Stock Price Process Research Articles

Time-changed generalized mixed fractional Brownian motion and application to arithmetic average Asian option pricing

In this paper we present a novel model to analyze the behavior of random asset price process under the assumption that the stock price pro-cess is governed by time-changed generalized mixed fractional Brownian motion with an inverse gamma subordinator. This model is con-structed by introducing random time changes into generalized mixed fractional Brownian motion process. In practice it has been observed that many different time series have long-range dependence property and constant time periods. Fractional Brownian motion provides a very general model for long-term dependent and anomalous diffusion regimes. Motivated by this facts in this paper we investigated the long-range dependence structure and trapping events (periods of prices stay motionless) of CSCO stock price return series. The constant time periods phenomena are modeled using an inverse gamma process as a subordinator. Proposed model include the jump behavior of price process because the gamma process is a pure jump Levy process and hence the subordinated process also has jumps so our model can be capture the random variations in volatility. To show the effectiveness of proposed model, we applied the model to calculate the price of an average arithmetic Asian call option that is written on Cisco stock. In this empirical study first the statistical properties of real financial time series is investigated and then the estimated model parameters from an observed data. The results of empirical study which is performed based on the real data indicated that the results of our model are more accuracy than the results based on traditional models.

Forward starting options pricing with double stochastic volatility, stochastic interest rates and double jumps

We present an extension of double Heston stochastic volatility model by introducing CIR stochastic interest rate and double exponential jumps in the stock price process. We derive the characteristic function and forward characteristic function of the log asset price and thereby forward starting options are well evaluated by the COS method. We also provide efficient simulation of the proposed model and Monte Carlo solutions to forward starting options pricing based on the QE scheme. Numerical results show that the COS method is fast and efficient for pricing forward starting options.

Optimal Investment in a Black-Scholes Model with a Bubble

Financial bubbles seem to exhibit a strong upward trend followed by a sharp decline when the bubble bursts. Starting from this descriptive perspective, we propose an extension of the Black--Scholes model which accommodates this effect. In addition to the standard instantaneous expected return in the Black–Scholes model, we allow a time-dependent positive excess return which is compensated by a negative jump at a random time representing the bursting of the bubble. Since the excess return and the distribution of the random time may be chosen almost arbitrarily, the model is flexible enough to display different types of bubbles, including the ones in the sense of Protter [22]. Moreover, the model is tractable enough to allow (semi-)explicit calculations and may thus be used as a toy example to study qualitative effects of financial bubbles.We study the problem of maximising expected utility from terminal wealth for a power utility investor. Using the convex duality approach, we determine the optimal strategy and the corresponding certainty equivalent up to the solution to an integral equation (or a first-order ODE). A decomposition of the optimal strategy into myopic and hedging demand allows to analyse the effects of the stochastic investment opportunities. On the one hand, investors with relative risk aversion below 1 speculate on an early bursting of the bubble in the sense that their optimal strategy lies below their myopic demand prior to the bursting; it might even involve short selling. On the other hand, investors with relative risk aversion above 1 hedge against a late bursting of the bubble in the sense that their optimal strategy lies above their myopic demand; it might even lie above the Merton proportion under extreme circumstances.Numerical examples reveal how the optimal strategy and its myopic and hedging demand parts depend on the model parameters. In particular, it is shown that the optimal strategy is not fundamentally different when the stock price process is a strict local martingale (as opposed to a true martingale) under a certain class of equivalent local martingale measures including the dual minimiser of the utility maximisation problem. In addition, we illustrate the welfare loss compared to the classical Black–Scholes model and its dependence on the model parameters.

Explicit Description of HARA Forward Utilities and Their Optimal Portfolios

This paper deals with forward performances of HARA type. Precisely, for a market model in which stock price processes are modeled by a locally bounded $d$-dimensional semimartingale, we elaborate a complete and explicit characterization for this type of forward utilities. In particular, under some mild technical assumptions of integrability, we prove that the risk aversion process of a power-type forward utility is constant. Furthermore, the optimal portfolios for all HARA forward utilities are explicitly described. Our approach is based on the minimal Hellinger martingale densities that are obtained from the important statistical concept of Hellinger process. These martingale densities were introduced recently and appeared herein tailor-made for these forward utilities. After outlining our parametrization method for the HARA forward, we provide illustrations on discrete-time market models.

Theoretical Framework for Stock Pricing Process Based on Micro-Economic Decision Model

The most common model for asset pricing (CAPM) is problematic and does not match the reality. In this article, I introduce a theoretical framework for a new model which aims at avoiding the problems of CAPM and keeping its advantages, therefore allowing universality of asset pricing. The model is built on the economic principles, using a budget constraint and a Risk Appetite (RA) function. It is based on the micro-economic decision model, involving an expected value and dividing a stock price to objective and subjective prices. As a result, rational based individuals, just like individuals with non-rational factors, may use the model to calculate a future price stock in exactly the same way.

Complexity and multifractal behaviors of multiscale-continuum percolation financial system for Chinese stock markets

A new financial agent-based time series model is developed and investigated by multiscale-continuum percolation system, which can be viewed as an extended version of continuum percolation system. In this financial model, for different parameters of proportion and density, two Poisson point processes (where the radii of points represent the ability of receiving or transmitting information among investors) are applied to model a random stock price process, in an attempt to investigate the fluctuation dynamics of the financial market. To validate its effectiveness and rationality, we compare the statistical behaviors and the multifractal behaviors of the simulated data derived from the proposed model with those of the real stock markets. Further, the multiscale sample entropy analysis is employed to study the complexity of the returns, and the cross-sample entropy analysis is applied to measure the degree of asynchrony of return autocorrelation time series. The empirical results indicate that the proposed financial model can simulate and reproduce some significant characteristics of the real stock markets to a certain extent.

American option valuation under time changed tempered stable Lévy processes

Abstract Given that the underlying assets in financial markets exhibit stylized facts such as leptokurtosis, asymmetry, clustering properties and heteroskedasticity effect, this paper presents a novel model for pricing American option under the assumptions that the stock price processes are governed by time changed tempered stable Levy process. As this model is constructed by introducing random time changes into tempered stable (TS) processes which specially refer to normal tempered stable (NTS) distribution as well as classical tempered stable (CTS) distribution, it permits infinite jumps as well as capturing random varying time in stochastic volatility, consequently taking into account the empirical facts such as leptokurtosis, skewness and volatility clustering behaviors. We employ the Fourier-cosine technique to calculate American option and propose the improved Particle Swarm optimization (IPSO) intelligent algorithm for model calibration. To demonstrate the advantage of the constructed model, we carry out empirical research on American index option in financial markets across wide ranges of models, with the time changing normal tempered stable distribution model yielding a superior performance than others.

Investment time horizon and multifractality of stock price process

We construct a multifractal random walk for a stock trades model with inverse power law interaction. Consider a stock in a stock market and introduce a discrete time model for pair trades [(first trade \(\rightarrow\) second (reverse) trade] transacted by various types of traders. The type of trader is characterized by the investment time horizon defined as a time difference of the pair trades. We assume that probability distributions of the investment time horizons are given by inverse power law interactions with different exponents depending on the types of traders, and define a discrete time log-volatility process of the stock. Using the method of abstract polymer expansion developed in the study of mathematical physics we obtain a continuous type log-volatility process as a scale limit, and define a log-return process from time 0 as a stochastic integral with respect to a Brownian motion. We then find a condition of the continuous cascade equation which derives a multifractality in the log-return process. Finally, we construct a multifractal random walk using the martingale convergence theorem.

Strong approximation of Black-Scholes theory based on simple random walks

A basic model in financial mathematics was introduced by Black, Scholes and Merton in 1973. A classical discrete approximation in distribution is the binomial model given by Cox, Ross and Rubinstein in 1979. In this work we give a strong (almost sure, pathwise) discrete approximation of the BSM model using a suitable nested sequence of simple, symmetric random walks. The approximation extends to the stock price process, the value process, the replicating portfolio, and the greeks. An important tool in the approximation is a discrete version of the Feynman-Kac formula as well. Our aim is to show that from an elementary discrete approach, by taking simple limits, one may get the continuous versions. We think that such an approach can be advantageous for both research and applications. Moreover, it is hoped that this approach has pedagogical merits as well: gives insight and seems suitable for teaching students whose mathematical background may not contain e.g. measure theory or stochastic analysis.

Uncertainty and inside information

In this paper, we study a robust-entropic optimal control problem in the presence of inside information. To be more precise, we consider an economic agent who is allowed to invest her wealth in a classical Black-Scholes type financial market. From the beginning of the trading interval, the agent exclusively possesses some inside information concerning the future realization of the stock price process. However, we assume that she is uncertain as to the validity of this information, thus introducing in this way robust aspects to our model. The aim of the economic agent is to solve an expected utility maximization problem under the worst-case scenario, taking into account her enlarged information set. By formulating this problem as a two-player, zero sum stochastic differential game, we are able to provide closed form solutions for the optimal robust strategies and the robust value function, in the case of the exponential and the power utility functions.

The mean correcting martingale measures for exponential additive processes

The mean correcting martingale measure for the stochastic process defined as the exponential of an additive process is constructed. Necessary and sufficient conditions for the existence of mean correcting martingale are also obtained. The investigation of this paper will establish a unified way that is applicable both to the case of Levy processes and that of the sums of independent random variables. As an application, we present the necessary and sufficient conditions that the discounted stock price process is a martingale.

Modeling High-Frequency Non-Homogeneous Order Flows by Compound Cox Processes*

A micro-scale model is proposed for the evolution of a limit order book in modern high-frequency trading applications. Within this model, order flows are described by doubly stochastic Poisson processes (also called Cox processes) taking account of the stochastic character of the intensities of order flows. The models for the number of order imbalance (NOI) process and order flow imbalance (OFI) process are introduced as two-sided risk processes that are special compound Cox processes. These processes are sensitive indicators of the current state of the limit order book since time intervals between events in a limit order book are usually so short that price changes are relatively infrequent events. Therefore price changes provide a very coarse and limited description of market dynamics at time micro-scales. NOI and OFI processes track best bid and ask queues and change much faster than prices. They incorporate information about build-ups and depletions of order queues and they can be used to interpolate market dynamics between price changes and to track the toxicity of order flows. The proposed multiplicative model of stochastic intensities makes it possible to analyze the characteristics of the order flows as well as the instantaneous proportion of the forces of buyers and sellers without modeling the external information background. The proposed model gives the opportunity to link the micro-scale high-frequency dynamics of the limit order book with the macroscale models of stock price processes of the form of subordinated Wiener processes by means of limit theorems for special random sums and hence, to give a deeper insight in the nature of popular models of statistical regularities of the evolution of characteristics of financial markets such as generalized hyperbolic distributions and other normal variance-mean mixtures.

Robust option pricing: Hannan and Blackwell meet Black and Scholes

We apply methods developed in the literature initiated by Hannan and Blackwell on robust optimization, approachability and calibration, to price financial securities. Rather than focus on asymptotic performance, we show how gradient strategies developed to minimize asymptotic regret imply financial trading strategies that yield arbitrage-based bounds for option prices. These bounds are new and robust in that they do not depend on the continuity of the stock price process, complete markets, or an assumed pricing kernel. They depend only on the realized quadratic variation of the price process, which can be measured and, importantly, hedged in financial markets using existing securities. Our results also apply directly to a new class of options called timer options. Finally, we argue that the Hannan–Blackwell strategy is path dependent and therefore suboptimal with a finite horizon. We improve it by solving for the optimal path-independent strategy, and compare the resulting bounds with Black–Scholes.

On Hannan and Blackwell's Approachability and Options - A Game Theoretic Approach for Option Pricing

We apply gradient strategy methods, developed in the literature on robust optimization, approachability and calibration, to develop new bounds for option prices. While this literature focuses on asymptotic performance, we provide a financial interpretation of these methods by demonstrating how the gradient strategies developed by Hannan and Blackwell to minimize asymptotic regret imply trading strategies that yield arbitrage-based bounds for option prices. These bounds are both new and robust in that they do not depend on the continuity of the stock price process, complete markets, or an assumed pricing kernel. Rather, they depend only on the realized quadratic variation of the stock price process, which can be measured and, more importantly, hedged in financial markets using existing securities. We then argue that the Hannan–Blackwell strategy is path dependent and therefore suboptimal with a finite horizon. We solve for the optimal path-independent strategy, and compare the bounds achieved with Black-Scholes.

Consumption Optimization and Equilibrium

This paper studies the problem of consumption optimization and equilibrium in discontinuous time financial markets. It is established that the behavior model of the stock pricing process is jump-diffusion driven by a count process. It is proved that the existence of unique optimal consumption and portfolio pair and unique equivalent martingale measure by stochastic analysis methods. The unique equivalent martingale measure, the unique optimal consumption and portfolio pair and the corresponding wealth process are deduced. Finally we provide a simple characterization of an equilibrium market.

OPTION PRICING WITH A LEVY-TYPE STOCHASTIC DYNAMIC MODEL FOR STOCK PRICE PROCESS UNDER SEMI-MARKOVIAN STRUCTURAL PERTURBATIONS

In this work, we consider a stock price process subjected to idiosyncratic Lévy jumps and global structural changes attributed to interventions due to a semi-Markov process. The semi-Markov process decomposes both the time and state domains of the price process into sub-intervals and price state sub-domains respectively, where a Lévy–Ito process operates. The Lévy jumps decompose the space domain of the currently operating Lévy process. We derive an infinitesimal generator for a stock price process and a closed form expression for the conditional characteristic function of a log price. The former result is used to derive a PIDE satisfied by option prices, while the latter could be used to retrieve risk neutral densities via Fourier transform and price European vanilla options. In the sequel, we derive the characteristic function of the residence time of a semi-Markov process. Incompleteness of the market is exhibited through a general change of measure. For pricing purpose, the minimum entropy martingale measure is defined as an Esscher transform.

ASYMPTOTIC ARBITRAGE IN THE HESTON MODEL

In this paper, we introduce a new form of asymptotic arbitrage, which we call a partial asymptotic arbitrage, half-way between those of Föllmer & Schachermayer (2007) [Mathematics and Financial Economics 1 (34), 213–249] and Kabanov & Kramkov (1998) [Finance and Stochastics 2, 143–172]. In the context of the Heston model, we establish a precise link between the set of equivalent martingale measures, the ergodicity of the underlying variance process and this partial asymptotic arbitrage. In contrast to Föllmer & Schachermayer (2007) [Mathematics and Financial Economics 1 (34), 213–249], our result does not assume a suitable condition on the stock price process to allow for (partial) asymptotic arbitrage.

Pricing geometric Asian power options under mixed fractional Brownian motion environment

It has been observed that the stock price process can be modeled with driving force as a mixed fractional Brownian motion with Hurst index H>34 whenever long-range dependence is possibly present. We obtain a closed form expression for the price of a geometric Asian option under the mixed fractional Brownian motion environment. We consider also Asian power options when the payoff function is a power function.

Volatility Analysis of Financial Agent-Based Market Dynamics from Stochastic Contact System

A financial agent-based time series model is developed and investigated by the stochastic contact systems. Multicolor contact system, as one of statistical physics systems, is applied to model a random stock price process for investigating the fluctuation dynamics of financial market. The interaction and dispersal of different types of investment attitudes in a financial market is imitated by viruses spreading in a multicolor contact system, and we suppose that the investment attitudes of market participants contribute to the volatilities of financial time series. We introduce a volatility duration analysis to detect the duration and intensity relationship of time series for both SSECI and the financial model. Furthermore, the empirical research is also presented to study the nonlinear behaviors of returns for the actual data and the simulation data.

Efficient Monte Carlo option pricing under CEV model

ABSTRACTOne of the financial model with nonconstant volatiltiy is the constant elasticity of varinace model, or CEV model for short. The CEV model is an altrnative to the Black–Scholes model of stock price movements. In this diffusion process, unlike the Black–Scholes model, the volatility is a function of the stock price and involves two parameters. In this article, we propose an efficient Monte-Carlo algorithm for pricing arithmetic Asian option under CEV model. In an earlier work by Mehrdoust, an efficient Monte Carlo simulation algorithm for pricing arithmetic Asian options under Black–Scholes model is proposed. The proposed algorithm has proved extremely successful in decreasing the standard deviation and the error of simulation in pricing of the arithmetic Asian options. In this article, we find that the proposed algorithm under the geometric Brownian motion assumption in the Black–Scholes model can effectively apply for pricing arithmetic Asian options when the stock price process follows the CEV model. Numerical experiments show that our algorithm gives very accurate results.