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Mathematical models and numerical methods for a capital valuation adjustment (KVA) problem

D. Trevisani, J. G. López-Salas, C. Vázquez, J. A. García-Rodríguez

Abstract

In this work we rigorously establish mathematical models to obtain the capital valuation adjustment (KVA) as part of the total valuation adjustments (XVAs). For this purpose, we use a semi-replication strategy based on market theory. We formulate single-factor models in terms of expectations and PDEs. For PDEs formulation, we rigorously obtain the existence and uniqueness of the solution, as well as some regularity and qualitative properties of the solution. Moreover, appropriate numerical methods are proposed for solving the corresponding PDEs. Finally, some examples show the numerical results for call and put European options and the corresponding XVA that includes the KVA.

Mathematical models and numerical methods for a capital valuation adjustment (KVA) problem

Abstract

In this work we rigorously establish mathematical models to obtain the capital valuation adjustment (KVA) as part of the total valuation adjustments (XVAs). For this purpose, we use a semi-replication strategy based on market theory. We formulate single-factor models in terms of expectations and PDEs. For PDEs formulation, we rigorously obtain the existence and uniqueness of the solution, as well as some regularity and qualitative properties of the solution. Moreover, appropriate numerical methods are proposed for solving the corresponding PDEs. Finally, some examples show the numerical results for call and put European options and the corresponding XVA that includes the KVA.

Paper Structure

This paper contains 17 sections, 3 theorems, 121 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mathematical modelling
  3. Asset-dividend dynamics
  4. Pricing by semi-replication
  5. PDEs with close-out conditions
  6. A different approach to the cost of capital
  7. Mathematical analysis of the PDE model
  8. Well-posedness of the PDEs formulation
  9. Numerical methods for the arising PDEs
  10. LDG space semidiscretization
  11. IMEX time semidiscretization
  12. Numerical results
  13. Admissible strategies and lack of arbitrage
  14. Definition of Capital
  15. EAD under Standardised method SACCR
  16. ...and 2 more sections

Key Result

Theorem 1

Let $F:[0,T]\times \mathbb{R}_{\geq 0}\times \mathbb{R} \longrightarrow \mathbb{R}$, and $g:\mathbb{R}_{\geq 0}\rightarrow \mathbb{R}$ be Lebesgue measurable and with linear growth. Suppose that there exists $L>0$ such that Then $J:\mathcal{X}\rightarrow \mathcal{X}$ is well-defined, and there exists $k\in \mathbb{N}$ such that the composition of $J$ k-times, $J^k$, satisfies Accordingly, eq:mil

Figures (7)

  • Figure 1: Plot of the XVA at time $t=0$ for Call (above) and Put (below) European options with the data in Table \ref{['tab:param']}. Both the linear and the nonlinear cases are considered.
  • Figure 2: Plot of $\Delta$ at time $t=0$ for Call (above) and Put (below) European options with the data in Table \ref{['tab:param']}. Both the linear and the nonlinear cases are considered.
  • Figure 3: Plot of $\Gamma$ at time $t=0$ for Call (above) and Put (below) European options with the data in Table \ref{['tab:param']}. Both the linear and the nonlinear cases are considered.
  • Figure 4: Variation of $\Delta$ with respect to the stock volatility $\sigma$. Nonlinear PDE for the Call (above) and Put (below) options.
  • Figure 5: Variation of $\Gamma$ with respect to the stock volatility $\sigma$. Nonlinear PDE for the Call (above) and Put (below) options.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Definition 1
  • Remark 1
  • Proposition 1
  • proof