A price alone is almost useless to a trading desk; what matters is how the price moves when the market moves — because that is what determines the hedge. The Greeks are the partial derivatives of the option value with respect to its inputs, and this note covers them twice over: first analytically (definitions, closed forms, parity relations, sign and shape facts), then numerically, for the realistic case where no closed form exists. The numerical half contains the deepest material: the bias/variance analysis of bump-and-revalue estimators, the common-random-numbers rule, and the pathwise estimator with its precise domain of validity. Knowing when the pathwise method fails (digitals, barriers, gamma) is a signature exam question.
Learning objectives. By the end of this note you should be able to:
- define all first-order Greeks and the key second-order Greeks, with their hedging interpretation;
- state the Black–Scholes closed forms and derive Greek parities by differentiating put–call parity;
- analyse the bias and variance of forward/central finite-difference Monte Carlo estimators, including the three variance regimes;
- derive pathwise estimators for delta and vega of European and Asian options;
- explain, with the digital-option counterexample, exactly why the pathwise method fails for discontinuous payoffs.
In this lesson
- 6.1 The Greeks: definitions and properties
- 6.2 Finite-difference (bump) estimators via Monte Carlo
- 6.3 The pathwise estimator
- 6.4 Greeks from trees and grids
The full lesson, with worked examples and exercise solutions, is for members.
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