A McKean-Vlasov game of commodity production, consumption and trading

TL;DR

This paper models a two-player stochastic game involving a producer and consumer affecting commodity prices, analyzing how risk aversion influences trading strategies and derivative pricing through a McKean-Vlasov framework.

Contribution

It introduces a novel linear-quadratic McKean-Vlasov game model for commodity trading with explicit Nash equilibrium strategies and indifference prices.

Findings

Risk aversion significantly impacts trading strategies.

Derived semi-explicit Nash equilibrium strategies.

Numerical analysis shows effects of risk aversion and volatility costs.

Abstract

We propose a model where a producer and a consumer can affect the price dynamics of some commodity controlling drift and volatility of, respectively, the production rate and the consumption rate. We assume that the producer has a short position in a forward contract on \lambda units of the underlying at a fixed price F, while the consumer has the corresponding long position. Moreover, both players are risk-averse with respect to their financial position and their risk aversions are modelled through an integrated-variance penalization. We study the impact of risk aversion on the interaction between the producer and the consumer as well as on the derivative price. In mathematical terms, we are dealing with a two-player linear-quadratic McKean-Vlasov stochastic differential game. Using methods based on the martingale optimality principle and BSDEs, we find a Nash equilibrium and characterize the corresponding strategies and payoffs in semi-explicit form. Furthermore, we compute the two indifference prices (one for the producer and one for the consumer) induced by that equilibrium and we determine the quantity \lambda such that the players agree on the price. Finally, we illustrate our results with some numerics. In particular, we focus on how the risk aversions and the volatility control costs of the players affect the derivative price.