Question

Suppose that the price today (t = 0) of a stock S is 50. With four periods, you know that in each period the price goes up 20% or down 20%. In each period you also have access to a bank account, which pays constant interest of 5% (a dollar invested in the bank account at time t returns 1.05 at time t + 1). 1. Find the stock price tree up to date t = 4. What are the possible payoffs of a call option with strike price of 42 that matures at date t = 4? What is the price of this option at date t = 0?

Answer 1

Suppose that the price today (t = 0) of a stock S is 50. In four periods, the price can go up 20% or down 20%.

Therefore, the price after 1st Period=

50*1.20 and 50*0.80

=60 or 40

The price after 2nd Period if is 60,, The price after 3rd Period=

60*1.20 and 60*0.80

= 72 or 48

The price after 2nd Period if is 40,, The price after 3rd Period=

40*1.20 and 40*0.80

= 48 or 32

The price after 3rd Period if is72,, The price after 4th Period=

72*1.20 and 72*0.80

= 86.40 or 57.60

The price after 3rd Period if is 48,, The price after 4th Period=

48*1.20 and 48*0.80

= 57.60 or 38.40

The price after 3rd Period if is 32,, The price after 4th Period=

32*1.20 and 32*0.80

= 38.40 or 25.60

If the a call option with strike price of 42 that matures at date t = 4, the possible pay offs at various probable prices after 4th year shall be=

At,

86.40= 86.40- 42= 44.40

57.60 = 57.60- 42= 15.60

38.40 the call option will not be exercised as the price is below the strike price of 42 payoff =0

25.60 the call option will not be exercised as the price is below the strike price of 42 payoff =0

Since the Bank interest rate is 5% the current price of call option can be calculated as

Strike Price of option after 4 years pulled down to current year

that means the present value of the Strike price of the call option by the current market interest rate.

= 42 / (1+5%)4

= 34.55 (approx)