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Unformatted text preview: STAT 2820 Chapter 5 Risk Management - Hedging by K.C. Cheung 5.1 Basic risk management: example from producer’s perspective 5.1.1 Assume that a gold miner plans to mine and sell 400000 ounces of gold next year. The price of gold today is known: $400 per ounce. The cost of mining one ounce of gold is $380. The net income the gold miner will get next year depends on gold price next year ($ G ): Net income = 400000( G- cost) = 400000( G- 380) . If G < 380, the gold miner suffers a loss. 5.1.2 [Hedging using futures] The gold miner can lock in a selling price for gold in one year by shorting futures/forward contract on 400000 ounces of gold: agreeing to sell 400000 ounces of gold in one year at a price of $ K per ounces. Suppose that K = 420. The payoff of the short futures position is Payoff of short futures = 400000( K- G ) = 400000(420- G ) , and hence Net income = 400000( G- 380) + 400000(420- G ) = 400000(40) = 1600000 . In other words, a profit of $40 per ounce is locked once a short position in the futures contract is taken. 5.1.3 [Hedging using call options] To get compensation when gold price is low, the gold miner can sell call options on gold. Suppose that the current market price for a 420-strike call option is $8.77 per ounce. The gold miner can receive ($8 . 77)(400000) when selling the calls. The payoff of the short call position is Payoff of short call =- 400000( G- 420) + , and hence Net income = 400000( G- 380)- 400000( G- 420) + = ( 400000( G- 380) , if G < 420 400000(40) , if G ≥ 420 . If the annual interest rate is 5%, then the future value of the call option is $(8 . 77)(1 . 05) = $9 . 21 per ounce, and hence P/L = Net income- FV (Initial investment) = ( 400000( G- 370 . 79) , if G < 420 400000(49 . 21) , if G ≥ 420 . STAT 2820 Chapter 5 2 When G < 420, the P/L is G- 370 . 79 per ounce, better than G- 380 per ounce when unhedged (cf. 5.1.1). It is because when G < 420, the gold miner can keep the call premium without any liability. On the other hand, if G ≥ 420, the P/L is locked at $49.21 per ounce, so (i) If 420 ≤ G ≤ 429 . 21 = ⇒ G- 380 ≤ 49 . 21 = ⇒ P/L of the hedged position is better (ii) If G > 429 . 21 = ⇒ G- 380 > 49 . 21 = ⇒ P/L of the unhedged position is better 5.1.4 [Hedging using put options] One disadvantage of using futures is that the gold miner can only earn $40 per ounce, no matter how high the gold price G is. Similar drawback also arises for using call options. Thus the potential for greater profit is lost. Put options on gold provide a way to have higher profit at gold prices G while being still protected against low prices. Suppose that the current market price for a 420-strike put option is $8.77 per ounce. The gold miner can use ($8 . 77)(400000) to buy the put. The payoff of the long put position is Payoff of long put = 400000(420- G ) + , and hence Net income = 400000( G- 380) + 400000(420- G ) + = ( 400000(40) , if G < 420 400000( G- 380) , if G ≥ 420 ....
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- Fall '11