgang J. Runggaldier,et al.  Hedging of Options under Discrete Observation on Assets with Stochastic Volatility , 1995 .

[16]  Walter Schachermayer,et al.  Attainable claims with p'th moments , 1996 .

[17]  Huyên Pham,et al.  Local Risk-Minimization Under Transaction Costs , 1998, Math. Oper. Res..

[18]  Huyên Pham,et al.  Dynamic programming and mean-variance hedging , 1999, Finance Stochastics.

[19]  Huyên Pham,et al.  Mean-variance hedging for continuous processes: New proofs and examples , 1998, Finance Stochastics.

[20]  M. Schweizer.  On the Minimal Martingale Measure and the Foellmer- Schweizer Decomposition , 1995 .

[21]  Closedness of some spaces of stochastic integrals , 1998 .

[22]  Christophe Stricker,et al.  Lois de martingale, densités et décomposition de Föllmer Schweizer , 1992 .

[23]  Some applications of L2-hedging with a non-negative wealth process , 1997 .

[24]  C. Stricker,et al.  Follmer-Schweizer Decomposition and Mean-Variance Hedging for General Claims , 1995 .

[25]  Damien Lamberton,et al.  Residual risks and hedging strategies in Markovian markets , 1989 .

[26]  F. Delbaen,et al.  A general version of the fundamental theorem of asset pricing , 1994 .

[27]  D. Sondermann.  Hedging of non-redundant contingent claims , 1985 .

[28]  J. Mémin,et al.  Espaces de semi martingales et changement de probabilité , 1980 .

[29]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

[30]  D. Duffie,et al.  Mean-variance hedging in continuous time , 1991 .

[31]  Wolfgang J. Runggaldier,et al.  Risk Minimizing Hedging Strategies Under Partial Observation , 1999 .

[32]  M. Schweizer.  Mean-Variance Hedging for General Claims , 1992 .

[33]  A.C.F. Vorst,et al.  Option Pricing and Hedging in Discrete Time with Transaction Costs and Incomplete Markets , 1997 .

[34]  L. Galtchouk.  Représentation des martingales engendrées par un processus à accroissements indépendants (cas des martingales de carré intégrable) , 1976 .

[35]  Christophe Stricker,et al.  Décomposition de Kunita-Watanabe , 1993 .

[36]  M. Frittelli.  The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets , 2000 .

[37]  Martin Schweizer,et al.  Variance-Optimal Hedging in Discrete Time , 1995, Math. Oper. Res..

[38]  Paolo Guasoni,et al.  Mean-Variance Hedging for Stochastic Volatility Models , 2000 .

[39]  Fabio Mercurio,et al.  Option pricing with hedging at fixed trading dates , 1996 .

[40]  Manfred Schäl,et al.  On Quadratic Cost Criteria for Option Hedging , 1994, Math. Oper. Res..

[41]  Local risk minimization and numéraire , 1999 .