The Black-Scholes Formula: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 33

Problem BSF3. The stock of Blackscholesian Co. currently sells for $1500 per share. The annual stock price volatility is 0.2, and the annual continuously compounded risk-free interest rate is 0.05. The stock's annual continuously compounded dividend yield is 0.03. Use the Black-Scholes formula to find the price of a call option on Blackscholesian Co. stock with strike price $1600 and time to expiration of 3 years.

Solution BSF3. We use the formula C(S, K, σ, r, T, ∂) = Se^-∂TN(d₁) - Ke^-rTN(d₂), where we know that S = 1500, K = 1600, r = 0.05, ∂ = 0.03, σ = 0.2, and T = 3. From Solution BSF1, d₁ = 0.1601034988. From Solution BSF2, d₂ = -0.1863066628.

We find N(d₁) via MS Excel using the input "=NormSDist(0.1601034988)" = N(d₁) = 0.563600238.

We find N(d₂) via MS Excel using the input "=NormSDist(-0.1863066628)" = N(d₂) = 0.426102153

Thus, C(S, K, σ, r, T, ∂) = 1500e^-0.03*30.563600238 - 1600e^-0.05*30.426102153 =

C(S, K, σ, r, T, ∂) = $185.8385153

Problem BSF4. The stock of Blackscholesian Co. currently sells for $1500 per share. The annual stock price volatility is 0.2, and the annual continuously compounded risk-free interest rate is 0.05. The stock's annual continuously compounded dividend yield is 0.03. Find the price of a put option on Blackscholesian Co. stock with strike price $1600 and time to expiration of 3 years.

Solution BSF4. We recall that, since the call price is known from Solution BSF3, we can use put-call parity to get the put price: P(S, K, σ, r, T, ∂) = C(S, K, σ, r, T, ∂) + Ke^-rT - Se^-∂T

where C(S, K, σ, r, T, ∂) = 185.8385153, S = 1500, K = 1600, r = 0.05, ∂ = 0.03, and T = 3.

Thus, P(S, K, σ, r, T, ∂) = 185.8385153 + 1600e^-0.05*3 - 1500e^-0.03*3 =

P(S, K, σ, r, T, ∂) = $192.0744997

Problem BSF5. The stock of Blackscholesian Co. currently sells for $1500 per share. The annual stock price volatility is 0.2, and the annual continuously compounded risk-free interest rate is 0.05. The stock's annual continuously compounded dividend yield is 0.03. Within the Black-Scholes formula for the price of a put option on Blackscholesian Co. stock with strike price $1600 and time to expiration of 3 years, find the value of N(-d₂).

Solution BSF5. We use the formula N(-x) = 1 - N(x). We know from Solution BSF3 that N(d₂) = 0.426102153. Thus, N(-d₂) = 1 - N(d₂) = 1 - 0.426102153 = N(-d₂) = 0.573897847

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.