Consider a producer exposed to output price risk. If price risk can be managed with futures contracts, a full hedge ensures that the producer's nancial position at the hedging horizon is almost risk-free. However, this is only true if the producing rm can always accommodate the liquidity needs that may arise from the marking to market of the futures position. Depending on the development of the futures price over time, marking to market may lead to interim cash in ows and/or cash out ows prior to the hedging horizon. Suppose that the producing rm faces a liquidity constraint in the sense that there is no free cash at hand. If the original futures position generates an interim loss, the producer will have to raise additional cash in order to maintain the position.
Usually, the borrowing rate is higher than the interest rate applicable to any excess cash that might be generated by marking to market. Hence, the producer faces liquidity risk: If the futures position creates an intermediate loss, additional cash has to be raised which is costly. The producer will anticipate the possibility of additional liquidity needs arising from the futures position when deciding about the optimal hedging position in futures contracts. If the producer can also trade options on futures, he might use these options to manage the liquidity risk borne by the futures position. This will also a ect the size of the optimal futures position and the optimal production decision.
This paper analyzes the impact of joint price risk and liquidity risk on optimal output and on the optimal positions in futures and in options on futures taken by a risk-averse producing rm. Hedging price risk with futures contracts creates liquidity risk through marking to market. Liquidity risks can be signi cant: In the extreme case, the entire derivatives position has to be liquidated. In less extreme cases, there is the opportunity cost of quickly raised cash. The rm modeled here faces a liquidity constraint in the following sense: There is no free cash available from within the rm, but the rm can borrow additional funds at a rm-speci c borrowing rate. In addition, the rm can trade options on futures. As the liquidity risk is an asymmetric risk it only materializes if the futures position creates an interim cash out ow options might be used to alleviate the impact of liquidity risk on the rm's nancial position.
The paper employs a two-period framework where futures contracts maturing at the end of the second period are traded at the beginning of each period. In addition, one-period options on futures are traded at the beginning of every period. The analytical results are as follows: If the derivatives position entered into in the rst period generates a loss by the end of this period, the rm will optimally sell fairly priced call options on futures in order to generate funds to cover (part of) this loss. As doing so changes the rm's exposure to price risk, the futures position is adjusted as well. If there is no loss by the end of the rst period, no options position will be taken and the rm fully hedges with futures contracts over the second period. The numerical results show that the rm under-hedges in the rst period as a result of the existence of the liquidity constraint. They also indicate that options are not used in the rst period.
The impact of liquidity risk on futures hedging has been studied by Lien (2003), Lien and Li (2003) Wong (2004a), Wong (2004b) and Wong and Xu (2006). Lien (2003) shows that the initial futures position depends upon the rm's ability to cope with losses arising from marking to market. Wong (2004a) proves that the optimal futures hedge is an underhedge if the rm is prudent in the sense of Kimball (1990; 1993). He also shows that production decreases if liquidity risk is introduced. Wong (2004b) analyzes the hedging problem of a rm that can trade futures contracts with two di erent maturities. All these papers focus on a particular type of liquidity risk where the rm has to liquidate the entire futures position if the interim cash out ow exceeds an exogenously given threshold.
This assumption is isomorphic to assuming that the cost of covering an interim loss caused by marking to market equals the risk-free rate for an amount up to the level of the threshold and then e ectively jumps to in nity such that raising external cash beyond the threshold is ruled out. In contrast, our model follows Korn (2004) by assuming that there is no such borrowing threshold but that all borrowing has to be done at a rm-speci c borrowing rate depending on the rm's credit standing. It seems more appropriate for most rms to assume that there is no such extreme jump in the cost of raising additional cash. As a consequence, the rm is able to maintain its futures position even if the interim losses are signi cant.
The two papers that are closest to ours are Korn (2004) and Wong and Xu (2006). Korn (2004) analyzes optimal forward hedging. His model is based on the assumption that the rm will have to provide cash as collateral if the forward position has a negative market value prior to the hedging horizon. Unlike our model, an interim cash in ow from the forward position is not permitted in Korn's (2004) model. More importantly, Korn (2004) does not allow for options whereas our model does. To our knowledge, Wong and Xu's (2006) model is the rst to incorporate options (on the rm's output) into the futures hedging problem.
In their model, however, the rm has to liquidate any derivatives position if the interim loss exceeds an exogenously given threshold. Our model, in contrast, uses a more exible approach by assuming a rm-speci c borrowing rate that allows the rm to borrow larger amounts, though always at a cost. In sum, the present paper's contribution is to combine the joint availability of options and futures contracts with the more exible, less extreme approach to model liquidity risk that does not require an exogenously xed borrowing threshold.
The paper is organized as follows: Section 2 presents the model. As a bench-mark for comparison, Section 3 characterizes the optimal decisions in the absence of liquidity risk. The main results of the paper are presented in Sections 4 and 5: The optimal decisions taken in the second period are characterized analytically in Section 4. Numerical results including the decisions taken in the rst period are presented in Section 5. Section 6 concludes. All proofs are relegated to the Appendix.
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