Extension of Short Rate Model Under a Lévy Process

A lot of abnormalities occur in real-life scenarios, thus leading to some difficulties in modelling such scenarios without deeper understanding of certain aspects of Lévy processes. In this paper, the short rate model of Hull-White (1990) is extended to a model for capturing possibilities of jumps in real-life situations using a class of Lévy processes called a variance gamma process.


Introduction
Lévy processes was named after a French mathematician, Paul Lévy (1886Lévy ( -1971) ) who carried out a lot of research involving central limit theorem, law of large numbers, Gaussian processes, stable and infinitely divisible laws, and processes of independent and stationary increments.Other major contribution in this aspect between 1930s and 1940s include a Russian mathematician named Aleksandr Khintchine (1894-1959) who contributed in the fields of probability theory and number theory, and a Japanese mathematician named Kiyoshi Itô (1915Itô ( -2008) ) who contributed in the aspect of random events.Lévy processes have become a popular tool for applications in physics, engineering, crude oil optionpricing, mathematical finance, etc. (Kyprianou, 2006;Papapantoleon, 2008;Swishchuk, 2008;Park et al., 2014;Park & Kim, 2015;Udoye et al., 2021;Ma, 2003;Yang & Zang, 2001).Moreover, Rhee & Kim (2004) discussed market price of risk under the process.Madan & Seneta (1990) proposed a special type of the Lévy processes called a variance gamma process which they applied in share market returns, while Madan et al. (1998) discussed its application in option pricing.This class of Lévy process has been applied in engineering, science and finance by different researchers, Hirsa & Madan (2004) derived a partial integro-differential equation for Amerian options' pricing using the variance gamma process.Udoye & Ekhaguere (2022) applied the process in an interest rate derivative to capture jumps due to unexpected happenings, while Udoye & Akinola (2022) extended the work to involve special greeks of the extended model involving the process.Salem et al. (2020) proposed modelling a water tank pump degradation driven by the process.Chan et al. (1992) discussed empirical comparison of different models of interest rate driven by Brownian motion without consideration for the possibility of jumps due to unexpected happenings.Rachev et al. (2005) provided financial models for portfolio choosing, option pricing, credit and market risk control; they emphasized that some empirical evidence does not agree with the use of Gaussian distributions and α-stable distributions; while Kim et al. (2008) observed that the distribution of returns for assets possess weightier tails than the tails of the Gaussian distribution but thinner tails when compared to the α-stable distributions.Hence, the need to consider improved models with the ability to take care of special occurrences due to abnormality in real-life scenarios Various researchers have contributed to the applications of the Lévy processes in the field of mathematical finance (Udoye et al., 2021(Udoye et al., & 2022;;Klinger et al., 2013).In this work, the Lévy process is applied in the extension of the interest rate model of Hull-White (1990) by considering the variance gamma process.

Methodology
The variance gamma process.Let   denote a variance gamma process.Then, Nicolleta (2021) provided a brief representation of the process as where w represents the cumulant generating function given by k is the variance of an arithmetic Brownian motion used in deriving the variance gamma process.(  ) is given by (  ) = √() such that   =  + ∆() +  ̂∆√()

∆𝐺(𝑡) = 𝐺(𝑡) − 𝐺(𝑡 − ).
Hull-White (1990) model: The model dynamics is given by   = ( −   )  +   where ,  σ and  denote constant parameters for long-run mean rate, mean reversion's speed and volatility of the short rate, respectively.  denotes the Brownian motion.The solution to the dynamics is given by Its mean is given by while as t tends to infinity, the mean becomes Its variance is given by

Result
In this section, the Hull-White short rate model is extended from a process driven by a Brownian motion to a process with the ability to capture different jumps that occur due to unexpected happenings in real-life situations.Thus, the modified model will help in selecting a better model in financial markets (Lang et al., 2018) Let the short rate dynamics be given by   = ( −   )  +   (3.1)where   denotes the special class of Lévy process called a variance gamma process.Solving the dynamics gives where   =  +   +  ̂(  ).
Here,  = ) which gives the solution for the modified short rate model under a variance gamma process.

Conclusion
Lévy processes have contributed a lot in providing improved model that considers unexpected scenarios in real-life.A special class of the process called a variance gamma process has been used to obtain a modified Hull-White short rate model.

Conflict of Interest:
There is no conflict of interest to be declared.