Effective Markovian projection: application to CMS spread options and mid-curve swaptions

Abstract

Pricing of interest rate derivatives, such as CMS spread or mid-curve options, depends on the modelling of the underlying single rates. For flexibility and realism, these rates are often described in the framework of stochastic volatility models. In this paper we allow rates to be modelled within a class of general stochastic volatility models, which includes the SABR, ZABR, free SABR and Heston models. We provide a versatile technique called Effective Markovian Projection, which allows a tractable model to be found that mimics the distribution of the more complex models used to price multi-rate derivatives. Three different numerical approaches are outlined and applied to relevant examples from practice. Finally, a new method that involves moment-matching of Johnson distributions is applied to facilitate closed-form pricing formulas.

Keywords:

• Stochastic volatility
• SABR
• ZABR
• Markovian projection
• Effective PDE
• Approximation formula

JEL Classification:

• G12
• C5
• C15

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Here T may be interpreted as the maturity of a claim. If several maturities are considered, T is set to the maximum of these times.

2 In the following applications, the explicit form of the measure H is not of importance—only the dynamics of the spread measure expressed in this measure is relevant.

3 Code for the examples presented in the paper is available at https://github.com/Lapsilago/EffectiveMarkovianProjection.

4 This actually also holds true when we replace the spread by a basket of nSABR models