Question 1:
The current price price of the share is $30 and may in one timestep go either up to $36 or down to $24. The strike price of a call option is $33. Assume an annual interest rate of 6%, and a time of 6 months to the exercise date.
Calculate the value of a call option using the one step binomial model. Use a portfolio of one option minus a shares. Show all working. Be sure to indicate how many shares should you buy or sell for each option in order to hedge.
Question 2:
Imagine that the current price price of a share of "Bonds Inc", So is $40. Using an annual interest rate of 6%, and a time of 6 months to die exercise date.
(a) Calculate the expected value of the share, Se, at the exercise date.
(b) Imagine that "GovBonds Pty Ltd" has no employees, no activities, and has assets consisting only of government bonds. In this case, what would be the volatility of the share, and what would be the value of a call option at a strike price of X = Se.
(c) Using a strike price X=Se, and choosing the upper value of the share, Su to be f Se, and the lower value of the share Sd to be (1/f) Se, where f = 1.02. Calculate the value of a call option Vco using a portfolio of one option minus a shares using the one step binomial model. Show all working.
(d) Caclulate the correct value of f, for an annual volitility of 10%, and repeat (c) using that value off.
Question 3:
Calculate the value of the call option from Question 2 (d) using the Black Scholes formula. Calculate this by hand and show all working. You should check your answers (including the subcalculations) using the function bs() given in "useful code snippets".
Question 4:
For this question your submission take the form an Ft code file called "q4.r", which the marker will inspect and run. Running the code by > source ( "q4 r" ) ; should produce the required plot. The X and Y scales on the plot should be suitable. The code should be properly indented and easy to read.
Plot the value of a put option and a call option together on the same plot varying the annual volatility between 0% and 20%. The number of days to the exercise date is 40 days. Use an annual interest rate of 4%, a strike price of $61, a current share price of $60. I strongly suggest using the Black-Scholes formula to calculate the option values.
Question 5:
Suppose that an equity starts at price So , and then increases or decreases by the ratios u > 1, or d < 1 respectively. Furthermore, suppose the risk-free interest rate is r and we wish to price an option with lifetime n with exercise price X. The specific values for So , u, d, r, and X are given in the following table. Use the binomial tree method to compute the values of both European and American put options, as specified by a row in the table.
You should should use the binomialput() function given in the week5 scripts. Show the input that you used, and the output that was generated.
|
|
n
|
So
|
u
|
d
|
X
|
r
|
|
a
|
3
|
90
|
1.1
|
0.9
|
98
|
0.05
|
|
b
|
3
|
90
|
l.3
|
0.7
|
98
|
0.05
|
|
|
4
|
85
|
1.2
|
0.8
|
80
|
0.06
|