Every strategy so far — bull spreads, bear spreads, butterflies, straddles, strangles — was drawn against a single stock. On a real derivatives desk, just as much volume trades on two other underlyings entirely: a broad stock index (the S&P 500, the FTSE 100) and a currency pair (EUR/USD, USD/JPY). The good news is you don't need new machinery. Every payoff diagram, every spread, every bound from the last four lessons carries straight over — the only change is a single number quietly inserted into the pricing formulas: a continuous yield the holder of the underlying gives up by holding the option instead of the underlying itself.

The one adjustment: a continuous yield q

Recall why a dividend lowers a call's value: the option holder doesn't collect dividends paid on the underlying before exercise, so the stock (and the call's claim on it) is effectively worth less to them than the quoted spot price suggests. A stock index and a foreign currency both have a direct analogue of "dividends" — a steady stream of value the asset earns that the option holder does not:

In every case, the fix to the pricing machinery you already know is the same substitution: wherever a plain stock formula uses spot S_0 "raw," an index or FX formula uses the yield-discounted spot S_0 e^{-qT} instead — the value of the asset with its continuous payout stripped out. Put–call parity, which you already proved for a plain stock, becomes

and the model-free bounds from the earlier lesson become c \ge \max(S_0 e^{-qT} - Ke^{-rT},\ 0) and c \le S_0 e^{-qT} — the ceiling itself drops, because the option can never be worth more than the yield-adjusted value of the asset it tracks.

Index options: q as a dividend yield

A broad index like the S&P 500 is really a basket of hundreds of stocks, each paying its own dividends on its own schedule. Rather than track hundreds of discrete payment dates, the standard practitioner approximation treats the whole basket as paying dividends continuously, at a blended annualized rate q — typically somewhere around 1\%\text{–}2\% for a major developed-market index. Every strategy from the earlier lessons still applies unchanged to index options — bull spreads on the S&P 500, straddles on the FTSE ahead of a rate decision — just priced with S_0 e^{-qT} standing in for spot.

Watch how the effective (yield-discounted) value used in pricing pulls away from raw spot as either the dividend yield or the time to maturity grows:

Currency options: two interest rates, not one

A currency option is subtler, because there is no single "the" risk-free rate — there are two, one per currency. Consider a call that gives the right to buy one euro for K U.S. dollars. Holding a euro today earns the foreign (euro) risk-free rate r_f; the strike is paid in dollars, discounted at the domestic (dollar) rate r_d. The foreign rate takes over exactly the role a dividend yield played for a stock — an interest stream the option holder gives up by holding the option instead of the currency itself — while the domestic rate plays the role of "the" risk-free rate everywhere else:

This is exactly the covered-interest-rate-parity relationship between spot and forward FX rates, wearing an options hat: it is why a currency's forward price sits above or below spot — and a currency option is, in practice, almost always priced and quoted off the forward rate rather than spot directly, since the forward already bakes in both rates cleanly.