Assume that all the assumptions which lead to the Black-Scholes-Merton PDE hold.

(a)Suppose that it is now ???? = 0 and ???? > 0 is some later time. Carefully show that the 

probability in a risk-neutral world satisfies the equation, ????(???????? > ????) = ????(????2). Here, ???????? is a 

stock’s price at time ????, ????(????) is the cumulative distribution of a standard normal random 

variable, and ????2 =     (ln(s0/k )+(????− ????^2/2  )???? )/ (????√???? )

(b)  Consider a European call with expiration date ????, underlying stock price ????(????), volatility ????, risk-free rate ????, and strike price ????. If ????(????0, 0) is its price at time ???? = 0, then compute 

lim????(????0,0). σ→0