Risk minimization for markets driven by pure-jump additive processes via Malliavin–Mancino–Taylor type formula

Abstract

We study quadratic risk minimization for discounted exponential asset prices driven by time-inhomogeneous pure-jump additive processes whose compensator factorizes through a deterministic time change and a family of Lévy kernels. Using Malliavin–Skorohod calculus on the canonical jump space, we derive a second-order Taylor–Clark–Haussmann–Ocone representation and combine it with the Galtchouk–Kunita–Watanabe decomposition to obtain an explicit representation of the risk-minimizing hedge ratio in incomplete markets. For exponential jump amplitudes, the denominator admits a cumulant representation, yielding closed-form expressions for this denominator in a model bank of time-varying jump kernels, including Merton, Kou, Variance Gamma, CGMY, Normal Inverse Gaussian, and double phase-type specifications. We also treat fixed- and floating-strike lookback options. A market-data-calibrated benchmark application to Henry Hub natural gas futures illustrates how the representation can be implemented from observed daily futures returns. In this calibrated experiment, the full second-order hedge reduces the terminal hedging error variance by 83.1% relative to the first-order truncation and performs competitively against stationary Kou and Black–Scholes benchmark hedges. These results lead to a practically computable second-order approximation for risk-minimizing hedges.