Bermudan Swaptions - Payer vs. Receiver (LGM)

Question

There is abundant literature discussing the pricing of Bermudan swaptions and the relevance of single-factor Markov-functional models (e.g. LGM) versus multi-factor market models (e.g. LMM).

From a famous paper by Andersen & Andreasen (and other research comparing the empirical hedging performances of various approaches), Bermudan swaption prices seem to depend only weakly on the number of factors of the underlying model. From what I've gathered, the market standard on the sell-side is then to use a LGM calibrated à la Hagan.

This simple framework effectively allows to express the price of a Bermudan swaption as a function of the relevant coterminal swaptions and a mean reversion term structure. The idea is that while the former determine the marginal future swap rates distributions, the latter can be tuned to impose their correlation, which seems to be the true relevant risk-factor here (compared to smile).

A question which I have never seen mentioned though is Can the same model and calibration strategy can be used to quote both payer and receiver Bermudan swaptions.

In an attempt to answer this question, I've calibrated a plain LGM model on coterminals with same strike as my Bermuda and compared output payer vs. receiver prices (expressed as a spread of the relevant most-expensive coterminal as per market standard). My conclusion is that I can never simultaneously fit both.

On one hand, it is not much of a surprise as:

• The exercise boundary of a payer vs. receiver Bermudan are not the same in the swap rate dimension (one being above, the other below, the strike level). I would therefore suspect smile to have a role to play here. But literature seems to indicate this effect is marginal when compared to the correlation effect, with 'local volatility' models à la Cheyette not adding much.
• Payer vs. receiver Bermudans do not really depend on the same risk factors at the end of the day. Stretching the comparison to the extreme, it would be like comparing an American call versus put with a steep forward curve. These two options would have very different optimal exercise time hence depend on smiles at two very different maturities. Again is the correlation between swap rates to capture this effect or does should this motivate us to use two distinct 'implied' LGM dynamics for payer vs. receiver. In the latter case, how to prove that it does not lead to arbitrage opportunities?

Anyway, I would be happy if you could share your expert thoughts on this one!

Answer 1

I’m guessing you are finding that your model overvalues Bermudan receiver options and probably undervalues Bermudan payer options. The rationale for this has more to do with supply and demand than theory. That’s because every time a callable bond is issued and swapped, dealers buy Bermudan receiver options, so there’s a huge supply. For Bermudan payers there is no such analogy.

Because of this supply and demand situation, Bermudan receivers tend to be overvalued by any reasonable model relative to the market. In order to deal with this, various market practices have evolved. Probably the most popular is to suppress a subset of the options such that the valuation is more in line with the market. For example , suppressing options expiring > N years. Or retaining only the maximal set of 2 coterminals. Such practices are not needed for Bermudan payers options.