Abstract
We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call nonlinear affine processes. This is done as follows: given a set Θ of parameters for the process, we construct a corresponding nonlinear expectation on the path space of continuous processes. By a general dynamic programming principle, we link this nonlinear expectation to a variational form of the Kolmogorov equation, where the generator of a single affine process is replaced by the supremum over all corresponding generators of affine processes with parameters in Θ. This nonlinear affine process yields a tractable model for Knightian uncertainty, especially for modelling interest rates under ambiguity.We then develop an appropriate Itô formula, the respective term-structure equations, and study the nonlinear versions of the Vasiček and the Cox–Ingersoll–Ross (CIR) model. Thereafter, we introduce the nonlinear Vasiček–CIR model. This model is particularly suitable for modelling interest rates when one does not want to restrict the state space a priori and hence this approach solves the modelling issue arising with negative interest rates.
Highlights
The modelling of a dynamic and unpredictable phenomenon like stock markets or interest rate markets is often approached via choosing an appropriate stochastic model
A promiment example in this direction is the role of affine short-rate models in the last 20 years: around 2000, the property of the Vasicek model that interest rates can become negative was heavily critizied and the non-negative Cox–Ingersoll–Ross (CIR) model was preferred, the consequences of the financial crises in 2007–2008 leading to negative interest rates in the Euro zone rendered the CIR model no longer applicable and led to a resurgence of the Vasicek model
This extends the existing class of nonlinear Levy processes to Markov processes where the interval for the parameter uncertainty may depend on the current state (in an affine way, Fig. 3 This figure shows the solution of the nonlinear Kolmogorov equation for the nonlinear Vasicekmodel with boundary condition f (x) = (x − K )+ and f (x) = (x − K1)+ −2(x − K2)+ +(x − K3)+, K = 0.1, K1 = −0.2, K2 = 0.3, and K3 = 0.8
Summary
The modelling of a dynamic and unpredictable phenomenon like stock markets or interest rate markets is often approached via choosing an appropriate stochastic model. Examples in this direction are the notions of g-Brownian motion and G-Brownian motion referring to a Brownian motion with drift or volatility uncertainty, see Peng (1997); Peng (2007a); Peng (2007b) and references therein Most recently, this theory has been extended to more general approaches, so-called nonlinear Levy processes, see Neufeld and Nutz (2017) and Denk et al (2017) in this regard. It is our aim to provide the appropriate tools for incorporating parameter uncertainty in the prominent class of affine models This naturally leads to a nonlinear version of affine processes and associated nonlinear expectations. We provide a number of examples: in addition to a nonlinear variant of the Black–Scholes model and nonlinear Vasicek and CIR models we introduce a nonlinear Vasicek–CIR model In the latter model one can incorporate negative interest rates in combination with a CIR-like behaviour, solving the problem raised in many practical applications when the state space needed to be restricted to positive interest rates (see Carver (2012)). (2003); Filipovic (2009) for details and applications in this regard
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