The Black–Scholes machinery was built for a single non-dividend-paying stock. Real desks trade options on indices (with a dividend yield), currencies (with two interest rates), futures (where the traded object is a forward price, not spot), and portfolios of correlated assets. Each case is a parameter tweak or a numeraire change to the same no-arbitrage spine — not a new theory. This topic makes those tweaks explicit, introduces the multi-dimensional PDE when payoffs depend on several underlyings, and closes the foundational unit by naming the volatility smile as the empirical failure mode that forces the next unit: advanced derivatives and exotic options.
By the end of this topic you should be able to:
- price index and FX options with the dividend-yield and Garman–Kohlhagen variants of Black–Scholes;
- state Black's model for options on futures and explain why the forward price enters with zero cost-of-carry drift;
- write the multi-asset Black–Scholes PDE and price exchange options via Margrabe's formula;
- describe basket and rainbow payoffs and choose between closed-form approximations and Monte Carlo;
- interpret the implied-volatility smile as a bridge out of constant- GBM without re-deriving surface calibration machinery.
In this lesson
- Contents
- The big picture
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