What the Black‑Scholes model is
– The Black‑Scholes model (also called Black‑Scholes‑Merton or BSM) is a closed‑form mathematical formula used to estimate the fair price of a European option. A European option is a contract that can only be exercised at expiration (not before).
– The standard BSM call formula assumes no dividends and a frictionless market; it produces a theoretical price you can use as a benchmark or starting point for trading and risk analysis.
Definitions (short)
– Option: a derivative contract giving the right, but not the obligation, to buy (call) or sell (put) an underlying asset at a specified price (strike) on or before a set date.
– European option: exercisable only at expiration.
– Volatility (σ): the annualized standard deviation of the underlying asset’s continuously compounded returns.
– Risk‑free rate (r): the continuously compounded return of a default‑free asset (often a short‑term government rate).
– N(·): cumulative distribution function (CDF) of the standard normal distribution (probability that a normal variable ≤ x).
Key formula (European call, no dividends)
C = S · N(d1) − K · e^(−rT) · N(d2)
where
d1 = [ln(S/K) + (r + 0.5·σ^2)·T] / (σ·√T)
d2 = d1 − σ·√T
Variables
– C = call option price (theoretical)
– S = current underlying price
– K = strike price
– r = risk‑free interest rate (continuous)
– σ = volatility (annual, continuous)
– T = time to expiry (in years)
– N(·) = standard normal CDF
Main assumptions (what BSM assumes)
– Underlying price follows geometric Brownian motion: continuous paths, no jumps; returns are normally distributed → prices are lognormal.
– Volatility (σ) and risk‑free rate (r) are constant over the option’s life.
– No dividends (original model) or continuous known dividends in some extensions.
– No transaction costs, no taxes, continuous trading, and unlimited borrowing/lending at r.
– Markets allow instantaneous rebalancing for continuous hedging.
What the model does—and does not—do
– Does: Gives a closed‑form price for European options, produces analytic risk sensitivities (“Greeks”), and serves as a common benchmark for implied volatility.
– Does not: Accurately price American options (early exercise), handle jumps or time‑varying volatility, or explain the observed patterns of implied volatility across strikes (the “volatility skew” or “smile”) without extensions.
Practical checklist before using Black‑Scholes
1. Confirm option type: Is it European? If American (early exercise possible), consider