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FNCE30007 Derivative Securities Tutorial 10: Futures Options

Question 1 Suppose that you buy a put option contract on October gold futures with a strike price of $400 per ounce. Each contract is for the delivery of 100 ounces. What happens if you exercise when the October futures price is $380? An amount (400 380) 100 2 000 = ,$ is added to your margin account and you acquire a short futures position obligating you to sell 100 ounces of gold in October. This position is marked to market in the usual way until you choose to close it out. Question 2 Consider a two-month call futures option with a strike price of 40 when the risk-free interest rate is 10% per annum. The current futures price is $47. What is a lower bound for the value of the futures option if it is (a) European and (b) American? Lower bound if option is European is 0 1 2 120( ) (47 40) 6 88

rTF K e e . / = = . Lower bound if option is American is 0 7F K = Question 3 A futures price is currently 60 and its volatility is 30%. The risk-free interest rate is 8% per annum. Use a two-step binomial tree to calculate the value of a six-month European call option on the futures with a strike price of 60? If the call were American, would it ever be worth exercising it early? In this case 0 3 1 4 1 161834u e . /= = . ; 1 0 860708d u= / = . ; and

1 0 860708 0 46261 161834 0 860708

p .= = .. .

In the tree shown in the figure below the middle number at each node is the price of the European option and the lower number is the price of the American option. The tree shows that the value of the European option is 4.3155 and the value of the American option is 4.4026. The American option should sometimes be exercised early.

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Question 4 A futures price is currently $25, its volatility is 30% per annum, and the risk-free interest rate is 10% per annum. What is the value of a nine-month European call on the futures with a strike price of $26? In this case 0 25F = , 26K = , 0 3= . , 0 1r = . , 0 75T = .

2

01

ln( ) 2 0 0211F K TdT

/ + /= = .

2

02

ln( ) 2 0 2809F K TdT

/ /= = .

0 075[25 ( 0 0211) 26 ( 0 2809)]c e N N .= . . 0 075[25 0 4916 26 0 3894] 2 01e .= . . = . Question 5 Suppose that a one-year futures price is currently $35. A one-year European call option and a one-year European put option on the futures with a strike price of $34 are both priced at $2 in the market. The risk-free interest rate is 10% per annum. Identify an arbitrage opportunity. In this case 0 1 12 34 32 76rTc Ke e . + = + = . 0 1 10 2 35 33 67

rTp F e e . + = + = . Put-call parity shows that we should buy one call, short one put and short a futures contract. This costs nothing up front. In one year, either we exercise the call or the put is exercised against us. In either case, we buy the asset for 34 and close out the futures position. The gain on the short futures position is 35 34 1 = .

80.9915 20.9915 20.9915

60 0 0

44.4491 0 0

67.7101 9.5178 9.7101

51.6425 0 0

60 4.3155 4.4026

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Question 6 The price of an at-the-money European call futures option always equals the price of a similar at-the-money European put futures option. Explain why this statement is true. The put price is 2 0 1[ ( ) ( )]

rTe KN d F N d Because ( ) 1 ( )N x N x = for all x the put price can also be written 2 0 0 1[ ( ) ( )]

rTe K KN d F F N d + Because 0F K= this is the same as the call price: 0 1 2[ ( ) ( )]

rTe F N d KN d This result can also be proved from putcall parity showing that it is not model dependent. Question 7 Suppose that a futures price is currently $30. The risk-free interest rate is 5% per annum. A three-month American call futures option with a strike piece of $28 is worth $4. Calculate bounds for the price of a three-month American put futures option with a strike price of $28. From equation (16.2), C P must lie between 0 05 3 1230 28 1 63e . / = . and 0 05 3 1230 28 2 35e . / = . Because 4C = we must have 1 63 4 2 35P. < < . or 1 65 2 37P. < < .