The Black$-$Scholes formula for a European Call option is given by

$C(S,t)=S(d(S,t))-E{e}^{-r(T-t)}(d(S,t)-\sqrt[\phantom{2}]{T-t}),$

where $d(S,t)=\frac{log(\frac{S}{E})+(r+\frac{1}{2}{}^2)(T-t)}{\sqrt[\phantom{2}]{T-t}}$ and $E,T,,$ and $r$ denote the exercise price,

expiry, volatility, and riskless interest rate, respectively.

$=\frac{de{l}^{2}C}{del{S}^2}2.$ A European asset$-$or$-$nothing call option CAoN is similar to a European call option, it also solves the Black$-$Scholes PDE but has the payoff function $($final condition$)($a$)$ Derive the Black$-$Scholes formula for the European asset$-$or$-$nothing call option. Pro$-$ ceed along the lines of the corresponding calculations for the European call option presented in the lectures, starting out from the Black$-$Scholes PDE with the payoff given above as final condition. $($b$)$ Use put$-$call$-$parity to derive the Black$-$Scholes formula for the European asset$-$or$-$ nothing put PAoN which is similar to a European put option but has the payof function $($final condition$)$The Black$-$Scholes formula for a European Call option is given by

$C(S,t)=S(d(S,t))-E{e}^{-r(T-t)}(d(S,t)-\sqrt[\phantom{2}]{T-t}),$

where $d(S,t)=\frac{log(\frac{S}{E})+(r+\frac{1}{2}{}^2)(T-t)}{\sqrt[\phantom{2}]{T-t}}$ and $E,T,,$ and $r$ denote the exercise price,

expiry, volatility, and riskless interest rate, respectively.

$=\frac{de{l}^{2}C}{del{S}^2}$