Black-Scholes Option Pricing

Stub note

This note states the key results needed by downstream notes (the-greeks, lp-as-short-vol). The full derivation — risk-neutral pricing, the replicating portfolio argument, assumptions and limitations — will be written separately. Reference: Hull, Options, Futures, and Other Derivatives, Ch. 13–15.

Prerequisites

• options-basics — calls, puts, payoff diagrams
• volatility — realized vs implied volatility

The Key Result

The Black-Scholes model (Black & Scholes 1973, Merton 1973) gives a closed-form price for a European option. For a call:

$C=SN(d_1)-K{e}^{-rT}N(d_2)$

where:

$d_1=\frac{\ln (S/K)+(r+{\sigma }^2/2)T}{\sigma \sqrt{T}},d_2=d_1-\sigma \sqrt{T}$

Notation

• $S$ — current price of the underlying
• $K$ — strike price
• $T$ — time to expiration (in years)
• $r$ — risk-free interest rate (continuously compounded)
• $\sigma$ — volatility of the underlying (annualized)
• $N(\cdot )$ — cumulative standard normal distribution function

Why This Formula Matters

The formula’s importance is not the number it produces — it’s the argument behind it:

1. Replicating portfolio. An option can be replicated by continuously adjusting a position in the underlying stock and a risk-free bond. The cost of maintaining this replicating portfolio equals the option price.

2. Delta-hedging. The replicating portfolio holds $\Delta =N(d_1)$ shares of the underlying at each instant. Continuously adjusting this hedge (buying when delta rises, selling when it falls) is delta-hedging. The cumulative cost of delta-hedging over the option’s life converges to the realized variance of the underlying.

3. The fundamental P&L equation. For a delta-hedged short option position, the P&L over each small interval is approximately:

   $\text{P\&L}\approx \frac{1}{2}\Gamma S^2({\sigma }_{\text{implied}}^2-{\sigma }_{\text{realized}}^2)dt$

   If realized vol < implied vol, the hedged seller profits. If realized vol > implied vol, the hedged seller loses. This is the bridge to the-greeks (where $\Gamma$ is defined) and to lp-as-short-vol (where LP losses are shown to equal the cost of replicating a short option).

Assumptions

The Black-Scholes formula assumes:

• Geometric Brownian motion — log-returns are normally distributed with constant volatility $\sigma$
• No transaction costs — continuous delta-hedging is free
• No dividends (basic version; extended versions handle dividends)
• Constant risk-free rate $r$
• European exercise only (no early exercise)
• No jumps in the underlying price

Every assumption fails in practice. The model remains the industry standard because it provides a common language (implied volatility) and a hedging framework that works approximately. Traders quote options in terms of implied vol precisely because the B-S formula provides the mapping between price and vol.

The Greeks Are Partial Derivatives of This Formula

The Greeks are defined as partial derivatives of the Black-Scholes price:

Greek	Definition	Closed form (call)
Delta ($\Delta$)	$\frac{\partial C}{\partial S}$	$N(d_1)$
Gamma ($\Gamma$)	$\frac{{\partial }^{2}C}{\partial S^2}$	$\frac{N^{\prime }(d_1)}{S\sigma \sqrt{T}}$
Theta ($\Theta$)	$\frac{\partial C}{\partial t}$	$-\frac{S{N}^{\prime }(d_1)\sigma }{2\sqrt{T}}-rK{e}^{-rT}N(d_2)$
Vega ($\nu$)	$\frac{\partial C}{\partial \sigma }$	$S\sqrt{T}{N}^{\prime }(d_1)$

where $N^{\prime }(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$ is the standard normal density.

See also

• options-basics — what calls and puts are
• volatility — realized vs implied vol
• the-greeks — how to use the partial derivatives above
• lp-as-short-vol — the DeFi application of the hedging argument