The Black-Scholes Formula for Options on Stocks with Discrete Dividends: Practice Problems and Solutions

This is Section 35 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here. See Section 9 here. See Section 10 here. See Section 11 here. See Section 12 here. See Section 13 here. See Section 14 here. See Section 15 here. See Section 16 here. See Section 17 here. See Section 18 here. See Section 19 here. See Section 20 here. See Section 21 here. See Section 22 here. See Section 23 here. See Section 24 here. See Section 25 here. See Section 26 here. See Section 27 here. See Section 28 here. See Section 29 here. See Section 30 here. See Section 31 here. See Section 32 here. See Section 33 here. See Section 34 here.

When stocks pay discrete dividends, we recall that the following relationship holds between the prepaid forward price and the stock price:
F^P_0,T(S) = S₀ - PV_0,T(Div)

For time to expiration T and annual continuously compounded risk-free interest rate, the following equality still holds with respect to the prepaid forward on the strike asset:

F^P_0,T(K) = Ke^-rT.

Making the substitutions above, we can now use the Black-Scholes formula from Section 34 - the formula that uses prepaid forward prices.

Thus, the Black-Scholes formula for the call price is

C(F^P_0,T(S), F^P_0,T(K), σ, T) = F^P_0,T(S)N(d₁) - F^P_0,T(K)N(d₂)

where d₁ = [ln(F^P_0,T(S)/F^P_0,T(K)) + 0.5σ²T]/[σ√(T)] and d₂ = d₁ - σ√(T)

The Black-Scholes formula for the put price is

P(F^P_0,T(S), F^P_0,T(K), σ, T) = F^P_0,T(K)N(-d₂)- F^P_0,T(S)N(-d₁)

We can also get the put formula via put-call parity:
P(F^P_0,T(S), F^P_0,T(K), σ, T) = C(F^P_0,T(S), F^P_0,T(K), σ, T) + F^P_0,T(K) - F^P_0,T(S)

Meaning of variables:

S = current stock price.

K = strike price of the option.

C = call option price.

P = put option price.

σ = annual stock price volatility.

r = annual continuously compounded risk-free interest rate.

T = time to expiration.

∂ = annual continuously compounded dividend yield.

PV_0,T(Div) = present value of future discrete dividends.

F^P_0,T(S) = price of prepaid forward on the stock (underlying asset).

F^P_0,T(K) = price of prepaid forward on the strike asset.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 12, p. 380.

Problem BSFOSDD1. The stock of Auspicious Co. currently trades for $221 per share. The stock will pay a dividend of $40 in 2 years and another dividend of $32 in 6 years. The annual continuously compounded risk-free interest rate is 0.05, and the annual price volatility relevant for the Black-Scholes equation is 0.3. Call options are written on Auspicious Co. stock with a strike price of $250 and time to expiration of 8 years. Calculate d₁ in the Black-Scholes formula for the price of one such call option.

Solution BSFOSDD1. First, we convert the asset and strike price terms into prepaid forward prices. F^P_0,T(S) = S₀ - PV_0,T(Div) = 221 - 40e^-2*0.05 - 32e^-6*0.05 = F^P_0,T(S) = 161.1003202

F^P_0,T(K) = Ke^-rT = 250e^-8*0.05 = 167.5800115. Here, T = 8 and σ = 0.3.

Now we use the formula d₁ = [ln(F^P_0,T(S)/F^P_0,T(K)) + 0.5σ²T]/[σ√(T)] =

[ln(161.1003202/167.5800115) + 0.5*0.3²*8]/[0.3√(8)] = d₁ = 0.3777910782

Problem BSFOSDD2. The stock of Auspicious Co. currently trades for $221 per share. The stock will pay a dividend of $40 in 2 years and another dividend of $32 in 6 years. The annual continuously compounded risk-free interest rate is 0.05, and the annual price volatility relevant for the Black-Scholes equation is 0.3. Call options are written on Auspicious Co. stock with a strike price of $250 and time to expiration of 8 years. Calculate d₂ in the Black-Scholes formula for the price of one such call option.

Solution BSFOSDD2. We use the formula d₂ = d₁ - σ√(T), where T = 8, σ = 0.3, and

d₁ = 0.3777910782 from Solution BSFOSDD1. So d₂ = d₁ - σ√(T) = 0.3777910782 - 0.3√(8) =

d₂ = -0.4707370592

Problem BSFOSDD3. The stock of Auspicious Co. currently trades for $221 per share. The stock will pay a dividend of $40 in 2 years and another dividend of $32 in 6 years. The annual continuously compounded risk-free interest rate is 0.05, and the annual price volatility relevant for the Black-Scholes equation is 0.3. Call options are written on Auspicious Co. stock with a strike price of $250 and time to expiration of 8 years. Use the Black-Scholes formula to find the price of one such call option.

Solution BSFOSDD3. We use the formula

C(F^P_0,T(S), F^P_0,T(K), σ, T) = F^P_0,T(S)N(d₁) - F^P_0,T(K)N(d₂). We know from Solutions BSFOSDD1-2 that d₁ = 0.3777910782, d₂ = -0.4707370592, F^P_0,T(S) = 161.1003202,

F^P_0,T(K) = 167.5800115.

In MS Excel, using the input "=NormSDist(0.3777910782)", we find that N(d₁) = 0.647207046

In MS Excel, using the input "=NormSDist(-0.4707370592)", we find that N(d₂) = 0.318914268

Thus, C(F^P_0,T(S), F^P_0,T(K), σ, T) = 161.1003202*0.647207046 - 167.5800115*0.318914268 =

C(F^P_0,T(S), F^P_0,T(K), σ, T) = $50.82160565

Problem BSFOSDD4. The stock of Auspicious Co. currently trades for $221 per share. The stock will pay a dividend of $40 in 2 years and another dividend of $32 in 6 years. The annual continuously compounded risk-free interest rate is 0.05, and the annual price volatility relevant for the Black-Scholes equation is 0.3. Put options are written on Auspicious Co. stock with a strike price of $250 and time to expiration of 8 years. Use the Black-Scholes formula to find the price of one such put option.

Solution BSFOSDD4. Since the corresponding call option price is known (from Solution BSFOSDD3), we can use put-call parity to find the put option price.

P(F^P_0,T(S), F^P_0,T(K), σ, T) = C(F^P_0,T(S), F^P_0,T(K), σ, T) + F^P_0,T(K) - F^P_0,T(S) =

50.82160565 + 167.5800115 - 161.1003202 = P(F^P_0,T(S), F^P_0,T(K), σ, T) = $57.30129695

Problem BSFOSDD5. You are given that d₁ in the Black-Scholes formula for the price of a particular call option on Unstoppable Co. is 0.4. You also know that the current stock price of Unstoppable Co. is $56, the relevant annual price volatility is 0.03, the price of the prepaid forward on the strike asset is $42, and the option's time to expiration is 2 years. The annual continuously-compounded interest rate is 0.03. The stock of Unstoppable Co. will pay one discrete dividend in 1 year. Find the size of the dividend.

Solution BSFOSDD5. We use the formula d₁ = [ln(F^P_0,T(S)/F^P_0,T(K)) + 0.5σ²T]/[σ√(T)], which we rearrange as follows: d₁σ√(T) = [ln(F^P_0,T(S)/F^P_0,T(K)) + 0.5σ²T]

d₁σ√(T) - 0.5σ²T = ln(F^P_0,T(S)/F^P_0,T(K))

exp(d₁σ√(T) - 0.5σ²T) = F^P_0,T(S)/F^P_0,T(K)

F^P_0,T(K)exp(d₁σ√(T) - 0.5σ²T) = F^P_0,T(S)

We are given that F^P_0,T(K) = 42, d₁ = 0.4, σ = 0.03, T = 2. Thus,

F^P_0,T(K)exp(d₁σ√(T) - 0.5σ²T) = 42exp(0.4*0.03√(2) - 0.5*0.03²*2) = F^P_0,T(S) = 42.68041633

Now we use the formula F^P_0,T(S) = S₀ - PV_0,T(Div) and rearrange it thus:
PV_0,T(Div) = S₀ - F^P_0,T(S) = 56 - 42.68041633 = 13.31958367

PV_0,T(Div) = Div*e^-1*0.03 = 13.31958367

So Div = 13.31958367e^0.03 = Div = 13.72522538

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