What Is the Black-Scholes Model? (Plain English)
May 24, 2025·3 min read
What Is the Black-Scholes Model? (Plain English)
The Black-Scholes model mathematically describes the variation of a financial instrument over time. It is used to price options. The formula is:
The Formula — what each variable means
- C — the price of the call option
- S₀ — the current price of the underlying asset
- K — the strike price of the option
- r — the risk-free interest rate
- t — the time to expiration
- N(·) — the cumulative distribution function of the standard normal distribution
- σ — the volatility of the underlying asset
- d₁ = [ln(S₀/K) + (r + σ²/2)t] / σ√t
- d₂ = d₁ − σ√t
And this is what it means:
For a call option
It is the present value of the expected payoff if the option is in the money, minus the present value of the strike price multiplied by the probability that the option is in the money. In essence, the price is the sum of each possible payout, weighted by the probability of that payout occurring in upside scenarios.
For a put option
It is the present value of the strike price multiplied by the probability that the option is in the money, minus the present value of the expected payoff if the option is in the money. In essence, the price is the sum of each possible payout, weighted by the probability of that payout occurring in downside scenarios.
The Black-Scholes model is slightly more complicated than the Black model because it includes dividends and other factors that affect the price of the underlying asset. The Black model is more straightforward — no dividends — and is therefore used to price options on futures contracts.
The bottom line
Black-Scholes is an options pricing model. It represents the fair value of an option based on the weighted probabilities of different outcomes — including zero payouts and payouts along the full probability distribution.
Did you know?
A swap is just a portfolio of options. If you buy a swap, you are buying a call and selling a put at the strike of the swap. If you sell a swap, you are selling a call and buying a put at the strike of the swap.
Swaps have a delta lower than 1. The delta of the swap is the sum of the delta of the call and the delta of the put — and that sum is lower than 1. Why? Because of the present value component of the options price. The delta of the swap is the present value of its strike price.
This is fascinating stuff. If you want to discuss more about options pricing, reach out any time. In the meantime, if you want to play around with the Black-Scholes model, check out our options calculator.