(), where Jamshidian decomposition is used for pricing credit default swap options under a CIR++ (extended Cox-Ingersoll-Ross) stochastic intensity model . Jamshidian Decomposition for Pricing Energy Commodity European Swaptions. Article (PDF Available) · January with Reads. Export this citation. Following Brigo 1 p, we can decompose the price of a swaption as a sum of Zero-Coupon bond options (Jamshidian’s Trick). To do so, the.
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We have seen in a previous post how to fit initial discount curves to swap rates in a model-independent way. What if we want to control the volatility parameter to match vanilla rates derivatives as well? Our next choice is which vanilla rates options we want to use for the calibration. A common choice is the decompositipn rate swaption, which is the right to enter a swap at some future time with fixed payment dates and a strike.
These are fairly liquid contracts so present a good choice for our calibration.
For simplicity, for the rest of this post we will assume all payments are annual, so year fractions are ignored. A reciever swaption can be seen as a call option on a coupon-paying bond with fixed payments equal to at the same payment dates as the swap. To see this, consider the price of a swap discussed before:. So, the price of a swaption is an option on receiving a portfolio of coupon payments, each of which can be thought of as a zero-coupon bond paid at that time, and the value of the swaption is the positive part of the expected value decompositipn these:.
options – Jamshidian’s trick for Swaptions – Quantitative Finance Stack Exchange
Looking at this expression, we see that each term is simply the present value of an option to buy a ZCB at time that expires at one of the payment dates with strike. So the price of a swaption has been expressed entirely as the price of a portfolio of options on ZCBs! For the HWeV model, these are dceomposition and depend only on the initial rate, and calibrated time dependent parameters in the model.
Since rates are gaussian in HWeV this can be done analytically.
Calibrating time-dependent volatility to swaption prices
Calculating these for time-varying parameters is algebra-intensive and I leave it for a later post, but for constant parameters the calculation is described in Brigo and Mercurio pg and gives a price of. We can see how we could use the above to calibrate the volatility parameter to match a single market-observed swaption price.
When several are visible, the challenge becomes to choose a piecewise jamshician function to match several of them. In HWeV this can be done analytically, but for more general models some sort of optimisation would be required.
Since these contracts have an exercise date when the swap starts and the swaps themselves will have another termination date which define a 2-dimensinal spaceit will not be possible to fit all market-observable swaptions with a one factor model.
Many alternatives are discussed in the literature to deal with this concern, but the general procedure is the same. Practically, we should choose the most liquid swaptions and bootstrap to these, and only a few 5Y, 10Y etc will practically be tradable in any case. Your email address will not be published.
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