A Taylor-type formula for pure jump additive processes and its application to risk minimization

Abstract We derive a Clark-Haussmann-Ocone-type formula with second-order Malliavin derivatives for pure jump additive processes, thus providing new tools for representing financial derivatives in markets with jumps. Building on the framework established in Mancino (2001) and using the Malliavin-Skorohod calculus in $$ L^0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msup> </mml:math> and $$ L^1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> for additive processes, as developed in Di Nunno and Vives (2017), we derive a Taylor-type formula for $$ L^1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -canonical pure jump additive processes. Our result allows obtaining explicit expressions for risk-minimizing hedging strategies in terms of Malliavin derivatives in the context of market models driven by pure jump additive processes. Risk minimization, a fundamental hedging approach in incomplete markets, seeks strategies that minimize the conditional variance of the hedging error. The representation formulas provided in this work provide useful tools for managing risks in financial markets characterized by price jumps with $$ L^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -additive processes.