Equity Option

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.

Want to see what happens when you combine several options into one position — spreads, straddles, condors? The strategy builder draws the combined payoff live, leg by leg.

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

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.

The formulas above are standard textbook formulations, simplified for teaching. They explain the mechanism — they are not a valuation tool, and they will not reproduce a dealer’s price.