Integral representations of risk functions for basket derivatives

TL;DR

This paper derives explicit integral formulas for risk functions associated with basket derivatives in a multidimensional Black-Scholes model, aiding in risk assessment and pricing.

Contribution

It provides new explicit integral representations for risk functions for common payoff types in basket derivatives within the Black-Scholes framework.

Findings

Explicit formulas for minimal risk and cost reduction functions.

Integral representations for risk functions with specific loss functions.

Application to digital, quantos, outperformance, and spread options.

Abstract

The risk minimizing problem $\mathbf{E}[l((H-X_T^{x,\pi})^{+})]\overset{\pi}{\longrightarrow}\min$ in the multidimensional Black-Scholes framework is studied. Specific formulas for the minimal risk function and the cost reduction function for basket derivatives are shown. Explicit integral representations for the risk functions for $l(x)=x$ and $l(x)=x^p$, with $p>1$ for digital, quantos, outperformance and spread options are derived.