# Local risk-minimization with multiple assets under illiquidity with applications in energy markets

## TL;DR

This paper develops a hedging strategy for contingent claims considering multiple assets with varying liquidity levels, balancing risk minimization and liquidity costs, with applications demonstrated in energy markets.

## Contribution

It introduces a new local risk-minimization approach under illiquidity and transaction costs, providing a closed-form solution for linear supply curves and practical energy market applications.

## Key findings

- Closed-form solution for optimal hedging strategy under liquidity risk
- Tradeoff analysis between liquidity and hedging effectiveness
- Comparison showing improved strategies over classical methods

## Abstract

We propose a hedging approach for general contingent claims when liquidity is a concern and trading is subject to transaction cost. Multiple assets with different liquidity levels are available for hedging. Our risk criterion targets a tradeoff between minimizing the risk against fluctuations in the stock price and incurring low liquidity costs. Following \c{C}etin U., Jarrow R.A., and Protter P. (2004) we work in an arbitrage-free setting assuming a supply curve for each asset. In discrete time, following the ideas in Schweizer M. (1998) and Lamberton D., Pham H., Schweizer M. (1998) we prove the existence of a locally risk-minimizing strategy under mild conditions on the price process. Under stochastic and time-dependent liquidity risk we give a closed-form solution for an optimal strategy in the case of a linear supply curve model. Finally we show how our hedging method can be applied in energy markets where futures with different maturities are available for trading. The futures closest to their delivery period are usually the most liquid but depending on the contingent claim not necessary optimal in terms of hedging. In a simulation study we investigate this tradeoff and compare the resulting hedge strategies with the classical ones.