This exercise concerns some properties of the Black-Scholes formulas (2.2.3) and (2.2.4).

(a) Consider a call option. If T is very small, the option will be ready to be exercised practically immediately, and hence its price should be close to (S₀ −K)+. Show that, indeed, the price in (2.2.3) converges to this limit as T →0.

(b) If σ → 0, the stock is becoming riskless, and in the risk neutral world, its price will grow to S₀e^δT . Then, at time T, the value of the call option will be (S₀e^δT −K)+, and the present value is e−δT (S₀e^δT −K)+ = (S₀−Ke^−δT )+. Show that the Black-Scholes formula is consistent with this reasoning.

(c) Carry out the reasoning analogous to those in Exercises 22a and 22b for a put option and find the corresponding limits.

(d) If S₀ is very large while K is not, the put option is unlikely to be exercised, so the present value of the put option is close to zero. Show that this is consistent with the Black-Scholes formula.