Option Valuation: Black-Scholes

Option valuation: Black-Scholes

a. (15%) Suppose you have the following data:

A put option on a non-dividend paying stock, that currently trades at $43. X = $ 45. The risk- free interest rate is 1,5% per year. The yield curve is flat. Volatility is 38%. The option has 18 months to go.

Determine the value of this option using Black-Scholes, split this up into intrinsic value and time value. Then calculate the value of this option if it has only 3 months to go; also split this up. Assume the same inputs are still valid, except the time to expiration. In both cases, show all steps of your calculations!

Spotprice (S): 43

Strikeprice (X): 45

Riskfree rate (Rf): 0,015

Volatility: 0,38

Maturity: 1,5 years

d1 =(LN(43/45)+(0,015+0,5*0,38^2)*1,5)/(0,38*SQRT(1,5)) = 0,183363

N(d1) = =NORMSDIST(d1) = 0,572743

d2 = 0,183363-0,38*SQRT(1,5) = -0,282040

N(d2) = NORMSDIST(-0,282040) = 0,388956

Put Value =45*EXP(-0,015*1,5)*(1- 0,388956) - 42*(1-0,572743) = 8,51

Maturity changed to 0,25

d1 =(LN(43/45)+(0,015+0,5*0,38^2)*0,25)/(0,38*SQRT(0,25)) = -0,124539

N(d1) = =NORMSDIST(d1) = 0,450444

d2 = 0,183363-0,38*SQRT(0,25) = -0,314539

N(d2) = NORMSDIST(d2) = 0,376556

Put Value =45*EXP(-0,015*0,25)*(1- 0,376556)- 42*(1-0,450444) = 4,32

b. (15%) Now construct a graph with the value of the call options for the stock mentioned in part a, with share prices 20, 25, 30, 35, 40, 45, 50, 55, 60, 65 and 70. Do this twice, once with options with a time to maturity of 1 year, once with options with a time to maturity of

0.15 years. (volatility remains 0.38, the strike price 45, the risk-free rate 0.015). Comment on the differences between the two graphs.

Make sure you put the value of the share price on the horizontal axis, and that of the calls on the vertical axis. create a smooth line through the dots, and show intrinsic value in the graph as well.

Finally construct a new graph with only the time value of each option.

NB: for this part, the use of excel is allowed and even encouraged!

c. (20%) In each of the following cases, calculate first the current value of the option, and then determine how much of a chance in the inputs (current share price, risk-free rate, volatility, time to maturity)