Equity Option
Also known as: Call, Put, Vanilla option
The right — not the obligation — to buy or sell a stock at a fixed price. The atom of derivatives.
- Asset class
- Equity derivatives
- Instrument type
- Option (call / put)
- Traded
- Exchange (listed) and OTC
- Typical users
- Hedgers, income sellers, speculators, market makers
BeginnerWhat is it, really?
An option is a contract that gives you a choice. A call lets you buy a stock at a fixed price (the strike) until a set date (expiry); a put lets you sell at the strike. You pay a premium up front for that choice, and you only use ("exercise") it if it benefits you.
Buy a call with a $100 strike and the stock rockets to $130 — you buy at 100, an instant $30 of value. The stock falls to $80 instead? You walk away, losing only the premium. That asymmetry — capped loss, open-ended gain — is what people pay for.
The seller (writer) of the option takes the other side: they collect the premium and hope the choice expires worthless. Selling options is a business of collecting many small premiums while wearing rare large losses.
IntermediateHow it works in practice
The vocabulary
- Moneyness: in-the-money (exercising pays), at-the-money (strike ≈ spot), out-of-the-money.
- Intrinsic value = payoff if exercised now; time value = premium − intrinsic. Time value melts to zero at expiry ("theta decay").
- American vs. European: American options can be exercised any day; European only at expiry. Most single-stock listed options are American.
- Implied volatility (IV): the volatility number that makes a pricing model match the market premium — the market's forecast of turbulence, and the actual unit in which traders quote options.
The Greeks — how the price moves
| Greek | Sensitivity to | Long call sign |
|---|---|---|
| Delta (Δ) | Stock price | + (0 to 1) |
| Gamma (Γ) | Delta itself (convexity) | + |
| Vega | Implied volatility | + |
| Theta (Θ) | Passage of time | − (you bleed) |
| Rho (ρ) | Interest rates | + |
AdvancedPricing & valuation
Black–Scholes–Merton
Under the BSM assumptions (lognormal spot, constant volatility \(\sigma\), continuous hedging), the price of a European call on a non-dividend stock is
with \(N(\cdot)\) the standard normal CDF; the put follows from put–call parity \(C - P = S_0 - Ke^{-rT}\). The derivation's core is not the formula but the idea: a continuously rebalanced portfolio of \(\Delta = N(d_1)\) shares replicates the option, so its price is the cost of replication — independent of anyone's forecast of direction.
Where the model bends
- Volatility smile/skew: equity IV rises for low strikes (crash insurance). The market prices a whole surface \(\sigma(K,T)\), not one \(\sigma\); models like local vol (Dupire) or stochastic vol (Heston) fit it.
- Dividends: discrete dividends lower forward price; American calls on dividend payers may be exercised early just before ex-dates, American puts early when deep ITM (priced on binomial/finite-difference grids).
What a desk actually does
Market makers run delta-hedged books: buy the option, short \(\Delta\) shares, rebalance. Their P&L over a hedge interval is the classic gamma-theta tradeoff:
— long options make money when realised volatility beats the implied vol paid, and vice versa. Options are, at bottom, a market for volatility.