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Interest rate convexity in a Gaussian framework

Antoine Jacquier, Mugad Oumgari

Abstract

The contributions of this paper are twofold: we define and investigate the properties of a short rate model driven by a general Gaussian Volterra process and, after defining precisely a notion of convexity adjustment, derive explicit formulae for it.

Interest rate convexity in a Gaussian framework

Abstract

The contributions of this paper are twofold: we define and investigate the properties of a short rate model driven by a general Gaussian Volterra process and, after defining precisely a notion of convexity adjustment, derive explicit formulae for it.
Paper Structure (20 sections, 7 theorems, 76 equations, 8 figures)

This paper contains 20 sections, 7 theorems, 76 equations, 8 figures.

Table of Contents

  1. Introduction and notations
  2. Introduction
  3. Model and notations
  4. Empirical motivation
  5. Gaussian martingale driver
  6. Dynamics of the zero-coupon bond price
  7. Convexity adjustments
  8. Examples
  9. Exponential kernels
  10. Riemann-Liouville kernels
  11. Extension to smooth Gaussian Volterra semimartingale drivers
  12. Comments on the Bond process
  13. Specific example
  14. Pricing OIS products and options
  15. Simple compounded rate
  16. ...and 5 more sections

Key Result

Lemma 2.2

Under Assumption assu:Kernel, $\left(\Xi_{T}(t,\cdot)\circ \mathfrak{W}\right)_{t}$ is an $(\mathcal{F}_{t})_{t \in [0,T]}$ Gaussian semimartingale.

Figures (8)

  • Figure 1: Examples of rates curves over different days from the OptionMetrics rates data.
  • Figure 2: Time series of the OptionMetrics rates for different maturities.
  • Figure 3: Estimation of the Hurst exponent for the OptionMetrics rates data.
  • Figure 4: Time series of the US Treasury rates for different maturities.
  • Figure 5: Estimation of the Hurst exponent for the US Treasury rates.
  • ...and 3 more figures

Theorems & Definitions (21)

  • Remark 1.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • Corollary 2.5
  • Example 2.7
  • Example 2.8
  • proof : Proof of Proposition \ref{['prop:ZeroCoupon']}
  • proof : Proof of Corollary \ref{['cor:InstFwd']}
  • ...and 11 more