Derivatives Valuation – AS 4510Spring 2015

Homework 1: Chapters 1 through 3Due Monday, 01/28/2015

1. Read Chapters 1-3 in the textbook.

2. You are given the following information:

• The current price to buy one share of XYZ stock is 500.

• The stock does not pay dividends.

• The risk-free interest rate, compounded continuously, is 6%.

• A European call option on one share of XYZ stock with a strike price of K that expires in one year costs66.59.

• A European put option on one share of XYZ stock with a strike price of K that expires in one year costs18.64.

Using put-call parity, calculate the strike price, K.

(A) 449

(B) 452

(C) 480

(D) 559

(E) 582

Solution:Question 2 from SOA exam: edu-2014-10-exam-fm-ques.pdf

This is put–call parity (McDonald, page 70):

Call(K,T )− Put(K,T ) = PV(F0,T −K)

66.59− 18.64 = 500−Ke−0.06

K = (500− 66.59 + 18.64)e0.06

= 480

Answer: C

3. You are given the following information:

• One share of the PS index currently sells for 1,000.

• The PS index does not pay dividends.

• The effective annual risk-free interest rate is 5%.

You want to lock in the ability to buy this index in one year for a price of 1,025. You can do this by buyingor selling European put and call options with a strike price of 1,025. Which of the following will achieve yourobjective and also gives the cost today of establishing this position.

(A) Buy the put and sell the call, receive 23.81.

(B) Buy the put and sell the call, spend 23.81.

(C) Buy the put and sell the call, no cost.

(D) Buy the call and sell the put, receive 23.81.

(E) Buy the call and sell the put, spend 23.81.

Solution:“Lock in the . . . ” means you want to go long. Buy the call and sell the out. The forward price is F = S0(1+r) =1, 050. Since you want to buy at a price below the forward price, you have to pay something now. (You can buyat the forward price 1050 for 0 now.) You can determine answer the correct choice now (E). Use put–call parityto find the cost, if you like.

Call(K,T )− Put(K,T ) = PV(F0,T −K)

= (1, 050− 1, 025) (1.05)−1

= 23.81

Answer: E

Page 2

4. The current stock price is 40, and the effective annual interest rate is 8%. These are the current prices for 1-yearEuropean call options on the stock:

Strike Price35 9.1240 6.2245 4.08

Assuming that all call positions being compared are long, at what 1-year stock price range does the 45-strike callproduce a higher profit than the 40-strike call, but a lower profit than the 35-strike call?

(A) S1 < 38.13.

(B) 38.13 < S1 < 40.44.

(C) 40.44 < S1 < 42.31.

(D) S1 > 42.31.

(E) There is no price for S1 at which this situation occurs.

Solution:Draw the profit diagrams.

10 20 30 40 50 60Stock Price

s

-20

-10

0

10

20Profit

Profit Call(S1, 45) = (S1 − 45)+ − 4.08(1.08)

Profit Call(S1, 40) = (S1 − 40)+ − 6.22(1.08)

Profit Call(S1, 35) = (S1 − 35)+ − 9.12(1.08)

For S1 out of the money the graph is constant. At the money, the slope changes from 0 to 1. The 35-strike hits the40-strike at the value of S1 > 35 for which

(S1 − 35)− 9.12(1.08) = −6.22(1.08)

which is S1 = 40.44. (This is enough to choose C.) For S1 > 40.44, the 40-strike call is below the 35-strike call.The 40-strike call is below the 45-strike call until

(S1 − 40)− 6.22(1.08) = −4.08(1.08)

which is S1 = 42.31.

Answer: C

Page 3

5. Suppose that you short one share of a stock index for 50, and that you also buy a 60-strike European call optionthat expires in 2 years for 10. Assume the effective annual interest rate is 3%.

If the stock index increases to 75 after 2 years, what is the profit on your combined position, and what is analternative name for the call in this context?

Profit Name

(A) −22.64 Floor

(B) −17.56 Floor

(C) −22.64 Cap

(D) −17.56 Cap

(E) −22.64 Written Covered Call

Solution:The short stock and long call combination is a cap (choice C or D).

Profit = 50(1.03)2 − 75︸ ︷︷ ︸short stock

+(75− 60)− 10(1.03)2︸ ︷︷ ︸long call

= −17.56

Answer: D

6. The current price of a non-dividend paying stock is 40 and the continuously compounded risk-free rate of returnis 8%. You are given that the price of a 35-strike call option is 3.35 higher than the price of a 40-strike call option,where both options expire in 3 months.

How much does the price of an otherwise equivalent 40-strike put option exceed the price of an otherwise equiv-alent 35-strike put option?

Solution:Use put-call parity.

Put(K1, T )− Put(K2, T ) = Call(K1, T )− PV(F0,T −K1)− Call(K2, T )− PV(F0,T −K2)

= Call(K1, T )− Call(K2, T ) + (K1 −K2)e−rT

= 3.35 + 5e−0.08(0.25) = 1.55

Answer: 1.55

7. Which of the following positions have an unlimited loss potential from adverse price movement in the underlyingasset, regardless of the initial premium received?

I. Short 1 forward contract

II. Short 1 call option

III. Short 1 put option

(A) I only.

(B) I and II only.

(C) I and III only.

(D) II and III only.

(E) I, II, and III.

Page 4

Solution:Only the short put has a limited loss potential.

Answer: B

Page 5

8. Which of the following are true?

I. The value of flexibility in manufacturing and service processes can be modeled using option pricing theory.

II. The stock of a company can be viewed as a put option.

III. As the stock price rises, the value of a put option falls.

(A) III only

(B) I and II only

(C) I and III only

(D) II and III only

(E) I, II, and III

Solution:This question is from the SOA exam for course 2 in 2000.

I. True: Option Pricing Theory has many uses and is versatile; helping evaluate the risk of decisions involvingoptions. Some examples in manufacturing and servicing include options on raw materials, the option toabandon a project, and the option to vary the product mix as demand changes.

II. False: The stock of a company is in effect a call option on the assets of the firm. The stock conveys ownershipof the company. In the extreme, stockholders can direct management to sell the company’s assets and paythe proceeds to the owners (shareholders).

III. True: The future payoff is (K − ST )+ so the higher S0 is less likely the option is in the money at time T

and the smaller the expected payoff.

Answer: C

9. Determine which statement about zero-cost purchased collar is false.

(A) A zero-width, zero-cost collar can be created by setting both the put and call strike prices at the forwardprice.

(B) There is an infinite number of zero-cost collars.

(C) The put option can be at-the-money.

(D) The call option can be at-the-money.

(E) The strike price on the put option must be at or below the forward price.

Solution:This question is from the SOA online sample exam for FM.

A is correct as explained in the textbook (page 79.) B is correct (page 77). C and D are correct. The call strikehas to be higher (or equal to) the put strike, but either could be at the money. E is false.

Answer: E

10. Stock ABC has the following characteristics:

• The current price to buy one share is 100.

• The stock does not pay dividends.

• European options on one share expiring in one year have the following prices in the table below.

Page 6

Strike Price Call Option Price Put Option Price90 14.63 0.24

100 6.80 1.93110 2.17 6.81

A butterfly spread on this stock has the profit diagram below.

80 90 100 110 120 130 140

-10

-5

0

5

10

The annual risk-free interest rate compounded continuously is 5%.

Determine which of the following will NOT produce this profit diagram.

(A) Buy a 90 put, buy a 110 put, sell two 100 puts

(B) Buy a 90 call, buy a 110 call, sell two 100 calls

(C) Buy a 90 put, sell a 100 put, sell a 100 call, buy a 110 call

(D) Buy one share of the stock, buy a 90 call, buy a 110 put, sell two 100 puts

(E) Buy one share of the stock, buy a 90 put, buy a 110 call, sell two 100 calls.

Solution:McDonald (page 81) defines a butterfly spread as a written straddle combined with a purchased call and purchasedput. For the given data, the at-the-money straddle is a written 100 call and written 100 put. According to thisdefinition, (C) is a butterfly spread.

For S1 ≤ 90 or S1 ≥ 110, the combined payoff has to be

(6.80 + 1.93− 2.17− 0.24)e0.05 − (110− 100) = −3.36.

We know that the graph of a butterfly spread is piece-wise linear function with slope 0, 1 or -1. It has to look likethis:

Profit =

−3.36 S1 ≤ 90

S1 + constant 90 ≤ S1 ≤ 100

−S1 + constant 100 ≤ S1 ≤ 110

−3.36

Now scan the other choices. (A), (B), and (E) have exactly four positions that payoff 0 or a linear function of S1

so it is possible for them to net to zero for large or small values of S1. But (D) is suspicious because it has fivefunctions. So take a closer look at (D) with S1 ≤ 90. In this range the call is out of the money, so you have thestock, a long put, and short 2 puts. The long put is a floor for the stock so in this range they sum to a constant. Sobeing short 2 puts means the slope is 2 in this range. Here is the formula for S1 ≤ 90:

Profit = (−100− 14.63− 6.81 + 2(1.93)) e0.05 + S1 + 0 + 110− S1 − 2(100− S1)

= constant + 2S1

Answer: D

Page 7

For completeness, here are the formulas for the other choices.

Consider (A).

[−0.24− 6.81 + 2(1.93)]e0.05+(90− S1)+ + (110− S1)+ − 2(100− S1)+

=

−3.36 for S1 ≤ 90 or S1 ≥ 110

−3.36 + S1 − 90 for 90 ≤ S1 ≤ 100

−3.36 + 110− S1 for t100 ≤ S1 ≤ 110

We see that (A) is has the same profit as the given butterfly diagram.

Consider (B).

[−14.63− 2.17 + 2(6.80)]e0.05+(S1 − 90)+ + (S1 − 110)+ − 2(S1 − 100)+

=

−3.36 for S1 ≤ 90 or S1 ≥ 110

−3.36 + S1 − 90 for 90 ≤ S1 ≤ 100

−3.36 + 110− S1 for 100 ≤ S1 ≤ 110

So (B) is also correct.

Consider (D).

[−100− 14.63− 6.81 + 2(1.93)]e0.05+S1 + (S1 − 90)+ + (110− S1)+ − 2(100− S1)+

=

{some constant1 + 2S1 for S1 ≤ 90

does not matter for S1 ≥ 90

So (D) is not correct since it does not have the correct slope for S1 ≤ 90. For completeness we consider (E).

[−100− 0.24− 2.17 + 2(6.80)]e0.05+S1 + (90− S1)+ + (S1 − 110)+ − 2(S1 − 100)+

=

−93.36 + 90 for S1 ≤ 90 or S1 ≥ 110

−93.36 + S1 for 90 ≤ S1 ≤ 100

−93.36− S1 + 200 for 100 ≤ S1 ≤ 110

There are other approaches. I suggested an alternative in a recent email and applied it to show (A) has the sameprofit structure as (C). The approach uses on the identify

(s−K)+ − (K − s)+ = s−K

to show the payoffs differ by a constant. If the payoffs of two positions differ by a constant, then they have thesame profit functions (or else there is an arbitrage).

Page 8