Glossary

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Swaption

A Swaption is an option written on the Swap rate $y_{S_0}[\bar{S}]$ defined on the tenor structure $\bar{S}=[S_0,S_1,S_2,\ldots,S_n]$. The pay off at time $S_n$ is: \[\Phi^{\text{Swaption}}(y_{S_0}[\bar{S}])=p(S_o,\bar{S})(y_{S_0}[\bar{S}]-K)^+.\]

Black's formula for Swaptions

Under the Swap market model we have the following. \begin{eqnarray*} \Pi^{\text{SWAPtion}}_t&=&p(t,\bar{S})(y_t[\bar{S}]\text{N}(d_1)-K\text{N}(d_2))\\ d_1&=&\frac{\ln(y_t[\bar{S}]/K)+\Sigma_{t,\bar{S}}^2/2}{\Sigma_{t,\bar{S}}}\\ d_2&=&\frac{\ln(y_t[\bar{S}]/K)-\Sigma_{t,\bar{S}}^2/2}{\Sigma_{t,\bar{S}}}\\ \Sigma_{t,\bar{S}}^2&=&\int_t^{S_0}\sigma(u,\bar{S})^2\text{d} u \end{eqnarray*}

 

Questions: Magnus Wiktorsson
Last update: 2014 Aug 13 17:14:26. Validate: HTML CSS

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