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# Chapter 9

By Debbie Daniels,2014-07-10 06:14
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Chapter 9

Chapter 9

Solutions to Exercises 1 and 2

Solution to Exercise 3(a)

The intrinsic value of the call option is the greater of (a) the current share

price minus the present value of the exercise price and (b) zero. With the risk

free interest rate at 4% pa, the present value of the exercise price of 650p is

about 644p, so the intrinsic value is (697 - 644)p = 53p. The rest of the value

of the call option is made up of its time value, which is (58 - 53)p = 5p.

Solution to Exercise 3(b)

Using the principle of Put-Call Parity (equation 9.1 on p335), we can write:

SPCPVX?????

And

697P58644???

P5?

So the fair price of a put option with the same exercise price and date as the

call would be 5p.

Solution to Exercise 3(c)

Mr Wellie has an option to buy for 650p a share whose market value is now

only 625p, so he will not exercise it. If he is still interested in buying the

shares, he can buy them in the market for 625p each. His overall loss on the

option transaction is the 58p premium he paid, or ?580 on 1,000 shares.

Solution to Exercise 3(d)

Ms Stiefel has an option to sell for 650p a share whose market value is now

only 625p, so she will exercise it. She makes a profit of 25p per share on the

share itself, and a net profit of 20p per share after paying the premium, a total

net profit of ?200 on 1,000 shares.

Solution to Exercise 4(a)(i) and (ii)

If the investor buys 2,000 shares at 320p, and the price subsequently rises to

350p, he will make a profit of (2,000 x 30)p = ?600.

If the investor buys 280p call options on 2,000 shares at a premium of 50p each, and the price of the option subsequently rises to 80p, he will make the same profit of (2,000 x 30)p = ?600.

Solution to Exercise 4(b)

The principal advantage of buying options is that the investor cannot lose more than the premium of 50p per share, whereas if he buys the shares and the price goes down below 280p, he is exposed to the full amount of the loss. Conversely, if the share price goes up, his percentage return on the money actually invested is much greater for the option than for the share.

Solution to Exercise 5(a)

We follow the same method as explained in the text, starting with Step 2 on page 327

Step 2 Calculate spread of outcomes from purchase of call

option

If the share value goes up to 625p, the result is a profit of 125p. If the share goes down to 400p, the option will not be exercised and will expire worthless; the outcome is therefore neither profit nor loss. So the result will be either a profit of 125p or a profit of zero, and the spread between the possible outcomes is 125p.

Step 3 Calculate the equivalent share portfolio

We then calculate how many shares we would have to buy in order to replicate precisely the same spread of outcomes. If we were to buy one share at 500p, the upstep would produce a profit of 125p and the downstep a loss of 100p so the spread would be 225p. This is too wide a spread, so we need to buy only a fraction of a share. The size of the required fraction is obtained by dividing the spread of outcomes for one option by the spread of outcomes for one share.

Spread of option outcomes 1250???0.5556 125100??

We can prove this as follows:

Cost of 0.5556 shares at 500p per share = 277.8p

Value of 0.5556 shares at 625p (upstep) = 347.2p Value of 0.5556 shares at 400p (downstep) = 222.2p

Difference between upstep and downstep values = 125.0p

Step 4 Calculate bank loan

The next step is to calculate how much the share purchaser must borrow at

the outset, such that when it is repaid with interest in three months’ time, it will exactly use up all the proceeds of selling the fractional share at the lower

price leaving the purchaser in exactly the same position as the option buyer, with nothing at all. So we assume that our share purchaser partially finances

his fractional share purchase by borrowing the present value of 222.2p. Using

222.2pthe risk-free rate of interest, this gives us a value for the initial bank loan of ?222.0p 0.043? ??1???12??

The outcomes for the share purchaser are now identical with those for the

option purchaser. In the event of the downstep, the share purchaser will be

left with nothing after repaying his bank loan with interest. In the event of the

upstep, he will have a share worth 347.2p, he will have to pay out 222.2p to

redeem his borrowing, and will be left with exactly 125p profit.

Step 5 Calculate option premium from initial investment

We can now calculate the option value. The share purchaser has borrowed

222.0p from the bank and has bought a fractional share for 277.8p. The

shortfall of 57.8p he must provide from his own resources as his initial net

investment in the overall strategy. On the basis that the outcome of this

strategy is, on the assumptions we have made, in all cases identical with the

outcome of purchasing of a call option, the value of a call option must be the

same as the cost of the alternative strategy, otherwise arbitrage would bring

the prices together. So the value of the call option must also be 57.8p.

The risk-neutral approach

We assign the probability p to the upstep and the probability (1 p) to the

downstep. Since there is an absolute certainty (under our chosen

assumptions) that one of these outcomes or returns will happen, then the

probability-weighted sum of these returns must be equal to the risk-free rate

of interest for the period in question. So, in our example, with the risk-free

rate set at 4% pa or 1% for three months:

??p0.251p0.200.01????????????

0.25p0.200.20p0.01???

0.45p0.21?

p0.4667?

So the probability of the upstep is 0.4667 or 46.67%. We can now use this to

calculate the value of the option. There is a 0.4667 probability that it will have

a value of 50p on expiry, and a (1 0.4667) or 0.5333 probability that it will

125p0.4667?expire worthless. Its expected value is therefore the present value of 125p ?57.8p times 0.4667 or 0.043???1? ??12??

This is exactly the same result as we obtained from the replicating portfolio

approach.

Summary of results:

Fair value of the call: 57.8p

Delta or hedge ratio: 0.5556

Size of bank loan: 222p

Probability of exercise: 46.67%

Solution to Exercise 5(b)

As we would expect, the substantial increase in the volatility of the share has

increased the value of the at-the-money call from 26.2p to 57.8p.

Solution to Exercise 6(a)

We need to use three equations. First the Black-Scholes formula itself:

?rt (9.2) CSNdXeNd??????12

And then the equations for the N(d) values: i

?rtlnSXe??t? d?? (9.4) 12t?

?rtlnSXe???t d?? (9.5) 22?t

Solving the N(d) values first: i

??0.100.5ln30(40e)??0.40.5d??120.40.5

0.2376820.282843? ?? 0.2828432

0.698907??

Nd0.242305?1??

??0.100.5ln30(40e)??0.40.5d??220.40.5

0.2376820.282843??? 0.2828432

0.981750??

Nd0.163112? 2??

Now for the main formula:

C300.242305400.9512290.163112????? ????

C7.2691506.2062751.062875p???

So the fair value of the call is about 1.06p.

Solution to Exercise 6(b)

Substituting values of (i) 40p and (ii) 50p for the share price S into the above

equations gives values for the call option C as follows:

S = 40p: C = 5.43p

S = 50p: C = 13.04p

Note how the time value is highest at the point of maximum uncertainty -

when the option is at-the-money.

Solution to Exercise 7(a)

The Black-Scholes equation for the fair value of a European-style put on a

non-dividend paying option is:

?rtPXeNdSNd???? (9.13) ????21

Solving the N(-d) values first: i

??0.080.5ln20(30e)??0.360.5d??120.360.5

0.3654650.254558? ?? 0.2545582

1.308406??

Nd0.904632??1??

??0.080.5ln20(30e)??0.360.5d??220.360.5

0.3654650.254558??? 0.2545582

1.562964??

Nd0.940969??2??

So the fair value of the put P is

??0.080.5P30e0.940969200.9046329.02p?????

Solution to Exercise 7(b)(i)

For the classic case of using the Black-Scholes model to value options on

non-dividend-paying shares, the share’s forward price (price on expiry allowing for the interest rate) will lie above the spot (current) price because of

the time value of money. Hence, a put option with an exercise price well

above the spot price will be more in the money with respect to the spot price

than to the higher forward. Thus, one would expect the Black-Scholes

formula, when applied to European options only to significantly underestimate

the value of a substantially in the money American put option where the

intrinsic value with respect to the spot price can be immediately realized by

early exercise.

Solution to Exercise 7(b)(ii)

By contrast, the Black-Scholes model applied to substantially out of the

money put options will give prices close to the real value.

Solution to Exercise 8(a) and (b)

As noted in the text, in 2004 BP was paying regular dividends equivalent to a

dividend yield of approximately 3% p.a. on the current share price of 492.5p.

If we test the more actively traded option pairs (puts and calls with the same

exercise price and expiry), we find that - allowing for bid-offer spreads - the

left hand side of the put-call parity equation 9.1

SPCPVX??? ??

equates to a figure only very slightly below the undiscounted exercise price.

For instance, if we look at two December 2004 pairs of contracts (those with

exercise prices of 460p and 500p respectively) we find that (S + P - C) gives

us values of 458.5p and 498.5p - in each case this is 3.5 - 4p higher than the

discounted present value of the exercise price. What we have overlooked is

that the dividend stream at 3% pa is expected almost to neutralise the cost of

carry at the risk-free rate of 4% pa, effectively reducing the cost of holding the

share (and hence the value of the left hand side of the equation).

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