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The Turnbull–Wakeman (J Financ Quant Anal 26:377, 2003) formula is a well-known formula for continuous arithmetic average rate options. However, the Turnbull–Wakeman formula was originally only developed for Asian options when the cost-of-carry is different from zero. In many commodity and energy markets where Asian options frequently trade, the average is typically based on futures or forward prices, that is to say, the cost-of-carry for the underlying asset is zero. Options on stocks can also have a cost-of-carry of zero. If the continuous dividend yield is equal to the risk-free rate, then the extension given in this note can be used in that case as well.

Asian options with zero cost-of-carry: EEX options on freight and iron ore futures