Estimation of a CIR process with jumps using a closed form approximation likelihood under a strong approximation of orde
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Estimation of a CIR process with jumps using a closed form approximation likelihood under a strong approximation of order 1 Patrice Takam Soh1 · Eugene Kouassi2,3 · Renaud Fadonougbo4 · Martin Kegnenlezom1 Received: 8 June 2019 / Accepted: 6 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We propose here an approach in order to estimate parameters of the CIR model with jumps in the case where the distribution of jump amplitude is estimated nonparametrically. Since the knowledge of the exact distribution of the jump amplitude is a challenge, in this paper we choose not to fix this law in advance but to estimate it on the basis of the available observations. The method of estimation we propose here is based on the approximation of the closed form of transition density. Since the CIR does not have an explicit solution, it is approximated by the second order Milstein scheme in order to have a more accurate approximation. The method of estimation is then applied on real data, which are the Federal Funds rate and 3 Month T-Bill rate. These two sets of data are used to estimated parameters of the CIR model. We then compare our results to those obtained from Vasicek and Brennon–Swartz models with jumps. Results indicate that there is no clear winner of models competitions. Apparently depending on the nature and structural components of the data, there is a winner. The challenge here is that, there is a trade off between the sample size, the number of jumps and the efficiency of estimates. More data involves the likelihood to have more jumps and thereby less efficient are estimates. Keywords CIR with jumps · Saddle point · Milstein approximation · Likelihood method
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Patrice Takam Soh [email protected]
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Department of Mathematics, University of Yaoundé 1, Yaoundé, Cameroon
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Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
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Department of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
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Institute of Basic Sciences, Technology and Innovation, PAN African University, Nairobi, Kenya
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P. T. Soh et al.
1 Introduction We are interested here in the parameters estimation of the Cox-Ingersoll-Ross (CIR) process with jumps. The CIR process is a well-known process in finance. It has been defined by Cox (2005) in order to model short term interest rate in finance. The existence and uniqueness of the solution of the CIR process has been proved under such assumptions in Yamada (1971). It is also well known that the conditional distribution of the CIR process follows the decentralized chi-square (Ioffe 2010). This distribution has been used to write the transition density of the process and also the likelihood of observations, and the parameters of the CIR process without jumps are then estimated by maximizing the likelihood of observations. Another method of parameters estimation proposed in the literature is the approximation of the likelihood by a Gaussian distribution (Shoji and Ozaki 1998). In fact this second approa
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