CMS Pricing - Convexity Adjustment by Replication












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I'm trying to learn CMS pricing, but didn't get the logic of this method. Previously cited articles about this method is pretty complex.
I'd be glad if you can provide me with simpler articles or spreadsheets to give an idea about replication of swaptions.






fixed-income interest-rates derivatives swaption cms







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    $begingroup$


    I'm trying to learn CMS pricing, but didn't get the logic of this method. Previously cited articles about this method is pretty complex.
    I'd be glad if you can provide me with simpler articles or spreadsheets to give an idea about replication of swaptions.






    fixed-income interest-rates derivatives swaption cms







    share|improve this question







    New contributor




    user38753 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.







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      4












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      $begingroup$


      I'm trying to learn CMS pricing, but didn't get the logic of this method. Previously cited articles about this method is pretty complex.
      I'd be glad if you can provide me with simpler articles or spreadsheets to give an idea about replication of swaptions.






      fixed-income interest-rates derivatives swaption cms







      share|improve this question







      New contributor




      user38753 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.







      $endgroup$




      I'm trying to learn CMS pricing, but didn't get the logic of this method. Previously cited articles about this method is pretty complex.
      I'd be glad if you can provide me with simpler articles or spreadsheets to give an idea about replication of swaptions.






      fixed-income interest-rates derivatives swaption cms




      fixed-income interest-rates derivatives swaption cms






      share|improve this question







      New contributor




      user38753 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|improve this question







      New contributor




      user38753 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|improve this question




      share|improve this question






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      user38753 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      asked 9 hours ago









      user38753user38753

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      user38753 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






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          $begingroup$

          The CMS represents the value of a swap rate for any point in time, i.e. we are interested in extrapolating the density of the swap rate in a similar way as the IBOR rate.
          Let us start with the fair value of a swaption under the annuity measure $mathcal{A}$ with tenor at time $tau$:
          $$mathcal{A}(t)mathbb{E}^mathcal{A}_t[(mathcal{S}(tau)-k)^+]$$
          Instead of having it paid out as a annuity over time, we want to evaluate the flow for paying it out at any given time $T$. We are, thus, in a change of measure from the annuity to a $T$-forward measure.
          Denoting the bond value value today maturing at $T$ by $B_{t,T}$, the CMS flow at time $t<T$ under the annuity measure is
          $$mathcal{A}(t)mathbb{E}^mathcal{A}_tleft[frac{mathcal{S}(tau)}{B_{tau,T}mathcal{A}(tau)}right]$$
          whilst under the $T$-forward measure it is:
          $$mathbb{E}_t^T[mathcal{S}(tau)]=mathbb{E}^mathcal{A}_tleft[mathcal{S}(tau)frac{dT}{dmathcal{A}}right]$$ with the Radon-Nykodim derivative $frac{dT}{dmathcal{A}}=frac{mathcal{A}(t)B_{t,T}}{B_{tau,T}mathcal{A}(tau)}$.
          The CMS convexity adjustment is the difference between the expectations under these two measures. In order to compute this convexity adjustment, one has to find an approximation for $mathbb{E}^S_t[1/B_{S,T}|mathcal{S}_tau]$, which can be done following Cedervall and Piterbarg (2012) CMS: covering all bases either assuming a non-stochastic Libor-OIS basis spread or finding an explicit expression of T-bonds in terms of swap rates, which allows to obtain the swap density.






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            $begingroup$

            The CMS represents the value of a swap rate for any point in time, i.e. we are interested in extrapolating the density of the swap rate in a similar way as the IBOR rate.
            Let us start with the fair value of a swaption under the annuity measure $mathcal{A}$ with tenor at time $tau$:
            $$mathcal{A}(t)mathbb{E}^mathcal{A}_t[(mathcal{S}(tau)-k)^+]$$
            Instead of having it paid out as a annuity over time, we want to evaluate the flow for paying it out at any given time $T$. We are, thus, in a change of measure from the annuity to a $T$-forward measure.
            Denoting the bond value value today maturing at $T$ by $B_{t,T}$, the CMS flow at time $t<T$ under the annuity measure is
            $$mathcal{A}(t)mathbb{E}^mathcal{A}_tleft[frac{mathcal{S}(tau)}{B_{tau,T}mathcal{A}(tau)}right]$$
            whilst under the $T$-forward measure it is:
            $$mathbb{E}_t^T[mathcal{S}(tau)]=mathbb{E}^mathcal{A}_tleft[mathcal{S}(tau)frac{dT}{dmathcal{A}}right]$$ with the Radon-Nykodim derivative $frac{dT}{dmathcal{A}}=frac{mathcal{A}(t)B_{t,T}}{B_{tau,T}mathcal{A}(tau)}$.
            The CMS convexity adjustment is the difference between the expectations under these two measures. In order to compute this convexity adjustment, one has to find an approximation for $mathbb{E}^S_t[1/B_{S,T}|mathcal{S}_tau]$, which can be done following Cedervall and Piterbarg (2012) CMS: covering all bases either assuming a non-stochastic Libor-OIS basis spread or finding an explicit expression of T-bonds in terms of swap rates, which allows to obtain the swap density.






            share|improve this answer









            $endgroup$


















              4












              $begingroup$

              The CMS represents the value of a swap rate for any point in time, i.e. we are interested in extrapolating the density of the swap rate in a similar way as the IBOR rate.
              Let us start with the fair value of a swaption under the annuity measure $mathcal{A}$ with tenor at time $tau$:
              $$mathcal{A}(t)mathbb{E}^mathcal{A}_t[(mathcal{S}(tau)-k)^+]$$
              Instead of having it paid out as a annuity over time, we want to evaluate the flow for paying it out at any given time $T$. We are, thus, in a change of measure from the annuity to a $T$-forward measure.
              Denoting the bond value value today maturing at $T$ by $B_{t,T}$, the CMS flow at time $t<T$ under the annuity measure is
              $$mathcal{A}(t)mathbb{E}^mathcal{A}_tleft[frac{mathcal{S}(tau)}{B_{tau,T}mathcal{A}(tau)}right]$$
              whilst under the $T$-forward measure it is:
              $$mathbb{E}_t^T[mathcal{S}(tau)]=mathbb{E}^mathcal{A}_tleft[mathcal{S}(tau)frac{dT}{dmathcal{A}}right]$$ with the Radon-Nykodim derivative $frac{dT}{dmathcal{A}}=frac{mathcal{A}(t)B_{t,T}}{B_{tau,T}mathcal{A}(tau)}$.
              The CMS convexity adjustment is the difference between the expectations under these two measures. In order to compute this convexity adjustment, one has to find an approximation for $mathbb{E}^S_t[1/B_{S,T}|mathcal{S}_tau]$, which can be done following Cedervall and Piterbarg (2012) CMS: covering all bases either assuming a non-stochastic Libor-OIS basis spread or finding an explicit expression of T-bonds in terms of swap rates, which allows to obtain the swap density.






              share|improve this answer









              $endgroup$
















                4












                4








                4





                $begingroup$

                The CMS represents the value of a swap rate for any point in time, i.e. we are interested in extrapolating the density of the swap rate in a similar way as the IBOR rate.
                Let us start with the fair value of a swaption under the annuity measure $mathcal{A}$ with tenor at time $tau$:
                $$mathcal{A}(t)mathbb{E}^mathcal{A}_t[(mathcal{S}(tau)-k)^+]$$
                Instead of having it paid out as a annuity over time, we want to evaluate the flow for paying it out at any given time $T$. We are, thus, in a change of measure from the annuity to a $T$-forward measure.
                Denoting the bond value value today maturing at $T$ by $B_{t,T}$, the CMS flow at time $t<T$ under the annuity measure is
                $$mathcal{A}(t)mathbb{E}^mathcal{A}_tleft[frac{mathcal{S}(tau)}{B_{tau,T}mathcal{A}(tau)}right]$$
                whilst under the $T$-forward measure it is:
                $$mathbb{E}_t^T[mathcal{S}(tau)]=mathbb{E}^mathcal{A}_tleft[mathcal{S}(tau)frac{dT}{dmathcal{A}}right]$$ with the Radon-Nykodim derivative $frac{dT}{dmathcal{A}}=frac{mathcal{A}(t)B_{t,T}}{B_{tau,T}mathcal{A}(tau)}$.
                The CMS convexity adjustment is the difference between the expectations under these two measures. In order to compute this convexity adjustment, one has to find an approximation for $mathbb{E}^S_t[1/B_{S,T}|mathcal{S}_tau]$, which can be done following Cedervall and Piterbarg (2012) CMS: covering all bases either assuming a non-stochastic Libor-OIS basis spread or finding an explicit expression of T-bonds in terms of swap rates, which allows to obtain the swap density.






                share|improve this answer









                $endgroup$



                The CMS represents the value of a swap rate for any point in time, i.e. we are interested in extrapolating the density of the swap rate in a similar way as the IBOR rate.
                Let us start with the fair value of a swaption under the annuity measure $mathcal{A}$ with tenor at time $tau$:
                $$mathcal{A}(t)mathbb{E}^mathcal{A}_t[(mathcal{S}(tau)-k)^+]$$
                Instead of having it paid out as a annuity over time, we want to evaluate the flow for paying it out at any given time $T$. We are, thus, in a change of measure from the annuity to a $T$-forward measure.
                Denoting the bond value value today maturing at $T$ by $B_{t,T}$, the CMS flow at time $t<T$ under the annuity measure is
                $$mathcal{A}(t)mathbb{E}^mathcal{A}_tleft[frac{mathcal{S}(tau)}{B_{tau,T}mathcal{A}(tau)}right]$$
                whilst under the $T$-forward measure it is:
                $$mathbb{E}_t^T[mathcal{S}(tau)]=mathbb{E}^mathcal{A}_tleft[mathcal{S}(tau)frac{dT}{dmathcal{A}}right]$$ with the Radon-Nykodim derivative $frac{dT}{dmathcal{A}}=frac{mathcal{A}(t)B_{t,T}}{B_{tau,T}mathcal{A}(tau)}$.
                The CMS convexity adjustment is the difference between the expectations under these two measures. In order to compute this convexity adjustment, one has to find an approximation for $mathbb{E}^S_t[1/B_{S,T}|mathcal{S}_tau]$, which can be done following Cedervall and Piterbarg (2012) CMS: covering all bases either assuming a non-stochastic Libor-OIS basis spread or finding an explicit expression of T-bonds in terms of swap rates, which allows to obtain the swap density.







                share|improve this answer












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                answered 6 hours ago









                FunnyBuzerFunnyBuzer

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