We analyze European options on CMS spreads, obtaining closed form formulas for the values of these instruments. There are three key steps in this analysis. The first step is to create a hybrid numeraire in which the spread ${\tilde{R}}_1\left(T\right)-{\tilde{R}}_2\left(T\right)$ is a Martingale. Like other CMS calculations, this reduces valuation to the valuation of standard European options plus quadratic convexity corrections. The second step is to combine the volatility smiles of the individual swap rates ${\tilde{R}}_1$ and ${\tilde{R}}_2$ to obtain the smile of the spread ${\tilde{R}}_1\left(T\right)-{\tilde{R}}_2\left(T\right)$. We do this by modeling the volatility smiles of both swap rates by normal $\left(\text{β}=0\right)$ SABR models. (In practice this may require using a shifter to convert SABR models with $\text{β}\ne 0$ into SABR models with $\text{β}=0$.) We then show that the volatility smile of the spread ${\tilde{R}}_1\left(T\right)-{\tilde{R}}_2\left(T\right)$ is also governed by a normal SABR model, and derive the SABR parameters for the spread. The third step is to use the closed-form formulae for quadratic options under the SABR model to obtain explicit formulae for the valuation of European options on CMS spreads. These formulas are not exact, but they are accurate up to $O\left({\epsilon }^2\right)$, the same accuracy as the original SABR formulas. They also satisfy call-put parity exactly, and are exactly consistent with the valuation of CMS options on the component swap rates ${\tilde{R}}_1$ and ${\tilde{R}}_2$.

我们分析 CMS 价差的欧式期权，获得这些工具价值的封闭式公式。这个分析有三个关键步骤。第一步是创建一个混合计算器，其中价差${\widetilde{\mathrm{电阻}}}_1\left(吨\right)-{\widetilde{\mathrm{电阻}}}_2\left(吨\right)$是马丁格尔。与其他 CMS 计算一样，这将估值降低到标准欧式期权的估值加上二次凸度修正。第二步是结合各个掉期利率的波动率微笑${\widetilde{\mathrm{电阻}}}_1$ 和 ${\widetilde{\mathrm{电阻}}}_2$ 获得传播的微笑 ${\widetilde{\mathrm{电阻}}}_1\left(吨\right)-{\widetilde{\mathrm{电阻}}}_2\left(吨\right)$. 我们通过对两个掉期利率的波动率微笑进行正常建模来做到这一点$\left(\text{β}=0\right)$SABR 模型。（在实践中，这可能需要使用移位器将 SABR 模型转换为$\text{β}\ne 0$ 进入 SABR 模型 $\text{β}=0$.) 然后我们证明点差的波动率微笑 ${\widetilde{\mathrm{电阻}}}_1\left(吨\right)-{\widetilde{\mathrm{电阻}}}_2\left(吨\right)$也受正常 SABR 模型的控制，并为价差导出 SABR 参数。第三步，利用SABR模型下二次期权的闭式公式，得到CMS价差欧式期权估值的明确公式。这些公式并不准确，但它们准确到$哦\left({\epsilon }^2\right)$，与原始 SABR 公式的精度相同。它们也完全满足看涨期权平价，并且与组件掉期利率上的 CMS 期权估值完全一致${\widetilde{\mathrm{电阻}}}_1$ 和 ${\widetilde{\mathrm{电阻}}}_2$.