arXiv subjects

Anindya Goswami

Publications and source records attributed to Anindya Goswami.

At least 19 recordsLinked to original sources

arXiv2026

A market resilient data-driven approach to option pricing

In this paper, we present a data-driven ensemble approach for option price prediction whose derivation is based on the no-arbitrage theory of option pricing. Using the theoretical treatment, we derive a common representation space for achieving domain adaptation. The success of an implementation of this idea is shown using some real data. Then we report several experimental results for critically examining the performance of the derived pricing models.

q-fin.MF↗

arXiv2024

Matrix method stability and robustness of compact schemes for parabolic PDEs

The fully discrete problem for convection-diffusion equation is considered. It comprises compact approximations for spatial discretization, and Crank-Nicolson scheme for temporal discretization. The expressions for the entries of inverse of tridiagonal Toeplitz matrix, and Gerschgorin circle theorem have been applied to locate the eigenvalues of the amplification matrix. An upper bound on the condition number of a relevant matrix is derived. It is shown to be of order $\mathcal{O}\left(\frac{δv}{δz^2}\right)$, where $δv$ and $δz$ are time and space step sizes respectively. Some numerical illustrations have been added to complement the theoretical findings.

q-fin.CP↗

arXiv2024

A novel difference equation approach for the stability and robustness of compact schemes for variable coefficient PDEs

Fourth-order accurate compact schemes for variable coefficient convection diffusion equations are considered. A sufficient condition for the stability of the fully discrete problem is derived using a difference equation based approach. The constant coefficient problems are considered as a special case, and the unconditional stability of compact schemes for such case is proved theoretically. The condition number of the amplification matrix is also analyzed, and an estimate for the same is derived. The examples are provided to support the assumption taken to assure stability.

math.NA↗

arXiv2024

Estimation of domain truncation error for a system of PDEs arising in option pricing

In this paper, a multidimensional system of parabolic partial differential equations arising in European option pricing under a regime-switching market model is studied in details. For solving that numerically, one must truncate the domain and impose an artificial boundary data. By deriving an estimate of the domain truncation error at all the points in the truncated domain, we extend some results in the literature those deal with option pricing equation under constant regime case only. We differ from the existing approach to obtain the error estimate that is sharper in certain region of the domain. Hence, the minimum of proposed and existing gives a strictly sharper estimate. A comprehensive comparison with the existing literature is carried out by considering some numerical examples. Those examples confirm that the improvement in the error estimates is significant.

q-fin.CP↗

arXiv2022

On Merging of Stochastic Flow of Semi-Markov Dynamics

Given a semi-Markov law, using an additional parameter, we consider a family of stochastic flows corresponding to that law. Then we suitably select a particular flow, for which we obtain expressions of the meeting and merging probabilities of a pair of semi-Markov processes, solving the same equation but having two different initial conditions. A set of sufficient conditions are also obtained under which any two solutions of the flow eventually merge with probability one.

math.PR↗

arXiv2022

A semi-Markovian approach to model the tick-by-tick dynamics of stock price

We model the stock price dynamics through a semi-Markov process obtained using a Poisson random measure. We establish the existence and uniqueness of the classical solution of a non-homogeneous terminal value problem and we show that the expected value of stock price at horizon can be obtained as a classical solution of a linear partial differential equation that is a special case of the terminal value problem studied in this paper. We further analyze the market making problem using the point of view of an agent who posts the limit orders at the best price available. We use the dynamic programming principle to obtain a HJB equation. In no-risk aversion case, we obtain the value function as a classical solution of a linear pde and derive the expressions for optimal controls by solving the HJB equation.

q-fin.MF↗

arXiv2022

Semimartingle Representation of a class of Semi-Markov Dynamics

We consider a class of semi-Markov processes (SMP) such that the embedded discrete time Markov chain may be non-homogeneous. The corresponding augmented processes are represented as semi-martingales using stochastic integral equation involving a Poisson random measure. The existence and uniqueness of the equation are established. Subsequently, we show that the solution is indeed a SMP with desired transition rate. Finally, we derive the law of the bivariate process obtained from two solutions of the equation having two different initial conditions.

math.PR↗

arXiv2022

Inference of Binary Regime Models with Jump Discontinuities

Identifying the instances of jumps in a discrete-time-series sample of a jump diffusion model is a challenging task. We have developed a novel statistical technique for jump detection and volatility estimation in a return time series data using a threshold method. The consistency of the volatility estimator has been obtained. Since we have derived the threshold and the volatility estimator simultaneously by solving an implicit equation, we have obtained unprecedented accuracy across a wide range of parameter values. Using this method, the increments attributed to jumps have been removed from a large collection of historical data of Indian sectorial indices. Subsequently, we have tested the presence of regime-switching dynamics in the volatility coefficient using a new discriminating statistic. The statistic has been shown to be sensitive to the transition kernel of the regime-switching model. We perform the testing using the Bootstrap method and find a clear indication of presence of multiple regimes of volatility in the data. A link to all Python codes is given in the conclusion. The methodology is suitable for analyzing high-frequency data and may be applied for algorithmic trading.

q-fin.ST↗

arXiv2022

Regime recovery using implied volatility in Markov modulated market model

In the regime switching extension of Black-Scholes-Merton model of asset price dynamics, one assumes that the volatility coefficient evolves as a hidden pure jump process. Under the assumption of Markov regime switching, we have considered the locally risk minimizing price of European vanilla options. By pretending these prices or their noisy versions as traded prices, we have first computed the implied volatility (IV) of the underlying asset. Then by performing several numerical experiments we have investigated the dependence of IV on the time to maturity (TTM) and strike price of the vanilla options. We have observed a clear dependence that is at par with the empirically observed stylized facts. Furthermore, we have experimentally validated that IV time series, obtained from contracts with moneyness and TTM varying in particular narrow ranges, can recover the transition instances of the hidden Markov chain. Such regime recovery has also been proved in a theoretical setting. Moreover, the novel scheme for computing option price is shown to be stable.

q-fin.CP↗

arXiv2021

Regime Switching Optimal Growth Model with Risk Sensitive Preferences

We consider a risk-sensitive optimization of consumption-utility on infinite time horizon where the one-period investment gain depends on an underlying economic state whose evolution over time is assumed to be described by a discrete-time, finite-state, Markov chain. We suppose that the production function also depends on a sequence of i.i.d. random shocks. For the sake of generality, the utility and the production functions are allowed to be unbounded from above. Under the Markov regime-switching model, it is shown that the value function of optimization problem satisfies an optimality equation and that the optimality equation has a unique solution in a particular class of functions. Furthermore, we show that an optimal policy exists in the class of stationary policies. We also derive the Euler equation of optimal consumption. Furthermore, the existence of the unique joint stationary distribution of the optimal growth process and the underlying regime process is examined. Finally, we present a numerical solution by considering power utility and some hypothetical values of parameters in a regime switching extension of Cobb-Douglas production rate function.

math.OC↗

arXiv2020

Data-Driven Option Pricing using Single and Multi-Asset Supervised Learning

We propose three different data-driven approaches for pricing European-style call options using supervised machine-learning algorithms. These approaches yield models that give a range of fair prices instead of a single price point. The performance of the models are tested on two stock market indices: NIFTY$50$ and BANKNIFTY from the Indian equity market. Although neither historical nor implied volatility is used as an input, the results show that the trained models have been able to capture the option pricing mechanism better than or similar to the Black-Scholes formula for all the experiments. Our choice of scale free I/O allows us to train models using combined data of multiple different assets from a financial market. This not only allows the models to achieve far better generalization and predictive capability, but also solves the problem of paucity of data, the primary limitation of using machine learning techniques. We also illustrate the performance of the trained models in the period leading up to the 2020 Stock Market Crash (Jan 2019 to April 2020).

q-fin.ST↗

arXiv2019

Option Pricing in a Regime Switching Jump Diffusion Model

This paper presents the solution to a European option pricing problem by considering a regime-switching jump diffusion model of the underlying financial asset price dynamics. The regimes are assumed to be the results of an observed pure jump process, driving the values of interest rate and volatility coefficient. The pure jump process is assumed to be a semi-Markov process on finite state space. This consideration helps to incorporate a specific type of memory influence in the asset price. Under this model assumption, the locally risk minimizing price of the European type path-independent options is found. The Föllmer-Schweizer decomposition is adopted to show that the option price satisfies an evolution problem, as a function of time, stock price, market regime, and the stagnancy period. To be more precise, the evolution problem involves a linear, parabolic, degenerate and non-local system of integro-partial differential equations. We have established existence and uniqueness of classical solution to the evolution problem in an appropriate class.

q-fin.PR↗

arXiv2018

Testing of Binary Regime Switching Models using Squeeze Duration Analysis

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.

q-fin.CP↗

arXiv2018

Option Pricing in a Regime Switching Stochastic Volatility Model

In the classical model of stock prices which is assumed to be Geometric Brownian motion, the drift and the volatility of the prices are held constant. However, in reality, the volatility does vary. In quantitative finance, the Heston model has been successfully used where the volatility is expressed as a stochastic differential equation. In addition, we consider a regime switching model where the stock volatility dynamics depends on an underlying process which is possibly a non-Markov pure jump process. Under this model assumption, we find the locally risk minimizing pricing of European type vanilla options. The price function is shown to satisfy a Heston type PDE.

q-fin.PR↗

arXiv2018

Risk Sensitive Portfolio Optimization in a Jump Diffusion Model with Regimes

This article studies a portfolio optimization problem, where the market consisting of several stocks is modeled by a multi-dimensional jump-diffusion process with age-dependent semi-Markov modulated coefficients. We study risk sensitive portfolio optimization on the finite time horizon. We study the problem by using a probabilistic approach to establish the existence and uniqueness of the classical solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We also implement a numerical scheme to investigate the behavior of solutions for different values of the initial portfolio wealth, the maturity, and the risk of aversion parameter.

q-fin.PM↗

arXiv2016

Pricing Derivatives in a Regime Switching Market with Time Inhomogeneous Volatility

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 analyze 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.

q-fin.PR↗

arXiv2016

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

This paper includes a proof of well-posedness of an initial-boundary value problem involving a system of degenerate non-local parabolic PDE which naturally arises in the study of derivative pricing in a generalized market model. In a semi-Markov modulated GBM model the locally risk minimizing price function satisfies a special case of this problem. 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 uniqueness.

math.AP↗

arXiv2016

Convergence of Estimated Option Price in a Regime switching Market

In an observed generalized semi-Markov regime, estimation of transition rate of regime switching leads towards calculation of locally risk minimizing option price. Despite the uniform convergence of estimated step function of transition rate, to meet the existence of classical solution of the modified price equation, the estimator is approximated in the class of smooth functions and furthermore, the convergence is established. Later, the existence of the solution of the modified price equation is verified and the point-wise convergence of such approximation of option price is proved to answer the tractability of its application in Finance. To demonstrate the consistency in result a numerical experiment has been reported.

q-fin.PR↗