This article provides an analytical approximation formula for a swaption price when the instantaneous forward rate follows a Heath–Jarrow–Morton (HJM) model. The author’s approximation strategy, based on the chaos expansion approximation, is to replicate the probability density function of the complex quasi-Gaussian process from a simpler one, which has a semi-closed form solution. It is not restricted to the linear approximation, as is the technique proposed by the existing literature, but can be extended to higher order approximations. Moreover, computation of the approximation is fast; hence, it is suitable for calibration purposes. The author illustrates results through numerical implementation and calibration done using market data. TOPICS: Options, interest-rate and currency swaps, derivatives Key Findings • The authors approximation strategy, based on the chaos expansion approximation, is to replicate the probability density function of the complex quasi-Gaussian process from a simpler one, which has a semi-closed form solution. • It is not restricted to the liner approximation, as is the technique proposed by the existing literature, but can be extended to higher-order approximations. • Computation of the approximation is fast; hence, it is suitable for calibration purposes.

中文翻译：

---

Heath-Jarrow-Morton模型中的近似交换公式

当瞬时远期汇率遵循Heath-Jarrow-Morton（HJM）模型时，本文提供了掉期价格的解析近似公式。作者基于混沌展开逼近的逼近策略是从具有半封闭形式的简单过程中复制复杂的拟高斯过程的概率密度函数。如现有文献所提出的那样，它不限于线性逼近，而是可以扩展到更高阶逼近。而且，近似值的计算是快速的。因此，它适合于校准目的。作者通过使用市场数据进行数值实施和校准来说明结果。主题：期权，利率和货币掉期，导数的主要发现•作者的近似策略基于混沌展开近似，是从一个具有半封闭形式的简单过程中复制复杂拟高斯过程的概率密度函数。•它不像现有文献所提出的那样局限于线性近似，而是可以扩展到高阶近似。•近似值的计算速度很快；因此，它适合于校准目的。•近似值的计算速度很快；因此，它适合于校准目的。•近似值的计算速度很快；因此，它适合于校准目的。