Question

Show that when T – t becomes 0, the Black and Scholes call price become max{0, S – K}  

Answer 1

The T-t > 0 means when the option is at expiration date.

The Call option is priced as Maximum of {0, Stock price (S) – Strike price (K)}

Further simplifying;

If Stock Price is greater than the Strike price that results in pay off for call option. Suppose, Stock price = 120 and Strike price = 100, then by formula of Call option > Max {0 , 120 – 100} = 20.

If Stock Price is lower than the Strike price that results no pay-off for call option i.e > 0. Suppose, Stock price = 80 and Strike price = 100, then by formula of Call option > Max {0, 80 – 100} = 0; Since -20 is lower than 0 hence pay-off is 0.

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For academic purpose only:

I am trying to show you how Black Scholes Model applied in real world for calculating the call option price or put option price with available variables as below:

For European options:
S = Stock price =	60
K = Strike price =	50
r = rate =	5%
e = exponential value = exp(.) =	2.718282
t = time =	3
s = standard deviation or volatility =	10%

* N(d1) is Normal distribution probability value

* N(d2) is Normal distribution probability value : Use normal distribution table             

C = Stock price – Strike price or C = Max (0,S – k) > Modified form: C = S*N(d1)-K*exp(-r*t)*N(d2)                                                                   

d1 = (Ln(S/(K*exp(-r*t))+0.5*s^2*t)/(s*t^0.5)                                                             

=(LN(60/((50*EXP(-0.05*3))))+0.5*0.1^2*3)/(0.1*3^0.5)                                                         

d2 =2.005261943           

Hence, N(d1) = 0.977532474     

                                                                       

d2 = d1 - (s*t^0.5)                                                                   

=2.005262-(0.1*3^0.5)                                                           

d2 =       1.832056919      Hence, N(d2) = 0.96652853       

                                                                       

C = S*N(d1)-K*exp(-r*t)*N(d2) = 60*0.977532-50*exp(-0.05*3)*0.966529                                                         

C = 17.057                                                                                               

Value of call option = 17.057                                 

                                                                       

Using put call parity formula:                                                             

P = C - S + K*exp(-r*t) = 17.057- 60 + 50*EXP(-0.05*3)                                                              

P =         0.092                  

Value of put option = 0.092