Efficient approximations for utility-based pricing

TL;DR

This paper introduces a novel deterministic decomposition of reservation prices in illiquid markets using Lambert functions, improving numerical approximation methods and aiding in optimal hedging strategy selection.

Contribution

It proposes a Lambert function-based decomposition for reservation prices, enhancing approximation accuracy and computational efficiency in illiquid market pricing.

Findings

Lambert Monte Carlo (LMC) improves reservation price estimation.

Deterministic approximations outperform traditional methods.

Optimal hedging asset selection reduces reservation price and cash invested.

Abstract

In a context of illiquidity, the reservation price is a well-accepted alternative to the usual martingale approach which does not apply. However, this price is not available in closed form and requires numerical methods such as Monte Carlo or polynomial approximations to evaluate it. We show that these methods can be inaccurate and propose a deterministic decomposition of the reservation price using the Lambert function. This decomposition allows us to perform an improved Monte Carlo method, which we name Lambert Monte Carlo (LMC) and to give deterministic approximations of the reservation price and of the optimal strategies based on the Lambert function. We also give an answer to the problem of selecting a hedging asset that minimizes the reservation price and also the cash invested. Our theoretical results are illustrated by numerical simulations.