Structured equity and fund-linked notes often pay on several underlyings at once: a basket of stocks, the spread between two indices, or the worst performer in a portfolio. Under multivariate GBM, the price satisfies a multi-dimensional Black–Scholes PDE with cross-derivative terms . Closed forms exist for the exchange option (Margrabe); baskets and rainbows generally require approximation or Monte Carlo. This note classifies the main payoffs, states the PDE, derives Margrabe's formula, and gives a simulation recipe.
Learning objectives. By the end of this note you should be able to:
- write the -dimensional Black–Scholes PDE with correlation and dividend yields;
- explain why an option on a basket is cheaper than a basket of options (Jensen);
- price a European exchange option with Margrabe's formula;
- simulate correlated paths for basket and rainbow payoffs and estimate standard errors.
In this lesson
- 3.1 The multi-asset Black–Scholes model
- 3.2 Basket options
- 3.3 Exchange options — Margrabe's formula
- 3.4 Rainbow options
Before you start
- The Black-Scholes PDE, Formula and Implied Volatility
- Correlated Randoms and Low-Discrepancy Sequences
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