FRM Part I · FRM Exam Part I

The Black-Scholes-Merton Model: formula sheet

Full chapter guide

Key formulas

Stock price process (GBM): dS = μS dt + σS dz; μ is expected return, σ is volatility, dz ~ Normal(0, dt). Percentage change dS/S is normal.
Log price process: d(ln S) = (μ − σ²/2) dt + σ dz; Follows from Ito's lemma. The drift of ln S is μ − σ²/2, not μ.
Distribution of ln S_T: ln S_T ~ Normal( ln S₀ + (μ − σ²/2)T , σ²T ); So S_T is lognormal. Standard deviation of ln S_T is σ√T.
Expected future price: E(S_T) = S₀ e^(μT); Mean of the lognormal variable. Under risk-neutral valuation use r in place of μ.
Variance of S_T: Var(S_T) = S₀² e^(2μT) (e^(σ²T) − 1); Standard lognormal result.
BSM differential equation: ∂f/∂t + r S ∂f/∂S + ½σ²S² ∂²f/∂S² = r f; Holds for any derivative on a non-dividend-paying stock. μ does not appear.
Riskless hedge portfolio: Π = −f + (∂f/∂S) S; Short one derivative, long ∂f/∂S shares. The portfolio is riskless only for a short interval and must be rebalanced.
Risk-neutral valuation: f = e^(−rT) E*[payoff]; E* assumes the stock drifts at r, so E*(S_T) = S₀ e^(rT).
European call price: c = S₀ N(d1) − K e^(−rT) N(d2); Non-dividend-paying stock. r is the continuously compounded risk-free rate.
European put price: p = K e^(−rT) N(−d2) − S₀ N(−d1); Uses N(−d) = 1 − N(d).
d1: d1 = [ln(S₀ ÷ K) + (r + σ²÷2) T] ÷ (σ √T); Use σ as a decimal (0.20, not 20).
d2: d2 = d1 − σ √T; Equals [ln(S₀ ÷ K) + (r − σ²÷2) T] ÷ (σ √T).
Put-call parity: c + K e^(−rT) = p + S₀; European options, same strike and expiry, no dividends.
Interpretation of N(d): N(d2) = risk-neutral P(S_T > K); N(d1) = call delta; Put delta is N(d1) − 1, and N(−d2) = risk-neutral P(S_T < K).
Log (continuously compounded) return: uᵢ = ln(Sᵢ ÷ Sᵢ₋₁); Use this for historical volatility. Ignore dividends unless the question says to adjust for them.
Sample standard deviation of returns: s = √[ Σ(uᵢ − ū)² ÷ (n − 1) ]; n is the number of returns, which is one fewer than the number of prices. Many questions set ū ≈ 0, which gives s = √[Σuᵢ² ÷ n]. Read the question to see which version is intended.
Annualizing volatility: σ per year = s × √252; Use the number of periods per year stated in the question. 252 trading days is the usual default, 12 for monthly returns, 52 for weekly.
Volatility over a horizon: Standard deviation over T years = σ × √T; Volatility scales with the square root of time, variance scales with time.
Standard error of a volatility estimate: Standard error ≈ σ̂ ÷ √(2n); Approximate, for n observations. More data gives a tighter estimate.
Implied volatility condition: c_BSM(S, K, r, q, T, σ_imp) = c_market; Solve for σ_imp numerically. Same for puts: p_BSM(σ_imp) = p_market.
Newton-Raphson update: σ_new = σ_old − (c_BSM(σ_old) − c_market) ÷ Vega; Vega is the derivative of the option price with respect to σ.
VIX (variance form): σ² = (2 ÷ T) Σ [ΔKᵢ ÷ Kᵢ²] e^(RT) Q(Kᵢ) − (1 ÷ T)(F ÷ K₀ − 1)²; VIX = 100 × σ; Q(Kᵢ) is the mid-quote of the out-of-the-money option at strike Kᵢ. K₀ is the first strike below the forward F. T is about 30 days. Know the structure, you are unlikely to compute it in full.
Merton call (continuous yield q): c = S0e^(-qT)N(d1) - Ke^(-rT)N(d2); Use for stock indices, and for stocks with a continuous yield. For a currency, q = rf. For a futures option, S0e^(-qT) becomes F0e^(-rT).
Merton put (continuous yield q): p = Ke^(-rT)N(-d2) - S0e^(-qT)N(-d1); Mirror image of the call. Note the signs on d1 and d2.
d1 and d2 with yield q: d1 = [ln(S0 / K) + (r - q + σ²/2)T] ÷ (σ√T); d2 = d1 - σ√T; The only change from plain BSM is r - q in place of r. Rates and σ are annual and continuously compounded.
Known discrete dividends: Use S* = S0 - PV(dividends) in place of S0, with q = 0; Discount each dividend at the risk-free rate from its payment date to today. Only dividends paid before expiry count. Volatility is that of S*, the stock price net of the dividends.
Currency options (Garman-Kohlhagen): c = S0e^(-rf T)N(d1) - Ke^(-r T)N(d2), with d1 = [ln(S0 / K) + (r - rf + σ²/2)T] ÷ (σ√T); S0 is the spot rate in domestic currency per unit of foreign currency. r is the domestic rate and rf the foreign rate.
Black's model for futures options: c = e^(-rT)[F0N(d1) - KN(d2)]; p = e^(-rT)[KN(-d2) - F0N(-d1)]; d1 = [ln(F0 / K) + σ²T/2] ÷ (σ√T); F0 is the futures price today. The option expires at T, and the formula assumes the futures matures at or after T. σ is the volatility of the futures price.
Put-call parity with yield q: c + Ke^(-rT) = p + S0e^(-qT); For futures options, c + Ke^(-rT) = p + F0e^(-rT). For discrete dividends, replace S0e^(-qT) with S0 - PV(dividends).
Forward price link: F0 = S0e^((r - q)T); Substituting this shows that q = r gives Black's model with F0 as the underlying price.
Dilution factor: Dilution factor = N ÷ (N + M); N = existing shares, M = new shares created if all warrants or options are exercised. Always between 0 and 1.
Warrant value: Warrant value = [N ÷ (N + M)] × c; c is the Black-Scholes-Merton value of an otherwise identical ordinary European call, using the current stock price S0 as the input.
Share price after exercise: S after = (N × S + M × K) ÷ (N + M); S is the pre-exercise price per share. The firm receives M × K in cash and issues M new shares.
Warrant payoff at exercise: Payoff = N × (S − K) ÷ (N + M), if S > K; Equals S after − K. Zero if S ≤ K.
BSM call value: c = S0 × N(d1) − K × e^(−rT) × N(d2); d1 = [ln(S0 ÷ K) + (r + σ²÷2) × T] ÷ (σ × √T); d2 = d1 − σ × √T; For ESOs, set T equal to expected life, not contractual life.
Total cost of an option grant: Total cost = number of options × fair value per option; If dilution is to be reflected, use the diluted per-option value.
American vs European value: C ≥ c and P ≥ p; The early exercise right can only add value. It is never negative.
No-dividend call: C = c; Never optimal to exercise early when the stock pays no dividend and r ≥ 0. Strictly, c > S − K when r > 0.
Lower bound for a European call, no dividends: c ≥ max(S − K·e^(−rT), 0); This bound is above S − K, which proves early exercise is not optimal.
Lower bound for a European put, no dividends: p ≥ max(K·e^(−rT) − S, 0); The American put can be worth K − S, which can exceed this bound. That is why early exercise may pay.
American put-call bounds, no dividends: S − K ≤ C − P ≤ S − K·e^(−rT); Parity is an inequality for American options, not an equality.
Bounds with dividends: S − D − K ≤ C − P ≤ S − K·e^(−rT), where D is the present value of dividends; Use this form when the stock pays known dividends during the option life.
Dividend-adjusted European call: Use S* = S − PV(dividends) in the Black-Scholes-Merton formula; Applies to European options with known discrete dividends.
Early exercise of a call before ex-dividend date: Exercise if D > K·(1 − e^(−r·Δt)) for the last ex-dividend date, roughly; A common rule: the dividend must exceed the interest earned on K until expiry. Treat it as a rule of thumb, since time value also matters.

Quick revision

The model assumes the stock price follows geometric Brownian motion, so returns are normal and prices are lognormal.
Volatility and the risk-free rate are assumed constant, and the model ignores transaction costs.
d1 = [ln(S0 ÷ K) + (r + σ²÷2)T] ÷ (σ√T), and d2 = d1 − σ√T.
European call: c = S0 N(d1) − K e^(−rT) N(d2).
European put: p = K e^(−rT) N(−d2) − S0 N(−d1).
Put-call parity for European options: c + K e^(−rT) = p + S0 for a non-dividend stock.
Implied volatility is the volatility that makes the model price equal the market price, and it is found by iteration.
Historical volatility is estimated from past returns, and it assumes the past is a guide to the future.
For a stock with known dividends, subtract the present value of the dividends from S0 before using the formula.
For options on indices and currencies, replace the dividend adjustment with a continuous yield or the foreign interest rate.
Warrants and employee options create new shares when exercised, which dilutes existing shareholders.
An American call on a non-dividend stock is never optimally exercised early, but an American put may be.

Common mistakes

Using μ as the mean of ln S_T. Fix: Always subtract σ²/2: the mean of ln S_T minus ln S₀ is (μ − σ²/2)T.
Saying the stock price is normally distributed. Fix: The log price is normal, so the price is lognormal. It is skewed and always positive.
Treating N(d2) as the real-world probability of finishing in the money. Fix: N(d2) is a risk-neutral probability, using drift r. Say risk-neutral whenever you interpret it.
Forgetting to discount the strike, using K instead of Ke^(−rT). Fix: Always compute Ke^(−rT) as its own step before combining terms.
Annualizing with √365 instead of √252, or multiplying by 252 instead of √252. Fix: Variance scales with time, volatility scales with its square root. Use the number of periods the question states, and 252 for daily data if none is given.
Dividing by n instead of n − 1, or using prices instead of returns. Fix: With n + 1 prices you have n returns. Use n − 1 in the sample variance unless the question tells you to assume a zero mean.
Using r - q in d1 but forgetting to multiply S0 by e^(-qT) in the price formula (or the reverse). Fix: q appears in two places: in d1 as r - q, and in the price as S0e^(-qT). Check both before you finish.
Applying F0 = S0e^((r - q)T) again to a futures price that is already given, or leaving out the e^(-rT) factor in Black's formula. Fix: In Black's model, F0 is already a forward price. Use it directly in d1 and discount the whole bracket by e^(-rT).
Using M ÷ (N + M) as the dilution factor. Fix: The factor scales the call value down to the holder's share of the gain. It is N ÷ (N + M), and it is always close to 1 when M is small.
Valuing an ESO using its full contractual life. Fix: ESOs are exercised early and forfeited on leaving. Use the expected life as T, or a tree with early exercise.

Exam tips

Expect a numeric question on the mean and standard deviation of ln S_T, then a normal probability. Remember μ − σ²/2 and σ√T.
Watch the wording: 'expected price' uses μ, 'expected log price' uses μ − σ²/2, and 'risk-neutral' uses r.
Conceptual questions often ask what is absent from the BSM equation. The answer is μ, and any risk preference.
Know the list of assumptions and which real-world features violate them: jumps, stochastic volatility, dividends, transaction costs, discrete trading.
Use a financial calculator or scientific mode for e^x and ln. Write each intermediate value to avoid rounding drift.
Expect numerical questions where you must compute d1, d2 and the price, often with N(d) values supplied. Practise the arithmetic until it is automatic.
Interpretation questions are common: N(d2) is the risk-neutral probability of exercise, N(d1) is the call delta, and N(−d2) applies to the put.
Use put-call parity to cross-check answers or to find a missing option price. It can save you two table lookups.