Let C be the value of a European call option given by the Black-Scholes formula.

Prove that each of the following limits exist and give a financial interpretation of each result.

(a) limσ→0 C(t, s,K, T, r, σ) = maxs − K−r(T−t), 0.

(b) limσ→∞ = C(t, s,K, T, r, σ) = s.

(c) limt→T C(t, s) = max(s − K, 0).

(d) Set d1,t = ln{S(t)/K}+r(T−t)+σ2(T−t)/2

σ

√

T−t

, and set d2,t = d1,t − σ

√

T − t.

Deduce from

(c) and the Black-Scholes formula that at time t, if one has N(d1,t) shares of the asset and if one borrows Ke−r(T−t)N(d2,t) at the risk-free rate, then the final value of the portfolio is the payoff, i.e., one can replicate exactly the payoff of the option.