Consider a European option on a non-dividend-paying stock when the stock price is $50, the exercise price is $45, continuously compounded risk-free interest rate is 3%, volatility is 35% per annum, and time to maturity is 3 months.

 a. Find the value of a put option at the strike price of $45 using the Black-Scholes option pricing model. Show all steps.

 b. Find the value of Delta for this option.  

c. Using just delta, what should be the change in the price of the put option if the price of the underlying stock decreases by $0.50? 

d. What is the change in the value of the put option if time changes by 1 day while all other variables remain the same in the Black-Scholes Option pricing model?