Question

$125.00, and r = 5%. Find the Black-Scholes formula for the option paying $10.00 in 3 months if S(T) S Ki or if S(T) 2 K2, and zero otherwise, in the 7. Let S(0) = $100.00, K = $92.00, K2 Black-Scholes continuous-time model.

Answer 1

Payoff = A = $ 10.00

Risk free rate, r = 5% = 0.05

T = 3 months = 3 / 12 = 0.25 year

PV(A) = Present value of A

P[S(T) ≤ K1] = Probability that stock price S(T) ≤ K1

P[S(T) ≥ K2] = Probability that stock price S(T) ≥ K2

Price of the call option = PV(A) x {P[S(T) ≤ K1] + P[S(T) ≥ K2]}

In the Black Scholes continuous model,

• PV(A) = Ae^-rT
• P[S(T) ≤ K1] = 1 − N(d₂(K1))
• P[S(T) ≥ K2] = N(d₂(K2))

Hence, Price of the call option = PV(A) x {P[S(T) ≤ K1] + P[S(T) ≥ K2]}

= Ae^-rT x [1 − N(d₂(K1)) + N(d₂(K2))] = 10e^{-0.05 x 0.25} x [1 − N(d₂(K1)) + N(d₂(K2))] =  9.8758 x [1 − N(d₂(K1)) + N(d₂(K2))]

Hence, the Black Scholes formula for price = 9.8758 x [1 − N(d₂(K1)) + N(d₂(K2))]