How much should someone pay for the right to buy a share at ₦165 in six months? Discounting expected cash flows does not work, because the option’s risk changes with every move in the share price. Black and Scholes found a way round the problem: build a portfolio of the share and borrowing that behaves exactly like the option, and value that instead.
This topic covers what drives an option’s value (intrinsic value, time value and the five factors), why options cannot be valued by ordinary discounting, the Black-Scholes model for a European call, how to use the normal distribution table, put-call parity to value a put, synthetic securities and parity arbitrage. The previous topic covered hedging with options. The next covers real options and further applications of Black-Scholes.
Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on option pricing.
In 30 seconds
Option value = intrinsic value + time value. Only in-the-money options have intrinsic value. Five factors: underlying price (Pa), exercise price (Pe), volatility (s), time to expiry (t) and the risk-free rate (r). Higher volatility and longer time raise value. Black-Scholes (European call): c = Pa·N(d1) − Pe·N(d2)·e−rt, with d1 = [ln(Pa/Pe) + (r + 0.5s²)t] ÷ (s√t) and d2 = d1 − s√t.
Look up N(d) in the table: add 0.5 if d is positive; subtract from 0.5 if negative. Put-call parity: p = c − Pa + Pe·e−rt, equivalently c + PV(Pe) = p + Pa. If parity fails, sell the dear side and buy the cheap side for a riskless profit. The binomial model is no longer in the syllabus.
Key terms in plain English
| Term | What it means |
|---|---|
| Intrinsic value | The gain if exercised now: market price less strike for a call; zero if out of the money |
| Time value | The extra value for the chance the option becomes (more) in the money |
| Volatility (s) | The annual standard deviation of returns on the underlying |
| N(d) | The cumulative normal probability for d standard deviations |
| Continuous discounting | Discounting with e−rt instead of 1 ÷ (1 + r)t |
| Put-call parity | The fixed link between the prices of a call, a put and the underlying |
Syllabus check
This topic is in syllabus area E (treasury and advanced risk management). It covers:
- explaining and using the Black-Scholes model to value options;
- explaining put-call parity.
CAMA 2020 and option pricing
The Companies and Allied Matters Act 2020 (CAMA 2020) is the company law of Nigeria.
| Section | What it says | Link to this topic |
|---|---|---|
| s. 305 | A director must act in good faith in the best interests of the company | Valuations should be prepared with care for the benefit of the company |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors should understand the assumptions behind the valuation of any derivatives held |
| s. 377(1) and s. 378 | Directors prepare statements that follow IFRS and give a true and fair view | Decisions later affect reported assets, profit and cash |
| s. 385 | The directors’ report covers performance and risk | Major plans and risks are reported to shareholders |
Option premium and value
The premium is the price paid for the option when it is written. Afterwards the value changes with the likelihood of exercise and the size of the gain. Valuation is needed for the writer’s pricing, for treasury decisions and for financial reporting, where derivatives are measured at fair value.
Value = intrinsic value + time value. Time value depends on:
- time to expiry: longer is worth more;
- closeness to the strike: at the money is worth more than out of the money, and slightly out of the money more than deeply out of the money;
- volatility: higher volatility raises the chance of a large move;
- the risk-free rate.
The five factors: underlying price, exercise price, volatility, time to expiry and the risk-free rate. ICAN’s comparison: an option on shares of Company X (strike 850, price 825, one year, volatility 35%) is worth much more than one on Company Y (strike 750, price 500, three months, volatility 10%), even though both are out of the money.
In published tables, call premiums fall as the strike rises, put premiums rise as the strike rises, and both rise with time to expiry.
Why options cannot be valued by discounting
A call is a geared position in the share, riskier than the share, and its risk changes as the price moves. ICAN’s illustration: a call with strike 70 on a share at 100 has intrinsic value 30. If the share rises 10% to 110, intrinsic value rises 33%; if it falls 10% to 90, it falls 33%. Since the risk changes with the price, no single discount rate can be found.
Black and Scholes valued the option through an option equivalent or replicating portfolio: some of the underlying share plus borrowing. Shares and borrowing are easy to value, so the option is too. This led to the binomial model (no longer in the syllabus) and the Black-Scholes model.
The Black-Scholes model
It values a European call and is the basis of banks’ over-the-counter premiums.
c = Pa·N(d1) − Pe·N(d2)·e−rt
d1 = [ln(Pa/Pe) + (r + 0.5s²)t] ÷ (s√t); d2 = d1 − s√t
- c = value of the call; Pa = price of the underlying; Pe = exercise price; r = annual risk-free rate (5% is 0.05); t = years to expiry (three months is 0.25); s = annual standard deviation (take the square root if the variance is given); ln = natural logarithm.
Using the table. Find d (to two decimals) in the normal table. For a positive d, N(d) = table value + 0.5; for a negative d, N(d) = 0.5 − table value. Calculate e−rt = 1 ÷ ert first; it is a continuous discount factor.
ICAN’s example. BS Company’s share price is ₦170. Value a European call at ₦165, six months to expiry, s = 12%, r = 7%.
- d1 = [ln(170/165) + (0.07 + 0.5×0.12²)×0.5] ÷ (0.12×√0.5) = (0.0299 + 0.0386) ÷ 0.0849 = 0.807
d2 = 0.807 − 0.085 = 0.722. - N(d1) = 0.5 + 0.2910 = 0.7910 (d1 ≈ 0.81)
N(d2) = 0.5 + 0.2642 = 0.7642 (d2 ≈ 0.72). - e−0.035 = 0.9656
c = 170 × 0.7910 − 165 × 0.7642 × 0.9656 = 134.5 − 121.8 = ₦12.7.
ICAN’s practice case (negative d). CB Company’s price is ₦400; strike ₦440; six months; s = 30%; r = 6%.
- d1 = [ln(400/440) + (0.06 + 0.045)×0.5] ÷ (0.30×√0.5) = (−0.0953 + 0.0525) ÷ 0.2121 = −0.202
d2 = −0.202 − 0.212 = −0.414. - N(d1) = 0.5 − 0.0801 = 0.4199
N(d2) = 0.5 − 0.1613 = 0.3387. - e−0.03 = 0.9704
c = 400 × 0.4199 − 440 × 0.3387 × 0.9704 = 167.96 − 144.62 = ₦23.34.
Put-call parity
Black-Scholes values a call. For a put with the same strike and expiry, use put-call parity: value of put = value of call + PV of the exercise price − current price of the underlying, that is p = c − Pa + Pe·e−rt. The PV of the exercise price uses continuous discounting. ICAN’s example: call ₦13, price ₦170, strike ₦165, six months, r = 7%. PV of the strike = 165 × 0.9656 = ₦159.3, so put = 13 + 159.3 − 170 = ₦2.3.
The formula c + PV(K) = p + S is the same relationship rearranged.
Synthetic securities
Rearranging parity gives positions that copy others. Synthetic long stock = buy a call, sell a put, lend PV(K). Synthetic short stock = buy a put, sell a call, borrow PV(K). Synthetic call = buy the stock, buy a put, borrow PV(K). Synthetic put = buy a call, lend PV(K), short the stock. Uses: hedging, speculation, reaching hard-to-access markets, arbitrage, customised risk and sometimes lower cost. Risks: counterparty, complexity, liquidity, model, regulatory and tracking error.
Arbitrage when parity fails
ICAN’s example. A non-dividend stock is at ₦100; an at-the-money one-year call trades at ₦12, the put at ₦5; r = 5% continuous. PV(K) = 100e−0.05 = ₦95.12.
- c + PV(K)
12 + 95.12 = 107.12
p + S = 5 + 100 = 105.
They differ, so parity is violated and the call is dear. - Sell the call (+12), buy the put (−5), buy the stock (−100), borrow 95.12 (+95.12): inflow ₦2.12 now.
- At expiry, whether the stock is ₦120 or ₦80, the positions and the loan repayment (₦100) net to nil, so the ₦2.12 is locked in.
With bid-ask spreads, trade at the side that is worse for you: you sell at the bid and buy at the ask. ICAN’s Zuba plc: stock bid/ask 199/201, call 13.50/14.80, put 9.00/10.00, strike 205, six months, r = 5%, PV(K) = 199.94. Mid prices give p + S = 209.50 against c + PV(K) = 214.09, so the call is dear.
Sell the call at 13.50, buy the put at 10.00, buy the stock at 201 and borrow 199.94: net ₦2.44, locked in whatever the stock price.
The rules in more depth
Why volatility raises value
The holder gains from large favourable moves but loses only the premium on adverse ones.
Why a replicating portfolio works
Shares and borrowing reproduce the option’s payoffs, so by no-arbitrage they must have the same value.
Why e−rt is used
The model assumes values change continuously, so discounting is continuous.
Why parity holds
A call plus cash equal to PV(K) has the same payoff as a put plus the stock. Otherwise arbitrage would be possible.
Effect on analysis
List the five inputs, compute d1, d2, the N values and the call price; then use parity for the put and say what the model assumes.
Worked examples
Example 1: intrinsic value. A call has a strike of ₦165 and the share is at ₦170. The option trades at ₦12.7.
- Intrinsic value.
170 − 165 = ₦5. - Time value = 12.7 − 5.
- Time value = ₦7.7.
Example 2: a continuous discount factor. r = 7% and t = 0.5.
- rt = 0.035.
- e0.035 = 1.0356.
- e−0.035 = 0.9656.
Example 3: a put from parity. Using the CB case, c = 23.34, Pa = 400, strike 440 and e−0.03 = 0.9704.
- PV of strike.
440 × 0.9704 = 426.98. - p = 23.34 − 400 + 426.98.
- p = ₦50.32.
Example 4: reading N(d). d1 = −0.20 and the table value is 0.0793.
- d1 is negative.
- N(d1) = 0.5 − 0.0793.
- N(d1) = 0.4207.
How to score full marks
- List Pa, Pe, r, t and s first, in annual terms.
- Calculate d1 then d2, showing the logarithm.
- Apply the add or subtract rule for the table.
- Compute e−rt separately.
- Use parity for a put.
- Comment on which inputs drive the answer.
Common mistakes in the exam
- Using the variance instead of the standard deviation.
- Forgetting to add 0.5 for positive d.
- Using months as t instead of years.
- Using ordinary discounting for the strike.
- Applying Black-Scholes to an American put.
- Trading at mid prices in arbitrage.
Quick revision
- Value = intrinsic + time; five factors.
- c = Pa N(d1) − Pe N(d2) e−rt.
- d1 = [ln(Pa/Pe) + (r + 0.5s²)t] ÷ s√t
d2 = d1 − s√t. - Positive d: add 0.5; negative d: subtract from 0.5.
- Put: p = c − Pa + Pe e−rt.
- Parity fails: sell dear, buy cheap.
- CAMA 2020: ss. 305, 308, 377(1), 378, 385.
Practice questions
These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer. Full model answers follow the written questions.
Recall check: multiple choice
Recent ICAN papers are written, although older papers such as May 2014 included multiple-choice questions. These questions are for quick recall. The written questions that follow are closer to the exam.
Question 1. Option value equals:
- Intrinsic value plus time value
- Intrinsic value only
- Time value only
- The premium minus the strike
Show the answer
A. Out-of-the-money options have time value only.
Question 2. Higher volatility generally:
- Raises option value
- Lowers option value
- Has no effect
- Removes time value
Show the answer
A. Larger moves help the holder.
Question 3. In Black-Scholes, s is the:
- Annual standard deviation
- Variance
- Share price
- Strike
Show the answer
A. Take the square root of a variance.
Question 4. If d1 is negative, N(d1) is:
- 0.5 minus the table value
- 0.5 plus the table value
- The table value
- 1
Show the answer
A. The table gives the area from the mean.
Question 5. The discount factor in Black-Scholes is:
- e⁻ʳᵗ
- 1 ÷ (1 + r)
- r × t
- eʳᵗ
Show the answer
A. Continuous discounting.
Question 6. Put-call parity gives the put as:
- c − Pa + Pe e⁻ʳᵗ
- c + Pa − Pe
- Pa − c
- c × Pa
Show the answer
A. With the same strike and expiry.
Question 7. If c + PV(K) exceeds p + S, you should:
- Sell the call and buy the put and stock
- Buy the call
- Do nothing
- Sell the stock
Show the answer
A. Sell the dear side.
Question 8. In an arbitrage with bid-ask prices, you sell at the:
- Bid
- Ask
- Mid price
- Strike
Show the answer
A. And buy at the ask.
Question 9. The binomial model is:
- No longer in the syllabus
- Required
- The only model
- Used for puts only
Show the answer
A. Black-Scholes is examined.
Question 10. Which section of CAMA 2020 sets the director’s duty of care and skill?
- Section 308
- Section 400
- Section 405
- Section 409
Show the answer
A. Directors should test the assumptions behind a decision.
Written questions
Question 11 (10 marks). (a) State the five factors affecting option value. (5 marks) (b) Explain why an option cannot be valued by discounting expected cash flows. (3 marks) (c) Define intrinsic value and time value. (2 marks)
Question 12 (20 marks). CB Company’s share is ₦400. A European call has a strike of ₦440 and six months to expiry. s = 30%; r = 6%. Given: ln(0.9091) = −0.0953; table value for d = 0.20 is 0.0793, and for 0.41 is 0.1591; e0.03 = 1.03045. (a) Calculate d1 and d2. (6 marks) (b) Calculate N(d1) and N(d2) (use 0.4207 and 0.3409 if you prefer to avoid interpolation) and the call value using 0.4207 and 0.3409. (8 marks) (c) Calculate the put value by parity.
(4 marks) (d) State the effect on the call value of a rise in volatility. (2 marks)
Question 13 (10 marks). (a) A stock is ₦100, the call is ₦12, the put ₦5, the strike ₦100, one year, r = 5% continuous. Test parity. (4 marks) (b) State the arbitrage trades and the profit. (4 marks) (c) State one risk of a synthetic position. (2 marks)
Model answers
Show the model answer to Question 11
(a) Factors (5 marks). Current price of the underlying; exercise price; volatility; time to expiry; risk-free interest rate. (1 each)
(b) Discounting (3 marks). A call is a geared and riskier position than the share. (1) Its risk changes as the share price moves, so no single discount rate exists. (2)
(c) Values (2 marks). Intrinsic value is the gain if exercised now (zero if not in the money). (1) Time value is the value of the chance of becoming (more) in the money. (1)
Show the model answer to Question 12
(a) d1, d2 (6 marks).
- d1 = [ln(400/440) + (0.06 + 0.5×0.09)×0.5] ÷ (0.3×0.7071) = (−0.0953 + 0.0525) ÷ 0.2121 = −0.202. (4)
- d2 = −0.202 − 0.2121 = −0.414. (2)
(b) Call (8 marks).
- For d1 = −0.20 the table value is 0.0793, so N(d1) = 0.5 − 0.0793 = 0.4207
- for d2 = −0.41 the table value is 0.1591, so N(d2) = 0.3409. (3)
- e−0.03 = 1 ÷ 1.03045 = 0.9704. (1)
- c = 400 × 0.4207 − 440 × 0.3409 × 0.9704 = 168.28 − 145.56 = ₦22.72. (4)
- (Rounding d to −0.202 and −0.414 gives ₦23.34.)
(c) Put (4 marks).
- PV of strike.
440 × 0.9704 = 426.98. (1) - p = 22.72 − 400 + 426.98 = ₦49.70. (3)
(d) Volatility (2 marks). A rise in volatility raises the call value, because larger moves raise the chance of a big gain while the loss is capped at the premium. (2)
Show the model answer to Question 13
(a) Parity (4 marks).
- PV(K).
100e−0.05 = 95.12. (1) - c + PV(K).
12 + 95.12 = 107.12. (1) - p + S.
5 + 100 = 105. (1) - They differ, so parity is violated and the call is dear. (1)
(b) Arbitrage (4 marks).
- Sell the call (+12), buy the put (−5), buy the stock (−100) and borrow 95.12 (+95.12). (3)
- Net inflow = ₦2.12, locked in. (1)
(c) Risk (2 marks). Counterparty risk, complexity, liquidity, model risk, regulatory risk or tracking error. Any one. (2)
Pathfinder on Option value, the Black-Scholes model, put-call parity and parity arbitrage
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on Black-Scholes. Work the two ICAN examples and the parity arbitrage example step by step.
Revise first: The Black-Scholes model
Exam questions give a call value and ask for a put. See the section on put-call parity.
Revise first: Put-call parity
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: No stand-alone question in the May 2014 or November 2014 Pathfinder.
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