## Optimal Dynamic Hedging Using Futures Under a Borrowing ConstraintBank for International Settlements, Monetary and Economic Department, 2002 - Bank liquidity - 30 pages Both financial and non-financial firms routinely implement hedging policies to mitigate their exposure to changes in asset prices. However, while these policies may perform satisfactorily in the limited sense of hedging the exposure under consideration, they might increase the overall likelihood of financial distress due to the liquidity risks that they create. This paper examines the case of hedging price risk using derivative contracts that are marked to market (such as futures contracts) and hence subject to margin calls. It is shown that liquidity risk, stemming from the need to meet margin calls on the futures position, can be a significant source of risk and can even lead to financial distress even though the firm remains 'hedged'. Such risks should therefore be taken into account in the formulation of an optimal hedging policy. This paper derives the dynamic hedging strategy of a firm that uses futures contracts to hedge a spot market exposure. The risk emanating from the margin requirement on futures contracts is incorporated into the hedging decision by restricting the borrowing capacity of the firm. It is shown that this leads to a substantial reduction in the firm's optimal hedge, especially if the hedging horizon is long. The results provide theoretical support for the low level of hedging observed empirically. |

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annum aversion of 0.5 Bank Bank runs binomial tree borrowing constraint boundary conditions date T-2 denoted derivative contracts dynamic programming Figure ﬁnal period wealth ﬁnancial distress ﬁnite ﬁrm hedges ﬁrm’s ﬁrst FT_1 fulﬁl margin account futures contracts futures markets futures price process geometric Brownian motion graph shows Hamilton-Jacobi-Bellman equation hedge on date hedge the spot hedging decision hedging horizon hedging policy hedging position hedging problem Hence inﬁnite large margin calls level of margin liquidity risk margin account obligations margin wealth margin wealth-exposure maximise Metallgesellschaft non-linear number of futures ODE solver optimal control optimal futures position optimal hedge ratio partial differential equation price on date relative risk aversion remaining to maturity riskless interest rate riskless rate shows the optimal solve speciﬁed spot asset spot position spot price exposure total exposure total hedge unconstrained hedge unconstrained problem unexpectedly large margin unhedged utility function value function wealth on date XT_1 XT_2