What is the difference between arithmetic and geometric Asian options?

Question

AcadiFi · Accepted Answer

An Asian option has a payoff based on the average price of the underlying over a monitoring period rather than the terminal spot. The two flavors differ in how the average is computed: arithmetic average uses the standard mean, while geometric average uses the n-th root of the product of observations. Geometric Asian options have closed-form pricing because, if spot prices are log-normal, the geometric average is also log-normal (product of log-normals is log-normal). Black-Scholes-style formulas apply with adjusted volatility and drift. Arithmetic averages of log-normals are not log-normal, so no closed-form exists and pricing requires Monte Carlo or approximation methods like moment-matching. Example: quantitative analyst Rosalind Fontaine at Meridian Arc Capital prices an average-rate Asian call on Borealis Copper with monthly observations over six months. The geometric version is priced analytically in seconds, giving 3.15. The arithmetic version requires 100,000 Monte Carlo paths and yields 3.28 — always slightly higher because the arithmetic average exceeds the geometric by Jensen's inequality. ```mermaid flowchart LR A[Asian Option] --> B[Arithmetic Avg] A --> C[Geometric Avg] B --> D[No closed form
Monte Carlo] C --> E[Closed form
log-normal] ``` Asian options cost less than vanilla because averaging reduces effective volatility, and they are popular in commodities and FX where manipulation of a single settlement price is a concern.

What is the difference between arithmetic and geometric Asian options?

Master Level II with our CFA Course

Related Questions

Practice Questions