Question

7) In this question the volatility is 0.4 and the risk free rate of return is 0.08. The payoff function at expiry is shown in the graph. At expiry the option pays nothing if the stock price is below the exercise price of $20 and it pays the stock price if the stock price is greater than or equal to $20. taunla with savo Set up the integral that will calculate the value of this option as a function of the stock price, S, and time until expiry, t. What does the integral evaluate to if there is 6 months left to expiry?

Answer 1

here we need to utilize the Black n Scholes Model for Options pricing

Option Price = Stock Price x Fn(d₁₎ - Strike Price x e^-rt x Fn(d₂)

d1 = (In(Si/x) + ((r +022)))/0 * vt

d2 = (In(Si/x) + ((r - 02/2)))/0 * vt

Si = Stock Price Initial

Strike Price = Price at end of expiry

r= risk free rate

t=time period

Fn = Cumulative Normal Distribution function as hilited above

As per problem Strike Price = $20 @=0.4 and t =6/12=0.5 so Integral = as below

if initial price = 20 too then to be solved as follows ( because initial price value not mentioned):

Si*[(log(Si/20)+((0.08+0.42/2)0.5))/0.4* 0.5 minus((20 * € (0.08 * 0.5)) *[(log(Si/20) + ((0.08 – 0.42/2)0.5))/0.4* 0.5

{20*(0+ ( 0.08+0.08)*0.5)/0.4*0.707 MINUS 20 e^-0.04  * (0+0)*0.5/0.4*0.707

{20 (0.08/0.2828) MINUS 0.96*0

Approx answer = 5.65 with the ASSUMPTION initial price = 20 same as price when it is exercised

If initial value is different please plug it into above formula instead of Si