The European asset-or-nothing call option pays one unit of underlying stock if the stock price at maturity is larger than the strike, and pays 0 otherwise. The European asset-or-nothing put option pays one unit of underlying stock if the stock price at maturity is smaller than the strike, and pays 0 otherwise.

(a) Use risk-neutral pricing to find the Black-Scholes value of an asset-ornothing call option.

(b) State the put-call parity for the asset-or-nothing options. Then, use it to find the Black-Scholes value of a asset-or-nothing put option

(c) Assuming the Black-Scholes model for the stock price, derive the formulae for the Delta of these asset-or-nothing options.

(d) Find the limit of the Black-Scholes values of these options as 0 and -> infiniti.