The End of the Month Option and Other Embedded Options in Futures Contracts

Futures contracts often contain several different kinds of embedded options related to the delivery of the underlying. The end of the month option allows the holder of the short end of a futures contract to deliver the underlying at any time during the last week of the contract period at a fixed price determined at the start of the last week. We derive a formula for this price in a general incomplete financial market, in which the process underlying the futures contract is a general adapted càdlàg process and the risk free interest rate process is a general adapted process, both satisfying certain integrability conditions. In a similar setting we also derive a formula for the futures price process of a futures contract which first has no option active, then it has an active timing option (which lets the holder of the short end deliver at any time during the last month of the contract period) and lastly it has an active end of the month option. This combination of delivery options is present in the real-world financial markets. We show that this futures price process is dominated by a standard futures price process with maturity at the time of the activation of the end of the month option. We also show that if the underlying is an asset with a non-positive convenience yield and the interest rate is non-negative then it is optimal to deliver when the end of the month option becomes active. The main contribution of this paper is to properly define the end of the month option and to derive a formula for the futures price process of a futures contract with the combination of options described above.