**CHAPTER 16**

**Futures Options and Black’s Model**

**Practice Questions**

**Problem 16.8.**

*Suppose you buy a put option contract on October gold futures with a strike price of $1,200 per ounce. Each contract is for the delivery of 100 ounces. What happens if you exercise when the October futures price is $1,160?*

You gain (1,200 −1,160)×100 = $4,000. This gain is made up of a) a short futures contract in October gold and b) a cash payoff you receive which is 100 times the excess of $1,200 over the previous settlement price. The short futures position is marked to market in the usual way until you choose to close it out.

**Problem 16.9.**

*Suppose you sell a call option contract on April live cattle futures with a strike price of 140 cents per pound. Each contract is for the delivery of 40,000 pounds. What happens if the contract is exercised when the futures price is 145 cents?*

In this case, you lose (1.45 −1.40)×40,000 = $2,000. The loss is made up of a) a cash payoff you have to make equal to 40,000 times the excess of the previous settlement price over the previous settlement price and b) a short April futures contract.

**Problem 16.10.**

*Consider a two-month futures call 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

Lower bound if option is American is

**Problem 16.11.**

*Consider a four-month futures put option with a strike price of 50 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

Lower bound if option is American is

**Problem 16.12.**

*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; *d *= 1/*u *= 0.8607; and

In the tree shown in Figure S16.1 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.

**Figure S16.1** Tree to evaluate European and American call options in Problem 16.12

**Problem 16.13.**

*In Problem 16.12 what value does the binomial tree give for a six-month European put option on futures with a strike price of 60? If the put were American, would it ever be worth exercising it early? Verify that the call prices calculated in Problem 16.12 and the put prices calculated here satisfy put–call parity relationships.*

The parameters *u*, *d*, and *p* are the same as in Problem 16.12. The tree in Figure S16.2 shows that the prices of the European and American put options are the same as those calculated for call options in Problem 16.12. This illustrates a symmetry that exists for at-the-money futures options. The American option should sometimes be exercised early. Because and , the European put–call parity result holds.

Also because , , and the result in equation (16.2) holds. (The first expression in equation (16.2) is negative; the middle expression is zero, and the last expression is positive.)

**Figure S16.2** Tree to evaluate European and American put options in Problem 16.13

**Problem 16.14.**

*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, , , , ,

**Problem 16.15.**

*A futures price is currently 70, its volatility is 20% per annum, and the risk-free interest rate is 6% per annum. What is the value of a five-month European put on the futures with a strike price of 65?*

In this case , , , ,

**Problem 16.16.**

*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

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 .

**Problem 16.17.**

*“The price of an at-the-money European futures call option always equals the price of a similar at-the-money European futures put option.” Explain why this statement is true.*

The put price is

Because for all the put price can also be written

Because this is the same as the call price:

This result can also be proved from put–call parity showing that it is not model dependent.

**Problem 16.18.**

*Suppose that a futures price is currently 30. The risk-free interest rate is 5% per annum. A three-month American futures call option with a strike price of 28 is worth 4. Calculate bounds for the price of a three-month American futures put option with a strike price of 28.*

From equation (16.2), must lie between

and

Because we must have or

**Problem 16.19.**

*Show that if ** is the price of an American call option on a futures contract when the strike price is ** and the maturity is **, and ** is the price of an American put on the same futures contract with the same strike price and exercise date,*

*where ** is the futures price and ** is the risk-free rate. Assume that ** and that there is no difference between forward and futures contracts. (Hint: Use an analogous approach to that indicated for Problem 15.12.)*

In this case we consider

*Portfolio A*: A European call option on futures plus an amount invested at the risk-free interest rate

*Portfolio B*: An American put option on futures plus an amount invested at the risk-free interest rate plus a long futures contract maturing at time .

Following the arguments in Chapter 5 we will treat all futures contracts as forward contracts.

Portfolio A is worth while portfolio B is worth . If the put option is exercised at time , portfolio B is worth

at time where is the futures price at time . Portfolio A is worth

Hence Portfolio A is worth more than Portfolio B. If both portfolios are held to maturity (time *T*), Portfolio A is worth

Portfolio B is worth

Hence portfolio A is worth more than portfolio B.

Because portfolio A is worth more than portfolio B in all circumstances:

Because it follows that

or

This proves the first part of the inequality.

For the second part of the inequality consider:

*Portfolio C*: An American call futures option plus an amount invested at the risk-free interest rate

*Portfolio D*: A European put futures option plus an amount invested at the risk-free interest rate plus a long futures contract.

Portfolio C is worth while portfolio D is worth . If the call option is exercised at time , portfolio C becomes:

while portfolio D is worth

Hence portfolio D is worth more than portfolio C. If both portfolios are held to maturity (time *T*), portfolio C is worth while portfolio D is worth

Hence portfolio D is worth more than portfolio C.

Because portfolio D is worth more than portfolio C in all circumstances

Because it follows that

or

This proves the second part of the inequality. The result:

has therefore been proved.

**Problem 16.20.**

*Calculate the price of a three-month European call option on the spot price of silver. The three-month futures price is $12, the strike price is $13, the risk-free rate is 4%, and the volatility of the price of silver is 25%.*

This has the same value as a three-month European call option on silver futures where the futures contract expires in three months. It can therefore be valued using equation (16.5) with , , , and . The value is 0.244.

**Further Questions**

**Problem 16.21.**

*A futures price is currently 40. It is known that at the end of three months the price will be either 35 or 45. What is the value of a three-month European call option on the futures with a strike price of 42 if the risk-free interest rate is 7% per annum?*

In this case and . The risk-neutral probability of an up move is

The value of the option is

**Problem 16.22.**

*The futures price of an asset is currently 78 and the risk-free rate is 3%. A six-month put on the futures with a strike price of 80 is currently worth 6.5. What is the value of a six-month call on the futures with a strike price of 80 if both the put and call are European? What is the range of possible values of the six-month call with a strike price of 80 if both put and call are American?*

Put call parity for European options gives

6.5+78*e*^{-0.03×0.5 }= *c*+80*e*^{-0.03×0.5}

so that *c*=4.53.

The relation for American options gives

78* e*^{-0.03×0.5 }– 80 < *C* − 6.5 < 78 − 80* e*^{-0.03×0.5}

so that

−3.16 < *C* −6.5 < −0.81

so that * C* lies between 3.34 and 5.69

**Problem 16.23.**

*Use a three-step tree to value an American futures put option when the futures price is*

*50, the life of the option is 9 months, the strike price is 50, the risk-free rate is 3%, and*

*the volatility is 25%.*

u=1.331, d = 0.8825, and =0.4688. As the tree in Figure S16.3 shows the value of the option is 4.59

**Figure S16.3:** Tree for Problem 16.23

**Problem 16.24.**

*Calculate the implied volatility of soybean futures prices from the following information concerning a European put on soybean futures:*

Current futures price | 525 |

Exercise price | 525 |

Risk-free rate | 6% per annum |

Time to maturity | 5 months |

Put price | 20 |

In this case , , , . We wish to find the value of for which where:

This must be done by trial and error. When , . When , . When , . When , . These calculations show that the implied volatility is approximately 15.2% per annum.

**Problem 16.25.**

*It is February 4. July call options on corn futures with strike prices of 260, 270, 280, 290, and 300 cost 26.75, 21.25, 17.25, 14.00, and 11.375, respectively. July put options with these strike prices cost 8.50, 13.50, 19.00, 25.625, and 32.625, respectively. The options mature on June 19, the current July corn futures price is 278.25, and the risk-free interest rate is 1.1%. Calculate implied volatilities for the options using DerivaGem. Comment on the results you get. *

There are 135 days to maturity (assuming this is not a leap year). Using DerivaGem with , , = 135/365, and 500 time steps gives the implied volatilities shown in the table below.

Strike Price | Call Price | Put Price | Call Implied Vol | Put Implied Vol |

260 | 26.75 | 8.50 | 24.69 | 24.59 |

270 | 21.25 | 13.50 | 25.40 | 26.14 |

280 | 17.25 | 19.00 | 26.85 | 26.86 |

290 | 14.00 | 25.625 | 28.11 | 27.98 |

300 | 11.375 | 32.625 | 29.24 | 28.57 |

We do not expect put–call parity to hold exactly for American options and so there is no reason why the implied volatility of a call should be exactly the same as the implied volatility of a put. Nevertheless it is reassuring that they are close.

There is a tendency for high strike price options to have a higher implied volatility. As explained in Chapter 19, this is an indication that the probability distribution for corn futures prices in the future has a heavier right tail and less heavy left tail than the lognormal distribution.

**Problem 16.26.**

*Calculate the price of a six-month European put option on the spot value of the S&P 500. The six-month forward price of the index is 1,400, the strike price is 1,450, the risk-free rate is 5%, and the volatility of the index is 15%.*

The price of the option is the same as the price of a European put option on the forward price of the index where the forward contract has a maturity of six months. It is given by equation (16.6) with , , , , and . It is 86.35.

**Problem 16.27**

*The strike price of a futures option is 550 cents, the risk-free rate of interest is 3%, the volatility of the futures price is 20%, and the time to maturity of the option is 9 months. The futures price is 500 cents.*

*What is the price of the option if it is a European call?**What is the price of the option if it is a European put?**Verify that put-call parity holds**What is the futures price for a futures style option if it is a call?**What is the futures price for a futures style option if it is a put?*

(a) The price given by equation (16.5) or DerivaGem is 16.20 cents

(b) The price given by equation (16.6) or DerivaGem is 65.08 cents

(c) In this case, the left hand side of equation (16.1) is 16.2+550*e*^{−0.03×0.75} = 553.96. The right hand side of equation (16.1) is 65.03+500 *e*^{−0.03×0.75}=553.96. This verifies that put-call parity holds.

(d) The futures price for a futures-style call is 16.20e^{0.03×0.75}=16.57 cents

(e) The futures price for a futures-style put is 65.08e^{0.03×0.75}=66.56 cents