17.3 The Black-Scholes-Merton Model

The Black-Scholes-Merton Model

The Black–Scholes–Merton (BSM) model is the continuous-time limit of the binomial model for European options when the underlying follows geometric Brownian motion. FRM Part I expects you to know the distributional story, the assumptions, how to use the call/put formulas conceptually, what implied volatility means, and how dividends, futures, FX, and warrants change the setup.

Lognormal Prices and Normal Returns

Under GBM, dS = μ S dt + σ S dW. Integrating gives

ln(S_T / S_0) ~ Normal( (μ − σ²/2) T , σ² T )

so S_T is lognormal. Equivalently, the continuously compounded return over horizon T is normal. The −σ²/2 adjustment is the Itô correction: E[S_T] = S_0 e^(μ T), not S_0 e^((μ−σ²/2)T).

Worked lognormal probability

S0 = 100, μ = 0.10, σ = 0.20, T = 1. Then ln(S_T/100) ~ N(0.10 − 0.5×0.04, 0.04) = N(0.08, 0.04). P(S_T > 100) = P(ln(S_T/100) > 0) = P( Z > (0 − 0.08)/0.20 ) = P(Z > −0.40) ≈ 0.655 where Z is standard normal. Median of S_T is 100 e^0.08 ≈ 108.3; mean is 100 e^0.10 ≈ 110.5.

Realized Return and Historical Volatility

Realized (continuously compounded) return over a sample interval is r_i = ln(S_i / S_(i−1)).

Historical volatility: compute the sample standard deviation s of the r_i series, then annualize. If observations are daily and there are 252 trading days,

σ_annual ≈ s × √252

For weekly data, multiply by √52. Use n − 1 in the sample variance denominator unless the question specifies otherwise.

Worked historical vol

Five daily log returns: 0.012, −0.008, 0.005, 0.010, −0.004. Mean m = 0.003. Squared deviations ≈ 0.000081, 0.000121, 0.000004, 0.000049, 0.000049; sum = 0.000304. Sample variance = 0.000304 / 4 = 0.000076; s ≈ 0.00872. Annualized σ ≈ 0.00872 × √252 ≈ 0.00872 × 15.87 ≈ 13.8%.

BSM Assumptions

Classic assumptions (exam checklist):

1. Underlying follows GBM with constant μ and constant σ.
2. No dividends (original Black–Scholes) or continuous yield q (Merton extension).
3. Constant risk-free rate r; continuous compounding.
4. No transaction costs or taxes; unlimited short selling; continuous trading.
5. No arbitrage; European exercise only (for the closed form).
6. Lognormal terminal prices / normal log returns.

Violations (stochastic vol, jumps, discrete dividends, early exercise) push you to trees, local/stochastic-vol models, or numerical PDE methods.

European Non-Dividend Call and Put

With continuous rates and no dividends,

c = S0 N(d1) − K e^(−rT) N(d2)

p = K e^(−rT) N(−d2) − S0 N(−d1)

where

d1 = [ln(S0/K) + (r + σ²/2) T] / (σ √T)

d2 = d1 − σ √T

N(·) is the standard normal CDF. Interpretation: N(d2) is itself the risk-neutral probability that the call finishes in-the-money (P*(S_T > K)); multiplying by e^(−rT) turns it into the price of a cash-or-nothing digital paying $1 if S_T > K. Do not conflate the two—N(d2) is the probability, e^(−rT)N(d2) is the discounted value of that bet. N(d1) is the corresponding probability under the stock numeraire and equals the call's delta for a non-dividend stock.

Worked conceptual calculation

S0 = 49, K = 50, r = 5%, σ = 20%, T = 0.3846 (≈ 20 weeks).

σ √T = 0.20 × √0.3846 ≈ 0.20 × 0.620 ≈ 0.124.

ln(S0/K) = ln(0.98) ≈ −0.0202.

(r + σ²/2)T = (0.05 + 0.02) × 0.3846 ≈ 0.0269.

d1 = (−0.0202 + 0.0269) / 0.124 ≈ 0.054; d2 ≈ 0.054 − 0.124 = −0.070.

N(d1) ≈ 0.5216; N(d2) ≈ 0.4721.

c ≈ 49 × 0.5216 − 50 e^(−0.05×0.3846) × 0.4721 ≈ 25.56 − 50 × 0.981 × 0.4721 ≈ 25.56 − 23.16 ≈ 2.40.

Put from parity: p = c + K e^(−rT) − S0 ≈ 2.40 + 49.05 − 49 ≈ 2.45.

You are not expected to memorize CDF tables beyond rough N(0) = 0.5, but you must know the formula structure and how inputs move price.

Implied Volatility

Implied volatility (IV) is the constant σ plugged into BSM that reproduces the observed market price. It is a quote convention: options are spoken in vol points, not dollars. Because one σ cannot fit all strikes, markets show a smile/skew: IV(K, T) varies with strike and maturity. Historical vol looks backward; IV looks forward through prices.

If two otherwise identical options imply different σ, at least one assumption (constant vol, diffusion-only) is wrong—or quotes are stale/misaligned.

Dividends and Early Exercise

Continuous yield q (Merton): replace S0 with S0 e^(−qT) in the formulas (or use d1, d2 with r − q):

c = S0 e^(−qT) N(d1) − K e^(−rT) N(d2)

Discrete known dividends: replace S0 with S0 − PV(dividends) for Europeans, then apply non-dividend BSM on the escrowed stock. This fails for Americans when early exercise around ex-dividend dates matters—use a tree.

Early exercise: BSM is for Europeans. American calls on non-dividend stocks equal Europeans (never exercise early). American puts (and American calls with large dividends) need numerical methods; BSM understates American put value.

Dividend, Futures, and FX Europeans

Black ’76 is the market standard for options on futures and many interest-rate options quoted on a forward underlying. Carry is embedded in F0; you do not separately estimate q.

Worked FX mapping

USD call on EUR (right to buy EUR for K dollars). Domestic rate = USD rate r, foreign rate = EUR rate r_f. Same algebra as an index with q = r_f. Higher foreign rates lower the FX call (EUR is “costlier to carry”), analogous to higher stock dividends.

Warrants and Dilution

A warrant is a long-dated call issued by the company on its own stock. When exercised, the firm issues new shares, diluting existing equity. If a firm has N shares outstanding and M warrants with strike K, upon exercise the firm receives M·K cash and the equity claim is split among N+M shares. A common dilution adjustment prices a claim on the firm’s assets with

effective underlying proportional to N/(N+M)

and scales the warrant value accordingly (Black–Scholes on firm value, multiply by N/(N+M) per warrant, under stylized assumptions). Exam cue: ordinary exchange-traded calls do not dilute; warrants do—so warrant value < undiluted BSM call on the stock, all else equal.

Worked dilution sketch

N = 10 million shares, M = 1 million warrants, stock price 50 as if no warrants. Rough dilution factor N/(N+M) = 10/11 ≈ 0.909. If an undiluted European call would be 8.00, a stylized diluted warrant might be worth about 8.00 × 0.909 ≈ 7.27 (actual formulas use asset value and simultaneous equity/warrant valuation—direction matters more than the exact 7.27).

Synthesis

BSM = lognormal S_T + European exercise + constant σ,r (+ optional continuous yield). Use N(d1), N(d2) formulas for non-dividend Europeans; invert for IV; switch to trees for Americans/discrete dividends; map q, r_f, or Black ’76 for indices/FX/futures; dilute for warrants.