Solutions to Assignment 5
Chapter 16
2. Only statement C is correct
4. Call option with X=75
cost = 8 1/8 ; payoff = 5; profit = -3 3/8
Put option with X=75
Cost = 3; payoff = 0; profit = -3
5. Walmart stock price = 100
Call option with X=100; six months till maturity; call premium (C) = $10
$10,000 available for investment
Value of investment at different stock prices:
80 100 110 120
Alt A: Buy 100 shares of W $8000 $10000 $11000 $12000
Alt B: Buy 100 C 0 0 $10,000 $20000
Alt C: Bills + Options
($9000 invested at 4%;
+ 100 calls) $9,360 $9,360 $10,360 $10,360
Rates of returns for the different strategies at different stock prices
80 100 110 120
Alt A: Buy 100 shares of W -20% 0% 10% 20%
Alt B: Buy 100 C -100% -100% 0% 100%
Alt C: Bills + Options
($9000 invested at 4%;
+ 100 calls) -6.4% -6.4% +3.6% +13.6%
For questions 9, 10, and 13, please see me if you have any problems.
Chapter 17:
4. Call option with the following characteristics:
X = 50; T=1 year; standard deviation = 20%; T-bill rate = 8%
What happens to the hedge ratio as stock price increases? Try three different stock prices: $45, $50, and $55.
Hedge ratio = N(d1) from the Black-Scholes equation (refer to book or notes)
If S=45, d1 = -0.0268 and N(d1) = 0.489309
If S = 50, d1 = 0.5 and N(d1) = 0.6915
If S = 55, d1 = 0.97655 and N(d1) = 0.8356
This means the price of the call becomes more sensitive to changes in the price of the underlying stock at higher stock prices.
5. Two-state put option
S = 100; X=110; 1+r = 1.1
The stock price today is $100, At the end of the year, stock price will be either $130 or $80
If the stock price increase to $130, put option will not be exercised so payoff =0
If the stock price decreases to $80, put option will pay $30 (i.e. buy the stock in the open market for $80 and exercise the put option to sell the stock for X=110)
The hedge ratio (ratio of put option payoffs to stock payoffs)
= (0-30)/(130-80) = -30/50 = -3/5
So I will create the following portfolio
CF today CF one year from today
If S=130 If S=80
Buy 3 Shares -300 3*130 = $390 3*80 = $240
Buy 5 puts -5P 0 5*30 = $150
TOTAL -(300+5P) $390 $390
Since the payoff is the same in either outcome, this is a riskless portfolio which should earn 10% rate of return. So the most I would be willing to pay for it today is the present value of $390 discounted at 10%
= 390/(1.1) = $354.54
In equilibrium, 300+5P = 354.54 So P = $10.91
7. T=0.5 years; X=50; S=50; r=10%; std dev = 0.5
Use the Black-Scholes equation for call premium (refer to book or notes).
The call premium = $ 8.13
[Steps involved are
1. calculate d1 (=0.3181) and N(d1) (=0.6248)
2. calculate d2 (=-0.03536) and N(d2) (=0.4859)
3. calculate C which turns out to be $8.13
14. Call option with a high exercise price will have a lower hedge ratio.
20. Hedge ratio of at-the-money call on IBM = 0.4
Hedge ratio of at-the-money put on IBM = -0.6
At-the-money straddle is formed by buying 1 call and 1 put. The sum of the two individual hedge ratios gives the hedge ratio of the straddle
0.4 + (-0.6) = -0.2
23. A collar is formed by buying 1 shares at S=50; buying one put with the exercise price of $45; and shorting 1 call with the exercise price of $55.
In addition, we are told that for X=45, the N(d1) = 0.6
And for X=55, the N(d1) = 0.35
What happens to the value of the collar if there is a small rise in the stock price?
We need to calculate the sum of the individual deltas to determine the delta of the portfolio.
Component of collar Individual delta
Stock 1
Long P (x=45) N(d1)-1 = 0.6-1 = -0.4
Short C (x=55) -N(d1) = -0.35
The delta of the share is 1 since this asset will increase in value by $1 for every $1 increase in the stock price
The delta of a put is given by N(d1) – 1. The exercise price of the put is $45 so the relevant N(d1) =0.6 and the delta of the put then is –0.4
The delta of a call is given by N(d1). The exercise price of the call is $55 so the relevant N(d1) = 0.35. However, we are shorting the call, so the delta is –0.35
Adding the three individual delta yields 0.25. So, if the stock price increases by $1, the value of the collar will increase by $0.25.
What happens to the delta of the collar if S becomes very large?
Delta of the collar approaches zero. Delta of the stock remains 1. However, at high stock prices, the put is unlikely to be exercised (so its delta will approach 0) while the call is very likely to be exercised (so its delta will approach –1).
What happens to the delta of the collar if S becomes very small?
Delta of the collar will be zero. Again the delta of the stock remains 1. At low prices, the put is very likely to be exercised (so its delta will approach 1) while the call is very unlikely to be exercised (so its delta will approach 0)