Let t_0:=June 2022 denote the current month and {t_k, k=1,, 12} be the twelve forthcoming months. For the sake of simplicity, we disregard the time value, that is r=0. A company has committed to buy N=1,000 widgets at times t_1, t_4, and t_7 for what will the widget spot prices be on those months. The array of these commitments defines the companys naked exposure. Prices are expressed in dollars per widget, namely $/u, where u stands for unit. Notation: S(t) denotes the spot price of a widget at time t and F_T (t) indicates the time t futures price of a widget to deliver at time T>=t. Clearly, F_t (t)=S(t). At each point in time t_0, , t_11 a term structure of futures prices is quoted for all forthcoming maturities. Each futures contract has a notional value of 25 widgets and liquidity decreases with an increasing maturity. The company is seeking for a rational hedging strategy of their naked exposure using futures contracts. This strategy must comply with a few requirements: 1) To profit from upside price movements, the strategy should hedge the 50% of naked exposure; liquidity is to be optimized while selecting futures maturity; in particular: 2) to avoid excessive liquidity spreads, only futures up to 6 months ahead should be adopted; 3) Futures positions are unwound in the month prior to maturity; 4) Trading entails transaction costs, hence its frequency ought to be minimized. Rollover is allowed.

a) Propose an ideal hedging strategy defined as when to go long/short how many and futures contracts of which kind (expressed as F(t=time of issuance, T=delivery date, x=delivery price, S = underlying asset)).

b) Write down a formula for the full-exposure P&L (=naked exposure + hedge P&Ls) of each of the three widget purchase commitments.