Martingale Method Research Articles

Pricing of A European Call Option in Stochastic Volatility Models

Volatility occupies a strategic place in the financial markets. In this context of crisis, and with the great movements of the markets, traders have been forced to turn to volatility trading for the potential gain it provides. The Black-Scholes formula for the value of a European option to purchase the underlying depends on a few parameters which are more or less easy to calculate, except for the realized volatility at maturity which makes a problem, because there is no single value, nor an established way to calculate it. In this article, we exploit the Martingale pricing method to find the expected present value of a given asset relative to a riskneutral probability measure. We consider a bond-stock market that evolves according to the dynamics of the Black-Scholes model, with a risk-free interest rate varying with time. Our methodology has effectively directed us towards interesting formulas that we have derived from the exact calculation, giving the present value of the volatility realized over a period of maturity for a European option in a stochastic volatility model.

On the extinction-extinguishing dichotomy for a stochastic Lotka–Volterra type population dynamical system

Applying the Foster–Lyapunov type criteria and a martingale method, we study a two-dimensional process (X,Y) arising as the unique nonnegative solution to a pair of stochastic differential equations driven by independent Brownian motions and compensated spectrally positive Lévy random measures. Both processes X and Y can be identified as continuous-state nonlinear branching processes where the evolution of Y is negatively affected by X. Assuming that process X extinguishes, i.e. it converges to 0 but never reaches 0 in finite time, and process Y converges to 0, we identify rather sharp conditions under which the process Y exhibits, respectively, one of the following behaviors: extinction with probability one, extinguishing with probability one or both extinction and extinguishing occurring with strictly positive probabilities.

Parameter estimation in branching processes with almost sure extinction

We consider population-size-dependent branching processes (PSDBPs) which eventually become extinct with probability one. For these processes, we derive maximum likelihood estimators for the mean number of offspring born to individuals when the current population size is z≥1. As is standard in branching process theory, an asymptotic analysis of the estimators requires us to condition on non-extinction up to a finite generation n and let n→∞; however, because the processes become extinct with probability one, we are able to demonstrate that our estimators do not satisfy the classical consistency property (C-consistency). This leads us to define the concept of Q-consistency, and we prove that our estimators are Q-consistent and asymptotically normal. To investigate the circumstances in which a C-consistent estimator is preferable to a Q-consistent estimator, we then provide two C-consistent estimators for subcritical Galton–Watson branching processes. Our results rely on a combination of linear operator theory, coupling arguments, and martingale methods.

A driven tagged particle in asymmetric exclusion processes

We consider the asymmetric exclusion process with a driven tagged particle on Z which has different jump rates from other particles. When the non-tagged particles have non-nearest-neighbor jump rates , we show that the tagged particle can have a speed which has a different sign from the mean derived from its jump rates. We also show the existence of some non-trivial invariant measures for the environment process viewed from the tagged particle. Our arguments are based on coupling, martingale methods, and analyzing currents through fixed bonds.

Мартингальный метод исследования ветвящихся случайных блужданий

Мартингальный метод исследования ветвящихся случайных блужданий

The Basic Reproduction Number for the Markovian SIR‐Type Epidemic Models: Comparison and Consistency

This paper is concerned with a well‐known epidemiological concept to measure the spread of infectious disease, that is, the basic reproduction number. This paper has two major objectives. The first is to examine Bayesian sensitivity and consistency of this measure for the case of Markov epidemic models. The second is to assess the Martingale method by comparing its performance to that of Markov Chain Monte Carlo (MCMC) methods in terms of estimating this parameter and the infection and removal rate parameters given only removal data. We specifically consider the Markovian SIR (Susceptible‐Infective‐Removed) epidemic model routinely employed in the literature with exponentially distributed infectious periods. For illustration, numerical simulation studies are performed. Abakaliki smallpox data are examined as a real data application.

Martingale method for studying branching random walks

Martingale method for studying branching random walks

Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets

Abstract The pricing and hedging of financial derivatives have become one of the hot research issues in mathematical finance today. In the case of non-risk neutrality, this article uses the martingale method and probability measurement method to study the pricing method and hedging strategy of financial derivatives. This paper also further studies the hedging strategy of financial derivatives in the incomplete market based on the BSM model and converts the solution of this problem into solving a vector on the Hilbert space to its closure. The problem of space projection is to use projection theory to decompose financial derivatives under a given martingale measure. In the imperfect market, the vertical projection theory is used to obtain the approximate pricing method and hedging strategy of financial derivatives in which the underlying asset follows the martingale process; the projection theory is further expanded, and the pricing problem of financial derivatives under the mixed-asset portfolio is obtained. Approximate pricing of financial derivatives; in the discrete state, the hedging investment strategy of financial derivatives H in the imperfect market is found through the method of variance approximation.

A restaurant process with cocktail bar and relations to the three-parameter Mittag–Leffler distribution

AbstractIn addition to the features of the two-parameter Chinese restaurant process (CRP), the restaurant under consideration has a cocktail bar and hence allows for a wider range of (bar and table) occupancy mechanisms. The model depends on three real parameters, $\alpha$ , $\theta_1$ , and $\theta_2$ , fulfilling certain conditions. Results known for the two-parameter CRP are carried over to this model. We study the number of customers at the cocktail bar, the number of customers at each table, and the number of occupied tables after n customers have entered the restaurant. For $\alpha>0$ the number of occupied tables, properly scaled, is asymptotically three-parameter Mittag–Leffler distributed as n tends to infinity. We provide representations for the two- and three-parameter Mittag–Leffler distribution leading to efficient random number generators for these distributions. The proofs draw heavily from methods known for exchangeable random partitions, martingale methods known for generalized Pólya urns, and results known for the two-parameter CRP.

Optimal asset allocation, consumption and retirement time with the variation in habitual persistence

In this paper, we study the individual's optimal asset allocation, consumption and retirement time under habitual persistence. To depict the phenomenon that the individual feels equally satisfied with a lower habitual level and is more reluctant to change the habitual level after retirement, we assume that both the level and the sensitivity of the habitual consumption decline at the time of retirement. We establish the concise form of the habitual evolutions, and obtain the optimal retirement time and consumption policy based on martingale and duality methods. The optimal consumption experiences a sharp decline at retirement, but the excess consumption raises because of the reduced sensitivity of the habitual level. This result is consistent with the evidence observed in the “retirement consumption puzzle”. Particularly, the optimal retirement and consumption policies are balanced between the wealth effect and the habitual effect. Larger wealth increases consumption, and larger growth inertia (sensitivity) of the habitual level decreases consumption and brings forward the retirement time.

Entropy dissipation estimates for inhomogeneous zero-range processes

Introduced by Lu and Yau (Comm. Math. Phys. 156 (1993) 399–433), the martingale decomposition method is a powerful recursive strategy that has produced sharp log-Sobolev inequalities for homogeneous particle systems. However, the intractability of certain covariance terms has so far precluded applications to heterogeneous models. Here we demonstrate that the existence of an appropriate coupling can be exploited to bypass this limitation effortlessly. Our main result is a dimension-free modified log-Sobolev inequality for zero-range processes on the complete graph, under the only requirement that all rate increments lie in a compact subset of (0,∞). This settles an open problem raised by Caputo and Posta (Probab. Theory Related Fields 139 (2007) 65–87) and reiterated by Caputo, Dai Pra and Posta (Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 734–753). We believe that our approach is simple enough to be applicable to many systems.

Concentration inequalities for suprema of unbounded empirical processes

Using martingale methods, we obtain some Fuk-Nagaev type inequalities for suprema of unbounded empirical processes associated with independent and identically distributed random variables. We then derive weak and strong moment inequalities. Next, we apply our results to suprema of empirical processes which satisfy a power-type tail condition.

Finite horizon portfolio selection problem with a drawdown constraint on consumption

In this paper we investigate the consumption and portfolio selection problem of a finitely-lived agent who faces drawdown constraint on consumption: the agent does not accept falling in her consumption below a fixed proportion of the historically highest level. We utilize the dual martingale method and study the dual problem, which can be transformed into an infinite series of optimal stopping problems. Based on a theory of partial differential equation (PDE), we analytically characterize a variational inequality arising from the optimal stopping problem. We provide a verification theorem that the value function for the original agent's problem is the Legendre-Fenchel transform of the integral of the value functions for the optimal stopping problems. Moreover, we derive integral equation representations for the optimal strategies and provide some numerical implications for the optimal strategies.

Optimal post-retirement consumption and portfolio choices with idiosyncratic individual mortality force and awareness of mortality risk

Considering the effects of mortality risk awareness and idiosyncratic mortality force on economic decision-making of a retiree, we study the optimal post-retirement consumption and portfolio problem with mortality risk awareness, idiosyncratic individual mortality force. We derive a closed optimal strategy in a complete financial market under stochastic interest rate and national population mortality force by martingale method. A numerical illustration is performed, and shows that mortality risk awareness and idiosyncratic individual mortality force have important effects on the consumption and investment of retiree. Especially, the demand of longevity bond is significantly affected by mortality risk awareness.

Optimal Control on Finite-Time Consensus of the Leader-Following Stochastic Multiagent System With Heuristic Method

In this paper, the optimization of the finite-time consensus of a multiagent system is studied when the leader is disturbed by white noise. Different from the consensus problem itself, this paper is devoted to optimizing the performance of the consensus and the constructive method and martingale method are used to guarantee the effectiveness of the optimization. Another contribution is that the problem of the optimal control with the piecewise constant controller gains under a stochastic environment is solved, and the reformed dynamic principle and the reformed Hamilton–Jacobi–Bellman (HJB) partial differential equations (PDEs) are proposed by using the splitting method and Feynman–Kac formula. Furthermore, since the multiagent system often runs on a high-dimensional space, a heuristic method is proposed based on the derivation of the reformed HJB PDEs and the scalar of the computational complexity of the PDEs is equivalent to that of a system of ordinary differential equations with the same dimensions when the multiagent system can be described as a piecewise linear system. Finally, the heuristic method is compared with usual filtering methods for solving engineering problems by a numerical example and it is found that the heuristic method achieves higher precision in finite time.

Implicit incentives for fund managers with partial information

We study the optimal asset allocation problem for a fund manager whose compensation depends on the performance of her portfolio with respect to a benchmark. The objective of the manager is to maximise the expected utility of her final wealth. The manager observes the prices but not the values of the market price of risk that drives the expected returns. Estimates of the market price of risk get more precise as more observations are available. We formulate the problem as an optimization under partial information. The particular structure of the incentives makes the objective function not concave. Therefore, we solve the problem by combining the martingale method and a concavification procedure and we obtain the optimal wealth and the investment strategy. A numerical example shows the effect of learning on the optimal strategy.

Control principle and error estimation for inverse trajectory method under locating error with optimization

Accurate trajectory control of wheeled robot is an unavoidable problem in path planning, especially in the environment of large positioning error. In this paper, by designing the reverse trajectory and using martingale method, the accurate execution of the target trajectory is realized, and the error estimation is given. Finally, through numerical simulation, it is shown that when the observation error is 0.5 units, the average trajectory error is about 0.1 units, and the probability of error exceeding 0.347 units is less than 2%.

Finite horizon portfolio selection with durable goods

We study the consumption and portfolio selection problem of a finitely lived agent who derives utility from the stock of durable goods. We show that the agent’s effective relative risk aversion implied by the optimal portfolio tends to decline and approaches zero, as the planning horizon approaches, whereas the agent exhibits constant effective relative risk aversion in the infinite horizon model of Hindy and Huang (1993). The existence of the stock of durable goods acts as buffer stock and induces the highly risk-tolerant attitude. We approach this problem using successive transformations. We transform our problem by applying an isomorphism proposed by Schroder and Skiadas (2002) to a singular control problem involving the choice of a monotone increasing consumption process. We next transform the problem into a dual singular control problem using the dual martingale method. We then transform the dual singular control problem into an optimal stopping problem. We analyze the variational inequality arising from the optimal stopping problem and provide an integral representation of optimal strategies.

On the pair correlations of powers of real numbers

A classical theorem of Koksma states that for Lebesgue almost every x > 1 the sequence (xn) =1 ∞ is uniformly distributed modulo one. In the present paper we extend Koksma’s theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every x > 1 the pair correlations of the fractional parts of (xn) =1 ∞ are asymptotically Poissonian. The proof is based on a martingale approximation method.

Normal deviation of synchronization of stochastic coupled systems

<p style='text-indent:20px;'>This paper will prove the normal deviation of the synchronization of stochastic coupled system. According to the relationship between the stationary solution and the general solution, the martingale method is used to prove the normal deviation of the fixed initial value of the multi-scale system, thereby obtaining the normal deviation of the stationary solution. At the same time, with the relationship between the synchronized system and the multi-scale system, the normal deviation of the synchronization is obtained.</p>