with the analysis of the European style and Bermudan style swaptions in this paper, is The mean reversion term ρ(t) depends on the interest rate local. It has become more common to use swap rates to model mortgage rates as the The volatility σ and mean reversion a can be analytically calibrated to. strategies on Bermudan swaptions. Our analysis will allow drawing several conclusions upon the behavior of the model under mean reversion uncertainty. 200 DOLLARS WORTH OF BITCOIN

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An interest rate swap is an agreement between counterparties, where one stream of future interest payments is exchanged for another. Interest rate swaps usually involve the exchange of a fixed interest rate for a floating rate or vice versa. The swap helps to reduce or increase exposure to fluctuations in interest rates. They may also offer the ability to obtain a marginally lower interest rate than would have been possible without the swap.

Only cash flows are exchanged in this swap. Bermuda vs. American and European Styles The exercise feature of Bermuda swaptions falls somewhere between American and European styles. Holders may exercise American-style options and swaptions at any time between the issue and the expiration dates.

Holders may utilize European-style options and swaptions only at maturity. Buyers and sellers determine the allowable expiration dates for Bermuda options and swaptions. Monthly expirations are customary, though the days are up to the counterparties.

There are also "Canary Swaptions," which can also be executed intermittently, but less frequently than "Bermuda Style. Bermuda swaptions have several advantages and disadvantages. Unlike American and European swaptions, Bermuda swaptions give writers and buyers the ability to create and purchase a hybrid contract. Writers of Bermuda swaptions can have more control over the exercising of the swaptions.

Pricing Bermuda Swaptions Pricing of such swaptions is more complex than vanilla swaptions. With the inclusion of more potential exercise dates, the calculations become more complicated. Therefore, counterparties use Monte Carlo Simulation pricing rather than other, more common, option and swaption pricing models. As the volatility is time-dependent, we obtain very close fits to the European swaptions whatever the mean reversion is.

For the 10Y Bermudan swaptions, we find the results in fig. As was also documented in [4], we observe a strong influence of the mean reversion on the Bermudan swaption prices, even though the relevant European swaptions are almost perfectly fit. This means that a trader can use this extra freedom to impose a view on the Bermudan swaptions. Alternatively, this can be viewed as an extension of the calibration process, in which Bermudan swaption information is also included.

This freedom may be considered as desirable and may motivate the choice of the local calibration method instead of the global one. Note that this is true for the particular parametric form that we have chosen for the mean reversion, with its corresponding number of degrees of freedom 4. One may think that allowing more degrees of freedom in the mean reversion and optimizing only on some of them using the European swaption information may lead to both a stable calibration procedure with good fit to the European swaptions while still allowing to impose a view on the Bermudan swaptions.

On the one hand, we provided a detailed documentation and numerical examples about the calibration methods of Hull-White model with time-dependent parameters, as we believe this was missing from the literature. We hope that the readers will find it useful as a starting point to implement their own calibration and that it will spare them the trouble of tedious trials and errors.

On the other hand, we showed that some common negative opinions and fears about the model are not necessarily justified. It is widely stated, for instance, that Hull-White model cannot fit the swaption matrix well, and that introducing time-dependent parameters to improve the fitting quality results in unstable behaviours. We believe this work has proved this is not the case, provided a suitable calibration strategy is adopted. In particular, we showed that the fully time-dependent model can achieve excellent fit in a stable manner.

In particular, we showed that the model can provide interesting insights in the analysis of Lehman crisis in the fall of It is not our purpose here to single out one best method. Instead we show that several calibration methods, such as local or global, with constant or time-dependent parameters, are acceptable, and that the choice depends on the user preferences as to fitting quality, runtime and ease of implementation.

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