Application of option pricing theory in investment decisions: formula sheet

Key formulas

Black-Scholes call value

c = Pa × N(d1) − Pe × e^(−rt) × N(d2)

Used for delay and expand options. Pa = PV of project inflows, Pe = exercise price (investment outlay), r = risk-free rate, t = years to decision.

d1

d1 = [ln(Pa ÷ Pe) + (r + 0.5 × s²) × t] ÷ (s × √t)

s is the annual volatility of project cash flows (standard deviation, as a decimal). ln is the natural log.

d2

d2 = d1 − s × √t

Read N(d1) and N(d2) from the normal distribution table. For negative d, N(−d) = 1 − N(d).

Put value via put-call parity

p = c − Pa + Pe × e^(−rt)

Used for the abandon option, where Pe is the abandonment value. The alternative is p = Pe × e^(−rt) × N(−d2) − Pa × N(−d1).

Expanded NPV

Expanded NPV = static NPV + real option value

Make sure the static NPV and the option do not double count the same cash flows.

Call option value

c = Pa × N(d1) − Pe × e^(−rt) × N(d2)

Pa = current share price, Pe = exercise price, r = continuously compounded risk-free rate, t = years to expiry. The AFM formula sheet gives this.

d1

d1 = [ln(Pa ÷ Pe) + (r + 0.5 × s²) × t] ÷ (s × √t)

s = annual volatility (standard deviation) as a decimal. Use the natural log.

d2

d2 = d1 − s × √t

Always derived from d1. Do not recalculate from scratch.

Put value via put-call parity

p = c − Pa + Pe × e^(−rt)

Valid for European options on a non-dividend-paying share with the same exercise price and expiry.

Negative d values

N(−d) = 1 − N(d)

Tables usually show positive values only. Use this for negative d1 or d2.

Put-call parity (European, no dividends)

c + Pe e^(−rt) = p + Pa

c = call value, p = put value, Pa = current share price, Pe = exercise price, r = risk-free rate, t = years to expiry. Both options must share the same Pe and expiry.

Put value from call value

p = c − Pa + Pe e^(−rt)

This is the form given in the ACCA formula sheet. Use it when you have a call value from Black-Scholes.

Call value from put value

c = p + Pa − Pe e^(−rt)

Rearrangement of parity. Useful if the question gives the put value.

Black-Scholes call value

c = Pa N(d1) − Pe N(d2) e^(−rt)

Where d1 = [ln(Pa ÷ Pe) + (r + 0.5s²)t] ÷ (s√t) and d2 = d1 − s√t. s = annual volatility (standard deviation).

Delta of a call and a put

Call delta = N(d1); Put delta = N(d1) − 1

Put delta is negative. A rise in share price lowers a put's value.

Delta hedge

Options needed = number of shares to hedge ÷ |delta of one option|

Shares held are hedged by writing calls (or buying puts). The hedge must be rebalanced as delta changes; gamma measures how fast.

Intrinsic value

Call = max(Pa − Pe, 0); Put = max(Pe − Pa, 0)

Option value = intrinsic value + time value, so value is never below intrinsic value for an American option.

Black-Scholes call value

c = Pa × N(d1) − Pe × e^(−rt) × N(d2)

Pa = PV of project inflows, Pe = investment cost, r = risk-free rate, t = years to expiry. Use the formula sheet and normal distribution tables ACCA gives you.

d1

d1 = [ln(Pa ÷ Pe) + (r + 0.5 × s²) × t] ÷ (s × √t)

s is annual volatility as a decimal, e.g. 0.25 for 25%.

d2

d2 = d1 − s × √t

Calculate d1 first and then deduct s√t.

Input mapping

Pa = PV of inflows; Pe = capital outlay to exercise; t = time to decide; r = risk-free rate; s = volatility of project value

Do not subtract Pe from Pa when you set Pa. The model does that through the formula.

Sample standard deviation of returns

s(period) = √[ Σ(x − x̄)² ÷ (n − 1) ]

x = periodic return, often ln(Pt ÷ Pt−1). Use n − 1 for a sample unless the question says otherwise.

Annualising volatility

s(annual) = s(period) × √(number of periods in a year)

Weekly: × √52. Monthly: × √12. Daily: × √(trading days), usually 250 or 252 as told. Volatility scales with the square root of time, not time.

Option value and volatility

Higher s → higher option value (for calls and puts)

Use this for comment marks on sensitivity.

Equity as a call

Equity = c(V, D, T, r, σ)

V is the market value of the firm's assets. D is the face value of zero-coupon debt. T is the time to debt maturity.

Risky debt value

Debt = V − Equity call

Valid when the firm's value is split between debt and equity only.

Debt as risk-free debt less a put

Debt = D × e^(−rT) − Put

Use D ÷ (1 + r)^T if the question uses annual compounding.

Put-call parity (no dividends, European)

c + X × e^(−rT) = p + S

Here S = V and X = D. Use it to find the put from the call. Under annual compounding, the PV of X is X ÷ (1 + r)^T, so use that in place of X × e^(−rT).

Black-Scholes call

c = S × N(d1) − X × e^(−rT) × N(d2)

ACCA gives the formula and normal tables. Use asset value as S for equity.

d1 and d2

d1 = [ln(S ÷ X) + (r + σ²÷2) × T] ÷ (σ × √T); d2 = d1 − σ × √T

σ is the volatility of the underlying. For equity-as-option it is asset volatility.

Convertible bond value

Convertible = Straight debt value + (Conversion ratio × Option value per share)

Straight debt value is the PV of coupons and redemption at the yield on similar non-convertible debt. Ignores dilution unless told otherwise.

Implied yield and spread on risky debt

Yield = (D ÷ Debt value)^(1÷T) − 1; Spread = Yield − risk-free rate

For zero-coupon debt only.

Quick revision

• A real option is a right, not an obligation, to take a future action on a project.
• Delay and expand options act like calls. An abandonment option acts like a put. Switch options can combine both, depending on the direction.
• Total project value = base NPV + value of the option.
• Black-Scholes call: c = Pa × N(d1) − Pe × e^(−rt) × N(d2).
• d1 = [ln(Pa ÷ Pe) + (r + 0.5σ²)t] ÷ (σ√t), and d2 = d1 − σ√t.
• Map inputs: Pa is the project's present value, Pe is the investment cost, t is the time the right lasts.
• Put-call parity for European options: c + Pe × e^(−rt) = p + Pa, assuming no income on the asset.
• Higher volatility raises the value of both calls and puts. Longer time raises the value of American options, and normally raises a European call on an asset with no dividends. For a European put the effect is ambiguous and can be negative.
• A higher risk-free rate raises call value and lowers put value.
• Volatility is the hardest input to find. Use history or a comparable firm and state the limits.
• Equity is a call option on the firm's assets, with the debt as the exercise price.
• Always state a decision and note that the model assumes things like constant volatility.

Common mistakes

• Using the full NPV as Pa instead of the PV of inflows. Fix: Pa is the PV of the cash inflows only. The outlay is Pe. NPV = Pa − Pe in the static case.
• Treating the abandon option as a call. Fix: The right to sell for a fixed value is a put. Compute the call, then use p = c − Pa + Pe × e^(−rt), or the direct put formula.
• Using t in months or r as a whole number, such as 5 instead of 0.05. Fix: Convert first. Write t = 0.5 and r = 0.05 at the top of your working, before any calculation.
• Using log base 10 instead of the natural log for ln(Pa ÷ Pe). Fix: Always press ln. As a check, ln of a number slightly above 1 is close to that number minus 1.
• Using Pe instead of its present value, Pe e^(−rt), in the parity formula. Fix: Always discount Pe at the risk-free rate over the option life. Write the PV as a separate line before substituting.
• Getting the signs wrong, giving p = c + Pa − Pe e^(−rt). Fix: Remember the pairs: call plus PV of exercise price equals put plus share. Then rearrange for the put and sense check that the put is positive.
• Setting Pa as the project NPV, or as the PV of inflows minus the investment. Fix: Pa is the PV of inflows only. The cost goes in as Pe. The model compares them itself. The option value is then added to, or compared with, the base NPV.
• Using weekly or monthly standard deviation directly as annual volatility. Fix: Check the data frequency. Multiply by √52 for weekly, √12 for monthly or √250 (or the stated figure) for daily data.
• Using the share price as S when valuing equity as a call. Fix: For equity-as-option, S is the value of the whole firm's assets and X is the debt's face value. The output is the value of all equity.
• Using equity volatility instead of asset volatility. Fix: Use the volatility the question labels as asset or firm volatility. If only equity volatility is given, say the model needs asset volatility and state your assumption.

Exam tips

• Look for trigger words in the scenario: wait, phase, pilot, scale up, withdraw, resale value and flexibility. They signal which option applies.
• Show the input mapping explicitly. Even if you slip in the arithmetic, you can still score for correct identification of Pa, Pe, t, s and r.
• Always separate the static NPV from the option value, then add them. Examiners expect to see the expanded NPV.
• Use the written commentary for professional skills marks: discuss estimation of volatility, exclusivity of the option, practical exercise conditions and what you would recommend to the board.
• If asked about the difference between NPV and real options, state that NPV assumes a fixed plan and a discount rate for risk, while real options value flexibility and reward uncertainty.
• Tables and the formula are given in the exam, but you must know what each symbol means. Write inputs down first so the marker can follow your logic.
• Round d1 and d2 to two decimals so you can use the table. Show N values clearly. Small rounding differences are normally accepted if the method is right.
• Questions often ask you to comment, not just calculate. Prepare two or three points on assumptions, volatility estimation and dividends.