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.

Turnbull-Wakeman (J Financ Quant Anal 26:377, 2003) 公式是连续算术平均利率期权的著名公式。然而，Turnbull-Wakeman 公式最初仅针对持有成本不为零的亚洲期权而开发。在许多亚洲期权交易频繁的商品和能源市场中，平均值通常基于期货或远期价格，也就是说，标的资产的携带成本为零。股票期权的持有成本也可能为零。如果连续股息收益率等于无风险利率，那么在这种情况下也可以使用本说明中给出的扩展。