Value a European option on a stock with no dividends through the Black-Scholes-Merton model.

FRM® Part I · Valuation and Risk Models · The Black-Scholes-Merton Model · FRM1.VR.15.d

The idea

For a European option on a stock paying no dividends, Black-Scholes-Merton values the call as c = S0 × N(d1) − K × e^(−rT) × N(d2). The put on the same stock, with the same strike and expiry, is p = K × e^(−rT) × N(−d2) − S0 × N(−d1). The normal distribution is evaluated at two points, and the first is d1 = [ln(S0 ÷ K) + (r + σ²/2) × T] ÷ (σ√T). d2 = d1 − σ√T. Here r is the continuously compounded risk-free rate, σ is the annualised volatility, T is the time to expiry in years, and N is the standard normal cumulative distribution. N(d2) is the risk-neutral probability of exercise for a call, and N(d1) is the call's delta. Put-call parity gives a check, or a shortcut from one price to the other, in the form p = c + K × e^(−rT) − S0. The common errors are forgetting to discount the strike, using N(d) where N(−d) is needed for a put, and mixing up the units of σ and T.

Formulas

Black-Scholes-Merton value of a European call, no dividends

c = S0 × N(d1) − K × e^(−r × T) × N(d2)

S0: the stock price today
N: the standard normal cumulative distribution function
d1: [ln(S0 ÷ K) + (r + σ² ÷ 2) × T] ÷ (σ × √T)
d2: d1 − σ × √T
K: the strike price
r: the continuously compounded risk-free rate
T: the time to expiry in years

European option on a stock paying no dividends during its life. Forgetting to discount the strike is the classic slip.

Black-Scholes-Merton value of a European put, no dividends

p = K × e^(−r × T) × N(−d2) − S0 × N(−d1)

K: the strike price
r: the continuously compounded risk-free rate
T: the time to expiry in years
N: the standard normal cumulative distribution function
d1: [ln(S0 ÷ K) + (r + σ² ÷ 2) × T] ÷ (σ × √T)
d2: d1 − σ × √T
S0: the stock price today

Check it against put-call parity, p = c + K × e^(−r × T) − S0. Using N(d1) and N(d2) instead of N(−d1) and N(−d2) is the usual error.

d1 in the Black-Scholes-Merton formula, no dividends

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

S0: the stock price today
K: the strike price
r: the continuously compounded risk-free rate
σ: the annualised volatility
T: the time to expiry in years

Then d2 = d1 − σ × √T. σ and T must be in matching units: annualise a daily volatility before using it.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which expression gives the Black-Scholes-Merton value of a European put on a non-dividend-paying stock?

S_0 x N(-d1) - K x e^(-rT) x N(-d2)Why not: Reverses the two terms, which gives the negative of the put value.
K x e^(-rT) x N(-d2) - S_0 x N(-d1) - correct
K x e^(-rT) x N(d2) - S_0 x N(d1)Why not: Uses N(d1) and N(d2) rather than N(-d1) and N(-d2) as the put requires.
K x N(-d2) - S_0 x N(-d1)Why not: Does not discount the strike price at the risk-free rate.

The put pays the strike and gives up the stock when exercised, weighted by N(-d2) and N(-d1) respectively, with the strike discounted: p = K e^(-rT) N(-d2) - S_0 N(-d1).

A harder one

A European call on a non-dividend-paying stock has S_0 = 60, K = 58, r = 4% (continuous), volatility 25% and maturity 9 months. Cumulative normal values: N(0.403) = 0.6567, N(0.187) = 0.5741. The BSM value of the call is closest to:

6.10Why not: Does not discount the strike: uses K instead of K x e^(-rT).
3.37Why not: Calculates the put value K e^(-rT) N(-d2) - S N(-d1) rather than the call.
2.44Why not: Applies N(d1) to both terms instead of N(d2) on the strike term.
7.09 - correct

d1 = [ln(60/58) + (0.04 + 0.03125) x 0.75] / (0.25 x sqrt(0.75)) = 0.403; d2 = 0.187. c = 60 x 0.6567 - 58 x e^(-0.03) x 0.5741 = 39.40 - 32.31 = 7.09.

The hardest tier

A stock trades at 75 and has a daily return volatility of 1.8%, annualised with 252 trading days. A 9-month European put has strike 80 and the continuously compounded risk-free rate is 3.5%. Cumulative normal values: N(0.031) = 0.5124, N(0.278) = 0.6097. The BSM value of the put is closest to:

10.6Why not: Annualises the daily volatility with sqrt(365) rather than sqrt(252).
2.9Why not: Uses the daily volatility of 1.8% directly in the formula without annualising it.
9.1 - correct
6.2Why not: Values the call rather than the put.

Annual volatility = 1.8% x sqrt(252) = 28.57%. With T = 0.75: d1 = -0.031, d2 = -0.278. p = 80 x e^(-0.02625) x 0.6097 - 75 x 0.5124 = 47.51 - 38.43 = 9.1.

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