Black & Scholes model Exercise 3.5. A cash-or-nothing option on a share S with strike K is an option where the holder receives a fixed amount K at maturity T if ST — K > 0. The price of the option at...

Black & Scholes model
Exercise 3.5. A cash-or-nothing option on a share S with strike K is an option where the holder receives a fixed amount K at maturity T if ST — K > 0. The price of the option at time 0 is given by EQ(CrT K • • • 1/47.>K1) = Keg. 4.(d_). Show that 4(d_) is the probability that the option is in the money at expiry. Under which probability distribution is the expectation taken?
Exercise 3.6. The objective of this exercise is to develop the Black-Scholes formula by deriving a closed-form solution for the risk-neutral expectation of the payoff of a call option. Let
(112)a2)t±crWt St = Soe(' be a Geometric Brownian motion with constant mean rate of return r and constant volatility a > 0.