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We propose a hedging approach for general contingent claims when liquidity isa concern and trading is subject to transaction cost. Multiple assets withdifferent liquidity levels are available for hedging. Our risk criteriontargets a tradeoff between minimizing the risk against fluctuations in thestock price and incurring low liquidity costs. Following \c{C}etin U., JarrowR.A., and Protter P. (2004) we work in an arbitrage-free setting assuming asupply curve for each asset. In discrete time, following the ideas in SchweizerM. (1998) and Lamberton D., Pham H., Schweizer M. (1998) we prove the existenceof a locally risk-minimizing strategy under mild conditions on the priceprocess. Under stochastic and time-dependent liquidity risk we give aclosed-form solution for an optimal strategy in the case of a linear supplycurve model. Finally we show how our hedging method can be applied in energymarkets where futures with different maturities are available for trading. Thefutures closest to their delivery period are usually the most liquid butdepending on the contingent claim not necessary optimal in terms of hedging. Ina simulation study we investigate this tradeoff and compare the resulting hedgestrategies with the classical ones.

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