An Asian option's payoff depends on the average price of the underlying over a window — not just the terminal spot. That single design choice makes it the standard hedge for corporate treasurers smoothing FX or commodity exposure over a quarter, and it is cheaper than the corresponding vanilla because volatility of an average is lower than volatility of a terminal price. Under GBM the price is still a risk-neutral expectation, but the payoff functional depends on the whole path through . This note classifies the contract zoo (average price vs average strike, arithmetic vs geometric, continuous vs discrete sampling), writes the two-factor Asian PDE that replaces the vanilla Black–Scholes equation, and states Hull's moment-matching shortcut when a closed form is wanted without solving the PDE. The similarity reduction that collapses one important case to one spatial dimension is carried in the derivation sub-note.
Learning objectives. By the end of this note you should be able to:
- write the four standard Asian payoffs and explain the desk motivation for each;
- distinguish arithmetic and geometric averaging and state which admits an exact lognormal reduction under GBM;
- derive the Asian PDE (6.8) from the integral state variable and Itô's lemma;
- describe the discrete-monitoring update and jump condition at sampling dates;
- apply Hull's two-moment lognormal approximation for an average-price call.
In this lesson
- Payoffs and why desks use them
- Arithmetic vs geometric averaging
- The integral state variable and the Asian PDE
- Discrete monitoring
- Hull's moment-matching approximation
Before you start
- The Black-Scholes PDE, Formula and Implied Volatility
- Stochastic Processes, Brownian Motion and Itô's Lemma
The full lesson, with worked examples and exercise solutions, is for members.
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