Interest Rate Modelling in the Multi-Curve Framework: Foundations, Evolution and Implementation (Applied Quantitative Finance) by Henrard Marc

Author:Henrard, Marc [Henrard, Marc]
Language: eng
Format: epub
ISBN: 9781137374677
Publisher: Palgrave Macmillan
Published: 2014-05-28T23:00:00+00:00

where κ is the -measurable random variable defined as the (unique) solution of

The price of the payer swaption is

Proof: The present value of the swaption is, in the numeraire,

Using Lemma A.2, the value becomes

The value inside the expected value is positive when X < κ. To see this, note that α(0,θ,θ, .) is increasing. The quantity is positive when

The large sum is a sum of negative exponentials with positive coefficients. It is a strictly decreasing function with limits +∞ at –;∞ and limit 0 at +∞. It crosses the value –d0 > 0 at a unique value (called κ). It is above –d0 when x < κ and below it when x > κ.

The expected value can be written in terms of integrals by

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