Exam-Style Questions on the Black-Scholes Formula

This is Section 38 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. See Section 35 here. See Section 36 here. See Section 37 here.

The problems in this section were designed to be similar to problems from past versions of Exam 3F / Exam MFE. They use original exam questions as their inspiration - and the specific inspiration for each problem is cited so as to give students a chance to see the original. All of the original problems are publicly available, and students are encouraged to refer to them. But all of the values, names, conditions, and calculations in the problems here are the original work of Mr. Stolyarov.

Problem ESQBSF1.

Similar to Question 20 from the Casualty Actuarial Society's Spring 2007 Exam 3:

You are aware of the following information for the dividend-paying stock of Astonishing Co. and a European call option on that stock. The current stock price is $600, and the call option strike price is $654. The stock's volatility is 0.4, and the expected annual return on the stock is 15%. The annual continuously-compounded risk-free interest rate is 0.12, and the stock's annual continuously-compounded dividend yield is 0.04. The call option expires in 2 years. Find the call price using the Black-Scholes formula.

Solution ESQBSF1.

We note that the given expected annual return on the stock is entirely superfluous to solving this problem.

First we find d₁ = [ln(S/K) + (r - ∂ + 0.5σ²)T]/[σ√(T)] =

[ln(600/654) + (0.12 - 0.04 + 0.5*0.4²)2]/[0.4√(2)] = d₁ = 0.4133433415

Now we find d₂ = d₁ - σ√(T) = 0.4133433415 - 0.4√(2) = d₂ = -0.1523420835

In MS Excel, using the input "=NormSDist(0.4133433415)", we find that N(d₁) = 0.660322421

In MS Excel, using the input "=NormSDist(-0.1523420835)", we find that N(d₂) = 0.43945855

Now we use the Black-Scholes formula:

C(S, K, σ, r, T, ∂) = Se^-∂TN(d₁) - Ke^-rTN(d₂) = 600e^-0.04*20.660322421 - 654e^-0.12*20.43945855 =

C(S, K, σ, r, T, ∂) = $139.6511706

Problem ESQBSF2.

Similar to Question 21 from the Casualty Actuarial Society's Spring 2007 Exam 3:

The stock of Intrepid LLC does not pay dividends. The stock currently trades for $5500 per share, with price volatility of 0.21. The annual continuously-compounded risk-free interest rate is 0.02. Use the Black-Scholes formula to find the price of a put option on Intrepid LLC stock with a strike price of $5430 and time to expiration of 1 year.

Solution ESQBSF2.

First, we find d₁ = [ln(S/K) + (r - ∂ + 0.5σ²)T]/[σ√(T)] =

[ln(5500/5430) + (0.02 - 0 + 0.5*0.21²)1]/[0.21√(1)] = d₁ = 0.2612331347

Now we find d₂ = d₁ - σ√(T) = 0.2612331347 - 0.21 = d₂ = 0.0512331347

In MS Excel, using the input "=NormSDist(-0.2612331347)", we find that N(-d₁) = 0.39695642

In MS Excel, using the input "=NormSDist(-0.0512331347)", we find that N(-d₂) = 0.479569806

Now we use the Black-Scholes formula: P(S, K, σ, r, T, ∂) = Ke^-rTN(-d₂) - Se^-∂TN(-d₁)

= 5430e^-0.020.479569806- 5500*0.39695642 = P(S, K, σ, r, T, ∂) = $369.2398137

Problem ESQBSF3.

Similar to Question 20 from the Casualty Actuarial Society's Fall 2007 Exam 3:

Which of these statements are assumptions of the Black-Scholes option pricing model? More than one answer may be possible.

(a) A risk premium must be paid for borrowing money.
(b) The volatility of continuously compounded returns is a random variable modeled by the normal probability distribution.
(c) All taxes and transaction costs are the same positive quantity for every transaction.
(d) The risk-free interest rate is known and constant.
(e) Continuously compounded returns on the stock are normally distributed and independent over time.
(f) The stock can only pay dividends on a continuously compounded basis; discrete dividends are not allowed.

Solution ESQBSF3.

According to R. L. McDonald, the six assumptions of the Black-Scholes model are

"1. Continuously compounded returns on the stock are normally distributed and independent over time.

"2. The volatility of continuously compounded returns is known and constant.

"3. Future dividends are known, either as a dollar amount or as a fixed dividend yield.

"4. The risk-free interest rate is known and constant.

"5. There are no transaction costs or taxes.

"6. It is possible short-sell costlessly and to borrow at the risk-free rate."

We note that assumption 4 corresponds to statement (d) and assumption 1 corresponds to statement (e). So (d) and (e) are true. Statement (a) contradicts assumption 6, so (a) is false. (b) contradicts assumption 2, so (b) is false. (c) contradicts assumption 5, so (c) is false. (f) is not true, as Section 35 describes how to apply the Black-Scholes model to stocks that pay discrete dividends. So (d) and (e) are correct answers.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 12, pp. 375-379.

Problem ESQBSF4.

Similar to Question 21 from the Casualty Actuarial Society's Fall 2007 Exam 3:

You know the following information about exchanging Platinum Icosagons (PI) for Titanium Dodecahedra (TD). The spot exchange rate is 1.23 PI/TD. The annual continuously-compounded PI-denominated interest rate is 0.01.The annual continuously-compounded TD-denominated interest rate is 0.005. The exchange rate volatility is 0.02. (These are commodity moneys, so exchange rate volatility should be low.) Polyneices wishes to buy 756 PI-demoniated TD call options. The options have a strike price of 1.10 PI/TD and expire in 3 years. Use the Garman-Kohlhagen formula to find the price of the block of 756 options.

Solution ESQBSF4. First we find d₁ = [ln(x/K) + (r - f + 0.5σ²)T]/[σ√(T)] =

[ln(1.23/1.10) + (0.01 - 0.005 + 0.5*0.02²)3]/[0.02√(3)] = d₁ = 3.674949633

Now we find d₂ = d₁ - σ√(T) = 3.674949633 - 0.02√(3) = d₂ = 3.640308616

In MS Excel, using the input "=NormSDist(3.674949633)", we find that N(d₁) = 0.99988102

In MS Excel, using the input "=NormSDist(3.640308616)", we find that N(d₂) = 0.99986381

Now we use the Garman-Kohlhagen formula: C(x, K, σ, r, T, f) = xe^-fTN(d₁) - Ke^-rTN(d₂) =

1.23e^-0.005*30.99988102 - 1.10e^-0.01*30.99986381 = C(x, K, σ, r, T, f) = 0.1441988137

We want to find 756C = 756*0.1441988137 = $109.0143031

Problem ESQBSF5.

Similar to Question 6 from the Society of Actuaries' Sample MFE Questions and Solutions:

Athena wishes to purchase 204 European call options on the stock of Mythology Co. The stock currently trades for $40 per share, and pays dividends at an annual continuously-compounded yield of 0.11. The annual continuously-compounded risk-free interest rate is 0.09, and the relevant measure of volatility is 0.1. The strike price of the options is $43, and the options will expire in 5 years. Use the Black-Scholes formula to find the price of the block of 204 call options.

Solution ESQBSF5.

First, we find d₁ = [ln(43/40) + (0.09 - 0.11 + 0.5*0.1²)5]/[0.1√(5)] = d₁ = -0.0119823657

Now we find d₂ = d₁ - σ√(T) = -0.0119823657 - 0.1√(5) = d₂ = -0.2355891634

In MS Excel, using the input "=NormSDist(-0.0119823657)", we find that N(d₁) = 0.495219816

In MS Excel, using the input "=NormSDist(-0.2355891634)", we find that N(d₂) = 0.406875787

Now we use the Black-Scholes formula:

C(S, K, σ, r, T, ∂) = Se^-∂TN(d₁) - Ke^-rTN(d₂) = 40e^-0.11*50.495219816 - 43e^-0.09*50.406875787 =

C(S, K, σ, r, T, ∂) = 0.2729545493

We want to find 204C = 204*0.2729545493 = $55.68272806

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