Listed equity options on Apple or Tesla are written on a single stock; a much larger notional sits in index options (SPX, SX5E, Nikkei) and FX options (EUR/USD, USD/JPY). Both are priced by the same Black–Scholes engine once you identify the correct continuous yield on the underlying: the index dividend yield for equities, or the foreign interest rate for currencies. The note develops the index case (including portfolio insurance), then the Garman–Kohlhagen FX formulas, and ends with the limits of the continuous-yield shortcut.
Learning objectives. By the end of this note you should be able to:
- state the risk-neutral SDE and PDE for a dividend-paying index and read the modified Black–Scholes call/put formulas;
- explain portfolio insurance via protective puts on an index and why dynamic replication breaks down in crashes;
- price a European FX option with Garman–Kohlhagen and compute the forward FX rate ;
- explain when the continuous-yield model is adequate and when discrete dividends or quanto adjustments are needed.
In this lesson
- 1.1 Index options and the dividend yield
- 1.2 Portfolio insurance on an index
- 1.3 Currency options — Garman–Kohlhagen
Before you start
- The Black-Scholes PDE, Formula and Implied Volatility
- Derivation: Black-Scholes PDE with Continuous Dividend Yield
The full lesson, with worked examples and exercise solutions, is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Derivatives Pricing and Computational Finance syllabus.