How Options Are Priced: From One Contract to One Formula
1. What an option actually gives you
A European call option is a contract: at expiry $T$ you have the right — the right, not the obligation — to buy the underlying at strike $K$. If the underlying is worth $S_T$ on that day, your payoff is:
$$\max(S_T - K,\ 0)$$
A put is the mirror image: the right to sell at $K$, with payoff $\max(K - S_T,\ 0)$.
The asymmetry of this payoff structure is the entire soul of options: your maximum loss is capped at the premium, while the upside (for a call) is unlimited. What that asymmetry is worth is the option pricing problem.
2. Why the naive idea fails
The most intuitive approach is “discounted expected payoff”:
$$C = e^{-rT},\mathbb{E}!\left[\max(S_T - K, 0)\right]$$
The problem appears immediately: the expectation $\mathbb{E}$ depends on your probability beliefs, and probabilities are subjective. An optimist sees big rallies and computes a high option value; a pessimist computes a low one. Price with real-world probabilities and you get a thousand prices from a thousand people — the market cannot converge.
Arbitrage-free pricing changes the question: don’t guess probabilities, ask instead — can the option’s expiry payoff be replicated dynamically using assets that already trade? If yes, the option’s fair value equals the cost of the replicating portfolio today, independent of anyone’s opinion.
3. A one-step binomial tree: the magic of replication
Simplify the world to one step: the underlying is at $S$ today; one period later there are only two states — up to $uS$ or down to $dS$. Build a replicating portfolio: hold $\Delta$ shares of the underlying and borrow cash $B$ (a negative bond position). Force it to match the option value in both states:
$$\begin{cases}\Delta \cdot uS + B,e^{r\Delta t} = C_u\[4pt] \Delta \cdot dS + B,e^{r\Delta t} = C_d\end{cases}$$
Two equations, two unknowns:
$$\Delta = \frac{C_u - C_d}{(u-d)S}, \qquad B = \frac{u,C_d - d,C_u}{u-d},e^{-r\Delta t}$$
The option’s price today is $C = \Delta S + B$.
The key observation: the “probability of an up move, $p$” never appears in the solution — replication eliminated it. Probability is just a bookkeeping device, usually packaged as the risk-neutral probability:
$$q = \frac{e^{r\Delta t} - d}{u - d}$$
so that $C = e^{-r\Delta t}\big[q,C_u + (1-q),C_d\big]$. Formally we are back to “discounted expectation”, but the expectation uses $q$, which has nothing to do with the real world — no-arbitrage pins it down.
Let the step size shrink to zero with up/down moves scaling as $\sigma\sqrt{\Delta t}$, and the binomial tree converges to continuous time — the Black–Scholes–Merton PDE:
$$\frac{\partial V}{\partial t} + \tfrac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$
4. The famous formula
For a European call, the BSM equation has a closed-form solution:
$$C = S,N(d_1) - K e^{-rT} N(d_2)$$
$$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2),T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}$$
where $N(\cdot)$ is the standard normal CDF. The put follows from parity:
$$P = K e^{-rT} N(-d_2) - S,N(-d_1)$$
Put–call parity, $C - P = S - K e^{-rT}$, is an identity that holds under no model assumptions whatsoever. It should be the first sanity check for any pricing code you write.
Two interpretations worth remembering for a lifetime:
- $N(d_1)$: the hedge ratio (delta) — one option is hedged with $N(d_1)$ units of the underlying. It is not the probability of exercise.
- $N(d_2)$: the risk-neutral probability of exercise.
- The whole formula therefore reads: $S,N(d_1)$ is the present value of the underlying you expect to receive; $K e^{-rT} N(d_2)$ is the present value of the strike you expect to pay.
5. Five inputs, one invisible
| Input | Symbol | Where it comes from |
|---|---|---|
| Spot price | $S$ | Quotes — visible |
| Strike | $K$ | Contract terms — visible |
| Time to expiry | $T$ | Calendar — visible |
| Risk-free rate | $r$ | Bond/repo market — visible |
| Volatility | $\sigma$ | Invisible |
Four inputs are public market data; the only unobservable one is $\sigma$. So the essence of option pricing is:
$$\text{given a }\sigma \to \text{get a price}$$
and the market performs the reverse operation every day: invert the traded price to recover $\sigma$ — the implied volatility (IV). Option markets appear to trade premiums; in reality they trade volatility. This is the key to understanding every derivatives market.
6. Greeks: derivatives of price as a portrait of risk
Differentiate with respect to the inputs and you get a set of risk measures:
| Greek | Definition | One-line meaning |
|---|---|---|
| Delta $\Delta$ | $\partial C/\partial S$ | Underlying exposure; the hedge ratio |
| Gamma $\Gamma$ | $\partial^2 C/\partial S^2$ | Rate of change of delta; drives rebalancing frequency and gamma scalping |
| Theta $\Theta$ | $\partial C/\partial t$ | Time value bleeding per day; the “rent” longs pay |
| Vega $\nu$ | $\partial C/\partial \sigma$ | Sensitivity to volatility; the option trader’s main battlefield |
| Rho $\rho$ | $\partial C/\partial r$ | Rate sensitivity; only material for long-dated options |
In practice, a market maker’s P&L is mostly the tug-of-war between vega and gamma: sell volatility (short vega), earn theta, wash out direction with delta hedging, and control gamma cost through rebalancing frequency.
7. Beyond the formula: the numerical toolbox
The BSM closed form covers only the idealized case. Real contracts are messier, each with its natural tool:
| Method | Best for | Character |
|---|---|---|
| Binomial tree / CRR | American early exercise, Bermudan | Intuitive; slow but robust convergence |
| Monte Carlo | Asian, barrier, basket and other path-dependent payoffs | Flexible and universal; error $O(1/\sqrt{N})$ |
| PDE finite differences | American/exotics with rich boundaries | Precise; dimension-limited (≤3) |
| FFT (Carr–Madan) | Pricing a whole strike axis at once | Pairs with characteristic-function models (Heston, Lévy) |
Rule of thumb for choosing: first ask whether the payoff is path-dependent, then whether early exercise matters, and finally whether you need one price or an entire curve.
8. A practical corrections checklist
- Dividends and carry: with a continuous dividend yield $q$, replace $S$ with $S e^{-qT}$ (for futures, use the forward $F$ directly).
- American options: the early-exercise right has value, mainly for deep-ITM puts or calls ahead of large dividends; handle the exercise boundary with trees or PDEs.
- Day counts: trading days or calendar days, 365 or 252 as the annualization base — conventions differ across exchanges; unify them before calibrating data.
- The volatility input: historical volatility, or IV interpolated from the surface? This is where pricing and risk management part ways — the subject of the next note.
9. Summary
Option pricing is relative pricing: no forecasting of up or down, only the requirement that “the option’s payoff can be replicated”. The greatness of the BSM formula is not that “stocks truly follow geometric Brownian motion”, but that it handed the industry a coordinate system that is hedgeable and communicable.
Within that coordinate system the only degree of freedom is $\sigma$. Plot the $\sigma$’s backed out from every strike and every expiry and you’ll find it is not a constant but a rolling surface — fitting that volatility surface is the subject of the next note.