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The Laplace transform of the integrated Volterra Wishart process

Eduardo Abi Jaber

Abstract

We establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and resolvent. Furthermore , we link the characteristic exponents to a system of non-standard infinite dimensional matrix Riccati equations. This leads to a second representation of the Laplace transform for a special case of convolution kernel. In practice, we show that both representations can be approximated by either closed form solutions of conventional Wishart distributions or finite dimensional matrix Riccati equations stemming from conventional linear-quadratic models. This allows fast pricing in a variety of highly flexible models, ranging from bond pricing in quadratic short rate models with rich autocorrelation structures, long range dependence and possible default risk, to pricing basket options with covariance risk in multivariate rough volatility models.

The Laplace transform of the integrated Volterra Wishart process

Abstract

We establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and resolvent. Furthermore , we link the characteristic exponents to a system of non-standard infinite dimensional matrix Riccati equations. This leads to a second representation of the Laplace transform for a special case of convolution kernel. In practice, we show that both representations can be approximated by either closed form solutions of conventional Wishart distributions or finite dimensional matrix Riccati equations stemming from conventional linear-quadratic models. This allows fast pricing in a variety of highly flexible models, ranging from bond pricing in quadratic short rate models with rich autocorrelation structures, long range dependence and possible default risk, to pricing basket options with covariance risk in multivariate rough volatility models.

Paper Structure

This paper contains 16 sections, 17 theorems, 177 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Quadratic Gaussian processes
  3. Fredholm's representation and first properties
  4. Conditional Laplace transform of the integrated quadratic process
  5. Approximation by closed form expressions
  6. Connection to Riccati equations
  7. The Volterra Wishart process and its Laplace transforms
  8. A first representation
  9. A second representation for certain convolution kernels
  10. Another approximation procedure
  11. Applications
  12. Bond pricing in quadratic Volterra short rate models with default risk
  13. Pricing options on volatility/variance for basket products in Volterra Wishart covariance models
  14. Wishart distribution
  15. Matrix tools
  16. ...and 1 more sections

Key Result

Theorem 2.2

Fix $t \leq s \leq T$. Conditional on ${\mathcal{F}}_t$, $Z_sZ_s^\top$ follows a Wishart distribution Further, for any $u \in {\mathbb S}^N_+$, the conditional Laplace transform reads

Figures (5)

  • Figure 1: Convergence of $I^n(H)$ with the Riemann sum (blue) and the Gauss--Legendre quadrature (green) towards the benchmark MC value $I(H)$ (red) for different values of $(H,n)$ from Table \ref{['tablewishartapprox']}. The dashed lines delimit the $90\%$ confidence interval of the Monte--Carlo simulation.
  • Figure 2: Left: Sensitivity of the yield curve $T \mapsto y_0(T)$ with respect to the Hurst index for $d=m=1$, $X=W^{H}$ and $\xi\equiv 0$. Right: yield curve $T \mapsto y_0(T)$ for $d=2$, $m=1$, $X=(W^{H_1},W^{H_2})^\top$, $(H_1,H_2)=(0.05,0.9)$, $(Q_{11}, Q_{12},Q_{22})=(0.4 , 0.05, 0.1)$ and $\xi\equiv 0$
  • Figure 3: Simulation of monthly short rates (top left) with $X_t=2.5+\frac{1}{\Gamma(H+1/2)}\int_0^t (t-s)^{H-1/2}dW_s$ and varying $H$ index: $H=0.1$ (red), $H=0.5$ (black) and $H=0.9$ (blue) with the corresponding autocorrelation plots.
  • Figure 4: Impact of the Hurst index on the term structure of the variance of yields with $X_t=2.5+\frac{1}{\Gamma(H+1/2)}\int_0^t (t-s)^{H-1/2}dW_s$.
  • Figure 5: Impact of the Hurst index on the principal component analysis of the covariance of the yields with $d=3$, $m=1$, $Q_{ii}=1$ and $Q_{12}=Q_{23}=0.5$ and $X^i_t=0.33+\frac{1}{\Gamma(H+1/2)}\int_0^t (t-s)^{H-1/2}dW^i_s$, $i=1,\ldots,3$.

Theorems & Definitions (40)

  • Remark 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Example 2.5
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • ...and 30 more