Stochastic Processes with Age-Dependent Transition Rates

2019

J. Finan. Eng.

We have developed a statistical technique to test the model assumption of binary regime switching extension of the geometric Brownian motion (GBM) model by proposing a new discriminating statistics. Given a time series data, we have identified an admissible class of the regime switching candidate models for the statistical inference. By performing several systematic experiments, we have successfully shown that the sampling distribution of the test statistics differs drastically, if the model assumption changes from GBM to Markov modulated GBM, or to semi-Markov modulated GBM. Furthermore, we have implemented this statistics for testing the regime switching hypothesis with Indian sectoral indices.

Section: Semi-markov Modulated Gbmmentioning

confidence: 99%

“…This discretization is obtained from the semi-martingale representation of the semi-Markov process, as in Ghosh and Saha (2011). The readers are referred to Ghosh and Saha (2011) for more details about this representation of semi-Markov process.…”

Section: Semi-markov Modulated Gbmmentioning

confidence: 99%

Testing of binary regime switching models using squeeze duration analysis

Das

2019

J. Finan. Eng.

“…. It is shown in [8] that F l (.|i) is the conditional c.d.f of the holding time of X l and p l ij (ȳ) is the conditional transition probability matrix. Let τ l (t) be the duration after which X l t would have a transition.…”

Section: )mentioning

confidence: 99%

Pricing derivatives in a regime switching market with time inhomogenous volatility

Das

Stochastic Analysis and Applications

Patankar

2018

This paper studies pricing derivatives in an age-dependent semi-Markov modulated market. We consider a financial market where the asset price dynamics follow a regime switching geometric Brownian motion model in which the coefficients depend on finitely many age-dependent semi-Markov processes. We further allow the volatility coefficient to depend on time explicitly. Under these market assumptions, we study locally risk minimizing pricing of a class of European options. It is shown that the price function can be obtained by solving a non-local B-S-M type PDE. We establish existence and uniqueness of a classical solution of the Cauchy problem. We also find another characterization of price function via a system of Volterra integral equation of second kind. This alternative representation leads to computationally efficient methods for finding price and hedging. Finally we analyse the PDE to establish continuous dependence of the solution on the instantaneous transition rates of semi-Markov processes. An explicit expression of quadratic residual risk is also obtained.

“…. It is shown in [10] that F (y | i) is the cumulative distribution function of holding time and p ij (y) is the conditional probability that X transits to j given the fact that it is at i for a duration of y. From (A1)(ii), lim y→∞ F (y | i) = 1.…”

Section: Market Modelmentioning

confidence: 99%

“…where Λ ij (y) are the consecutive (with respect to the lexicographic ordering on X × X ) left closed and right open intervals of the real line, each having length λ ij (y). We refer [10] for more details about this kind of pure jump processes. Under some smoothness and tail assumptions on λ ij (y) and independence of W and ℘, we obtain the following equation of locally risk minimizing price of call option…”

Section: Introductionmentioning

confidence: 99%

A system of non-local parabolic PDE and application to option pricing

Stochastic Analysis and Applications

Patel

Shevgaonkar

2016

This paper includes a proof of well-posedness of an initial-boundary value problem involving a system of non-local parabolic partial differential equation(PDE) which naturally arises in the study of derivative pricing in a generalized market model which is known as a semi-Markov modulated geometric Brownian motion(GBM) model. We study the well-posedness of the problem via a Volterra integral equation of second kind. A probabilistic approach, in particular the method of conditioning on stopping times is used for showing the uniqueness.