Delay Ait-Sahalia-type interest rate model with jumps and its strong approximation

Emmanuel Coffie

doi:10.1515/strm-2022-0013

Article

Delay Ait-Sahalia-type interest rate model with jumps and its strong approximation

Emmanuel Coffie

Published/Copyright: August 25, 2023

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From the journal Statistics & Risk Modeling Volume 40 Issue 3-4

Abstract

In this paper, we study the analytical properties of the true solution to the generalised delay Ait-Sahalia-type interest rate model with Poisson-driven jumps. Since this model does not have a closed-form solution, we employ several new truncated Euler-Maruyama (EM) techniques to investigate the finite-time strong convergence theory of the numerical solution under the local Lipschitz condition plus the Khasminskii-type condition. We justify the strong convergence result for Monte Carlo calibration and valuation of some debt and derivative instruments.

Keywords: Stochastic interest rate model; delay volatility; Poisson jumps; truncated EM scheme; strong convergence; Monte Carlo scheme

MSC 2020: 65C05; 65C30; 91G30; 91G60

Appendix

The proof for Lemma 3.2 is provided below.

Proof of Lemma 3.2.

If we assume that x < x ¯ , then the mean-value theorem gives us

| x ρ - x ¯ ρ | ≤ ρ ⁢ ( | x | ρ - 1 + | x ¯ | ρ - 1 ) ⁢ | x - x ¯ |

and

| x θ - x ¯ θ | ≤ θ ⁢ ( | x | θ - 1 + | x ¯ | θ - 1 ) ⁢ | x - x ¯ | .

It then follows that

= | α - 1 ⁢ ( 1 x - 1 x ¯ ) + α 1 ⁢ ( x - x ¯ ) - α 2 ⁢ ( x ρ + x ¯ ρ ) | + | x θ - x ¯ θ | + α 3 ⁢ | x - x ¯ |

≤ α - 1 ⁢ | 1 x ⁢ x ¯ | ⁢ | x - x ¯ | + α 1 ⁢ | x - x ¯ | - α 2 ⁢ ρ ⁢ ( | x | ρ - 1 + | x ¯ | ρ - 1 ) ⁢ | x - x ¯ | + θ ⁢ ( | x | θ - 1 + | x ¯ | θ - 1 ) ⁢ | x - x ¯ | + α 3 ⁢ | x - x ¯ |

≤ ( α - 1 ⁢ | 1 x ⁢ x ¯ | + α 1 - α 2 ⁢ ρ ⁢ ( | x | ρ - 1 + | x ¯ | ρ - 1 ) + θ ⁢ ( | x | θ - 1 + | x ¯ | θ - 1 ) + α 3 ) ⁢ | x - x ¯ | .

Noting from Assumption 2.3 that

ρ + 1 > 2 ⁢ θ ⟹ ρ - 1 > 2 ⁢ ( θ - 1 ) ⟹ ρ - 1 > θ - 1 ,

then for any R > 0 , there exists a constant K R > 0 such that for x , x ¯ ∈ [ 1 R , R ] , we get

| f ⁢ ( x ) - f ⁢ ( x ¯ ) | + | g ⁢ ( x ) - g ⁢ ( x ¯ ) | + | h ⁢ ( x ) - h ⁢ ( x ¯ ) | ≤ K R ⁢ | x - x ¯ | ,

as required, where

K R ≥ α 1 + α 3 + α - 1 ⁢ | 1 x ⁢ x ¯ | - α 2 ⁢ ρ ⁢ ( | x | ρ - 1 + | x ¯ | ρ - 1 ) + θ ⁢ ( | x | θ - 1 + | x ¯ | θ - 1 ) . ∎

Here is the proof for Lemma 3.3.

Proof of Lemma 3.3.

By Assumptions 2.1 and 2.3, we get

x ⁢ f ⁢ ( x ) + p - 1 2 ⁢ | φ ⁢ ( y ) ⁢ g ⁢ ( x ) | 2 = x ⁢ ( α - 1 ⁢ x - 1 - α 0 + α 1 ⁢ x - α 2 ⁢ x ρ ) + p - 1 2 ⁢ | φ ⁢ ( y ) ⁢ g ⁢ ( x ) | 2

≤ ( α - 1 - α 0 ⁢ x + α 1 ⁢ x 2 - α 2 ⁢ x ρ + 1 ) + p - 1 2 ⁢ σ 2 ⁢ x 2 ⁢ θ

≤ Q ⁢ ( p ) + α - 1 - α 0 ⁢ x + α 1 ⁢ x 2

≤ Q ⁢ ( p ) + α - 1 + α 1 ⁢ x 2

≤ 𝒥 4 ⁢ ( 1 + | x | 2 ) ,

where 𝒥 4 = ( α - 1 + Q ⁢ ( p ) ) ∨ α 1 and Q ⁢ ( p ) ≥ - α 2 ⁢ x ρ + 1 + p - 1 2 ⁢ σ 2 ⁢ x 2 ⁢ θ . ∎

Acknowledgements

The author would like to thank the University of Strathclyde for the scholarship and his first PhD supervisor, Professor Xuerong Mao.

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Received: 2022-06-19

Revised: 2023-03-26

Accepted: 2023-08-08

Published Online: 2023-08-25

Published in Print: 2023-09-01

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Articles in the same Issue

https://doi.org/10.1515/strm-2022-0013

Keywords for this article

Stochastic interest rate model; delay volatility; Poisson jumps; truncated EM scheme; strong convergence; Monte Carlo scheme