Solutions to Exercises 1 and 2
See chapter text for answers
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:
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
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??
Spread of share outcomes
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:
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
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
0.2376820.282843? ?? 0.2828432
Now for the main formula:
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
0.3654650.254558? ?? 0.2545582
So the fair value of the put P is
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
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
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).