The Black-Scholes Formula Using Prepaid Forward Prices: Practice Problems and Solutions

This is Section 34 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.

It is possible to express the Black-Scholes formula using prepaid forward prices for the stock and for the strike asset.

We note that F^P_0,T(S) = Se^-∂T and F^P_0,T(K) = Ke^-rT.

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)

This formula has two advantages over the standard Black-Scholes formula.

1. The interest rate and dividend yield do not appear explicitly in the formula; the formula requires only four parameters to be known rather than six.

2. This formula allows for calculating option prices for options where the strike asset is something other than cash.

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.

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, pp. 379-380.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem BSFUPFP1. Put options are written, using the stock of Ferocious LLC as the underlying asset and the stock of Timid Co. as the strike asset. The current prepaid forward price on Ferocious LLC stock is $330. The current prepaid forward price on Timid LLC stock is $310.

The annual underlying asset price volatility relevant for the Black-Scholes formula is 0.1, and the options' time to expiration is 4 years. Calculate d₁ in the Black-Scholes formula for the price of such a put option.

Solution BSFUPFP1. We use the equation d₁ = [ln(F^P_0,T(S)/F^P_0,T(K)) + 0.5σ²T]/[σ√(T)], where F^P_0,T(S) = 330, F^P_0,T(K) = 310, T = 4, and σ = 0.1.

Thus, d₁ = [ln(330/310) + 0.5*0.1²*4]/[0.1√(4)] = d₁ = 0.4126017849

Problem BSFUPFP2. Put options are written, using the stock of Ferocious LLC as the underlying asset and the stock of Timid Co. as the strike asset. The current prepaid forward price on Ferocious LLC stock is $330. The current prepaid forward price on Timid LLC stock is $310.

The annual underlying asset price volatility relevant for the Black-Scholes formula is 0.1, and the options' time to expiration is 4 years. Calculate d₂ in the Black-Scholes formula for the price of such a put option.

Solution BSFUPFP2. We use the equation d₂ = d₁ - σ√(T), where d₁ = 0.4126017849 from Solution BSFUPFP1, T = 4, and σ = 0.1. Thus, d₂ = 0.4126017849 - 0.1√(4) =

d₂ = 0.2126017849

Problem BSFUPFP3. Put options are written, using the stock of Ferocious LLC as the underlying asset and the stock of Timid Co. as the strike asset. The current prepaid forward price on Ferocious LLC stock is $330. The current prepaid forward price on Timid LLC stock is $310.

The annual underlying asset price volatility relevant for the Black-Scholes formula is 0.1, and the options' time to expiration is 4 years. Use the Black-Scholes formula to find the price of such a put option.

Solution BSFUPFP3. The Black-Scholes formula for the put price is

P(F^P_0,T(S), F^P0,T(K), σ, T) = F^P_0,T(K)N(-d₂)- F^P_0,T(S)N(-d₁)

We are given that F^P_0,T(S) = 330, F^P_0,T(K) = 310, T = 4, and σ = 0.1.

Furthermore, from Solution BSFUPFP1, d₁ = 0.4126017849. In MS Excel, using the input "=NormSDist(-0.4126017849)", we find that N(-d₁) = 0.339949236

Furthermore, from Solution BSFUPFP2, d₂ = 0.2126017849. In MS Excel, using the input "=NormSDist(-0.2126017849)", we find that N(-d₂) = 0.415818821

Thus, P(F^P_0,T(S), F^P0,T(K), σ, T) = 310*0.415818821 - 330*0.339949236 =

P(F^P_0,T(S), F^P0,T(K), σ, T) = $16.72058663

Problem BSFUPFP4. Call options are written, using the stock of Ferocious LLC as the underlying asset and the stock of Timid Co. as the strike asset. The current prepaid forward price on Ferocious LLC stock is $330. The current prepaid forward price on Timid LLC stock is $310.

The annual underlying asset price volatility relevant for the Black-Scholes formula is 0.1, and the options' time to expiration is 4 years. Use the Black-Scholes formula to find the price of such a call option.

Solution BSFUPFP4. Since the put price is known from Solution BSFUPFP3 to be 16.72058663, we can calculate the call price using 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), which implies that

C(F^P_0,T(S), F^P_0,T(K), σ, T) = P(F^P_0,T(S), F^P_0,T(K), σ, T) - F^P_0,T(K) + F^P_0,T(S), where F^P_0,T(K) = 310 and F^P_0,T(S) = 330. Thus, C(F^P_0,T(S), F^P_0,T(K), σ, T) = 16.72058663 - 310 + 330 =

C(F^P_0,T(S), F^P_0,T(K), σ, T) = $36.72058663

Problem BSFUPFP5. Use the Black-Scholes formula with prepaid forward prices to find the price of a call option whose underlying asset price is not volatile at all. (That is, σ = 0.)

Solution BSFUPFP5. 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)

Where σ = 0, d₁ = [ln(F^P_0,T(S)/F^P_0,T(K)) + 0]/[0√(T)]. This entails division of some nonzero quantity by zero, which means that d₁ = +∞. Likewise, d₂ = d₁ - σ√(T) = +∞ - 0 = d₂ = +∞

Thus, N(d₁) = N(d₂) = N(+∞) = 1. (The probability of x being less than positive infinity within the standard normal distribution is 1.)

Thus, C(F^P_0,T(S), F^P_0,T(K), 0, T) = F^P_0,T(S)*1 - F^P_0,T(K)*1 and so

C(F^P_0,T(S), F^P_0,T(K), 0, T) = F^P_0,T(S) - F^P_0,T(K)

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