Corporate Finance, Module 20: “Introduction to Options”

Homework Assignment

(The attached PDF file has better formatting.)

Put Call Parity Relation

A European call option that expires in three months and has a strike price of $80 has a price of $5. The underlying stock price is $77, the stock pays no dividends, and the risk-free interest rate is 8% per annum compounded quarterly, or 2% each three months.

A.    What is the price of a European put option on this stock that expires in three months and has a strike price of $80? Use the put call parity relation, that c + PV(X) = p + S.
B.    If the European put option also had a price of $5, which would you prefer to buy, the call option or the put option? (A consumer prefers the good that is underpriced, so the true value is more than the price paid.)
C.    If the European put option also had a price of $5, how might an investor make a risk-free profit? Assume the investor is a large financial intermediary that can borrow at the risk-free rate. The transactions at time 0 are:

●    The investor borrows $77 from a bank at the risk-free interest rate of 8% per annum, or 2% for three months.
●    The investors uses the $77 to buy one share of stock.
●    The investor sells a call option for $5.
●    The investor buys a put option for $5.

1.    What is the investor’s cash outflow at time 0? (The bank loan pays for the share of stock; the proceeds from selling the call option pay for the put option.)
2.    In three months time, how much does the investor repay the bank? ($77 × 1.02)
3.    If the stock price in three months time is less than $80, which option is exercised, the put or the call? (In any scenario, either the put option or the call option is exercised, not both.) What does the investor do with his share of stock, and how much cash does he receive? What is the investor’s net gain? (The amount received for the share of stock minus the repayment to the bank.)
4.    If the stock price in three months time is more than $80, which option is exercised, the put or the call? What does the investor do with his share of stock, and how much cash does he receive? (The investor wrote the call option; the buyer exercises it against the investor.) What is the investor’s net gain?

Question: If this problem gave the risk-free interest rate as 8% with (i) annual compounding, (ii) semi-annual compounding, or (iii) continuous compounding, how do we solve it?

Answer: We solve for the interest rate over three months:

(i) Three month rate = 1.08¼ –1 = 1.94%

(ii) The six month rate is 8% / 2 = 4%, so the three month rate = 1.04½ –1 = 1.98%

(iii) The three month rate with continuous compounding is 8% / 4 = 2%, so the effective three month rate is ln(1.02) = 1.98%

Question: The net profit is small; is it worth using this investment strategy?

Answer: The investor does this for 100,000 shares, 100,000 put options, and 100,000 call options. A profit of $1 per share become $100,000 for 100,000 shares.

Question: Don’t we have to consider expenses as well?

Answer: We have two answers:

●    The expenses eliminate some of the profit, but not all.
●    The investor is an options trader, for whom we are determining marketing strategy. The expenses are paid by the clients of the options trader.

We do not consider expenses in the corporate finance on-line course, though they are discussed in practical investment courses.

Thus far I have

A) Value of call =  77-80 = -3 so the put = -3 + 80*1.02 - 77= $1.60

B) Buy the put option because the value of the call = -3, also the put seller gains at prices at market share prices above $75=80-5

1) since the investor borrows $77 his outflow @ time 0 is 0

2) for 3 months the investor repays $235.62

3) put option is excercised because the excercise price will be greater than the market share of the price, so the net gain is ???

4) I'm stuck because I'm still confused about 3

1. you stated call value =-3 , this is nonsense.
A call option is an OPTION. Meaning that you need to pay extra to get this feature. It can't be <0.

c + pv(k) = p+ S

we are given c =5, S=77, K=80, then pv(K) = 80/1.02
=> value of put = c + pv(K) - S

[NEAS: Yes]

MC:

1) agree

2) They give us the equation: $77*1.02% = $78.54. Why did you multiply by 3?

3) You're correct that the put option is exercised. So he sells at 80$. Subtract what he pays to the bank, $78.54, get a net profit of $1.46.

4) The call option is exercised. Whoever bought the call will force the investor to sell his stock for $80. The net gain for the investor will thus be the same, either way.

This exercise demonstrates why an option will never be underpriced. If it were, we would have a quick money scheme on our hands. (I apologize if I just sounded like Brealey and Myers. I also apologize for the font changes. I'm still trying to figure this thing out.

For Part A, why are people calculating the price of the call when it is given in the information. It says the value is $5. I think this means the formula should be:

p + 77 = 5 + 80/1.02     p = 6.43

[NEAS: Yes]

Let me know what you think or why you are calculating the call. What is the $5 given supposed to be used for.

Also, how would this change the answer for part B, I have you would prefer the put becuase the ture value, 6.43 is greater than the price paid which is said to be 5.00

KPlunk:
C-4) At the end of 3 months, if stock prce rises > 80, say S. The put option is worthless. The buyer of call option exercise his right. The investor is FORCE to sell his share at market price S (remember S>80), and pay the buyer the difference (S-80). Investor get to keep $80. Repay the loan of 77*91.02) = 78.54, his net gain is $1.46

Or you can think of, since the stock price > 80. The buyer of the call option wants the share (he call sell it immediiately for S>80, or keep in his investment portfolio), he exercise his right, pay the investor $80 to exchange a share worth more than $80.
The investor get $80. pay off the loan of 77*(1.02), leave him with $1.46

I am still having some trouble with figuring out the net gain of C) 4.

The investor is forced to sell, but is it at $80 or the market price.

It would make more semse to me if it were at $80 because the investor looses and the buyer is the one that gains the market price - $80.

And if this is true then the investor's net gain is 80-78.54 = $1.46.

I agree with you here.

The investor has both (1) sold a call option and (2) bought a put option.

The investor has effectively insured herself against any volatility in the stock price.

If the stock price is less than $80 then she exercises the put option and sells the stock at $80. If the stock price is more than $80 then whoever she sold the call option to will want to buy it at $80. So again she sells it at $80.

In either case the investor sells the stock at $80 and she pays the bank $78.54. There is a risk free gain of $80-$78.54= $1.46.

I get the same answer for both C3 and C4 ($1.46)

The question and answer gives the wrong rate of interest.  It should be 2% per quarter (.08/4) giving a 80/1.02 for the PV of the exercise price.  This is clearly stated in the problem where the risk free rate is 8% per annum, or 2% per quarter.  When the answer uses (1.08^.25) -1 = 1.0194 as the PV discount factor, it incorrectly interprets 8% as the effective annual rate.  This will be very confusing to some who may not be familiar with this topic.

[NEAS: Correct. The answer explains how different compounding intervals affect the solution.]

Ken

For part B, if the put has a price of $5, then

c + PV(X) = p + s  =>  c = $3.57

So, would we prefer either the put (since 6.43 > 5.00) or the call (since 5.00 > 3.57)?  I'm a bit confused.

[NEAS: Use the following logic.

If the call is truly worth $5, the put is worth $Z. If it is selling for $5, it is either overpriced or underpriced. We should buy under-priced securities and sell over-priced securities.

If the put is truly worth $5, the call is worth $Z. If it is selling for $5, it is either overpriced or underpriced. We should buy under-priced securities and sell over-priced securities.

Substitute the figures. The result: we should buy one of the options and sell the other option.]