Abstract
Bermudan swaptions are options on interest rate swaps which can be exercised on one or more dates before the final maturity of the swap. Because the exercise boundary between the continuation area and stopping area is inherently complex and multi-dimensional for interest rate products, there is an inherent “tug of war” between the pursuit of calibration and pricing precision, tractability, and implementation efficiency. After reviewing the main ideas and implementation techniques underlying both single- and multi-factor models, we offer our own approach based on dimension reduction via Markovian projection. Specifically, on the theoretical side, we provide a reinterpretation and extension of the classic result due to Gyöngy which covers non-probabilistic, discounted, distributions relevant in option pricing. Thus, we show that for purposes of swaption pricing, a potentially complex and multidimensional process for the underlying swap rate can be collapsed to a one-dimensional one. The empirical contribution of the paper consists in demonstrating that even though we only match the marginal distributions of the two processes, Bermudan swaptions prices calculated using such an approach appear well-behaved and closely aligned to counterparts from more sophisticated models.
Highlights
OutlineInterest rate contracts constitute the largest chunk of the global over-the-counter derivatives market, with an estimated total notional outstanding of just over $480 trillion, of which roughly $350 trillion is traded in swaps and almost $50 trillion is traded in interest rate options
Bermudan swaptions are options on interest rate swaps which can be exercised on one or more dates before the final maturity of the swap
Since V (t, x ) is a martingale, its drift must be zero, and applying Ito’s lemma and writing down the dynamics dV, we find that V (t, x ) must satisfy the following parabolic partial differential equation (PDE):
Summary
Interest rate contracts constitute the largest chunk of the global over-the-counter derivatives market, with an estimated total notional outstanding of just over $480 trillion, of which roughly $350 trillion is traded in swaps and almost $50 trillion is traded in interest rate options. Apart from parsimony and consistency, short rate models have another considerable virtue that came to explain their initial success as tools-of-choice for the valuation of path-dependent interest rate instruments such as Bermudan swaptions: they are relatively easy to implement numerically in lattices, i.e., finite differences or trees, which can efficiently handle the early exercise boundary through backward induction These virtues do not come without vices to match. In what follows we try to tell the story—the odyssey—behind the evolution of Bermudan swaption pricing in greater detail, reviewing in a much more formal way the main ideas and solutions underlying both single- and multi-factor models We discuss their relative merits and drawbacks, presenting especially the crux of the consensus-shattering debate between. It has become market practice to quote prices of caps in terms of Black’s implied volatilities, i.e., parameters σα,β , which plugged into the formula:
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