What is static hedging?
“Spanning Multi-Asset Payoffs With ReLUs” by Sébastien Bossu, Stéphane Crépey, and Hoang-Dung Nguyen, dated March 22, 2024, presents a novel approach to the spanning problem of multi-asset payoffs using vanilla basket options. The authors propose a distributional formulation of this problem and establish that a unique solution exists if and only if the payoff function is even and absolutely homogeneous. They further provide a Fourier-based formula for calculating this solution.
A key insight of the paper is the recognition that financial payoffs are typically piecewise linear!
A continuous piecewise linear function
Which can make the derived solution explicit, yet challenging to utilize numerically. To address this, the authors turn to one-hidden-layer feedforward neural networks as a more efficient and practical method for discrete spanning.
A function (blue) and a piecewise linear approximation to it (red)
The paper’s significant contribution lies in its empirical testing of this approach on various archetypal payoffs. The results demonstrate better hedging outcomes with vanilla basket options compared to conventional single-asset vanilla hedges used in the industry. This finding challenges existing industry practices and offers a new perspective on hedging strategies.
The authors build upon the Carr and Madan (1998) spanning formula for single-asset options!
A piecewise linear function in two dimensions (top) and the convex polytopes on which it is linear (bottom)
Thus, expanding the theory to encompass European option payoffs and formulating a general solution for the continuum spanning problem. An interesting application of this theory is the derivation of the continuum solution replicating the industry ℓ1 dispersion call. The paper also provides explicit solutions for other absolutely homogeneous payoffs such as best-of and worst-of options.
One of the paper’s most intriguing aspects is its exploration of leveraging the parallel between vanilla basket calls and ReLU functions. The authors successfully apply neural networks to numerically solve the discrete spanning problem, identifying finite static hedges. The empirical study suggests that this unrestricted neural network approach yields superior static hedging results across dimensions ranging from 2 to 50, compared to other strategies and optimization schemes such as least-squares SVD.
The paper concludes with potential areas for future research, such as combining this approach with delta-hedging of residual payoff mismatches, and extending the method to non-homogeneous payoffs.
Moreover, an impressive contribution to the field of financial derivatives and offers practical implications for the exotic derivatives industry. The paper’s innovative use of neural networks in solving complex financial problems is particularly noteworthy and opens up new avenues for exploration in both theory and practice.
Read the full paper:
