Martingale Condition Research Articles

Feasible joint posterior beliefs with binary signals

We extend a quantitative Aumann’s agreement theorem in Arieli et al. (2021) from binary states to arbitrary finite states and provide a complete characterization of feasible joint posteriors beliefs when beliefs supports are binary. Using a network flow approach, we show that the martingale condition and the cut condition for a feasible flow are necessary and sufficient for feasible joint posterior beliefs. For symmetric supports, we show that the cut condition implies that the posteriors beliefs must be positively dependent. We also relate the cut condition to various characterization conditions in the literature.

Deep reinforcement learning of viscous incompressible flow

We develop a new computational method to solve viscous incompressible fluid flow. This method is an extension of the Derivative-Free Loss Method, which solves elliptic and parabolic partial differential equations using Brownian motion, the Feynman-Kac formula, and reinforcement learning. Herein we derive a martingale condition and directly incorporate a projection of the velocity to solve the incompressible Navier-Stokes equations. Our method is mesh-free, scalable, and flexible, as demonstrated with several test cases.

A generalized Esscher transform for option valuation with regime switching risk

A generalized Esscher transform is introduced for option valuation in a Markov regime-switching model. It is intended that the generalized Esscher transform might provide novel insights into pricing regime switching risk. A new pricing kernel and the related martingale condition are derived which might provide a convenient way to price both diffusion and regime switching risks. Numerical studies are provided to illustrate the proposed method.

The Probability Flow in the Stock Market and Spontaneous Symmetry Breaking in Quantum Finance

The spontaneous symmetry breaking phenomena applied to Quantum Finance considers that the martingale state in the stock market corresponds to a ground (vacuum) state if we express the financial equations in the Hamiltonian form. The original analysis for this phenomena completely ignores the kinetic terms in the neighborhood of the minimal of the potential terms. This is correct in most of the cases. However, when we deal with the martingale condition, it comes out that the kinetic terms can also behave as potential terms and then reproduce a shift on the effective location of the vacuum (martingale). In this paper, we analyze the effective symmetry breaking patterns and the connected vacuum degeneracy for these special circumstances. Within the same scenario, we analyze the connection between the flow of information and the multiplicity of martingale states, providing in this way powerful tools for analyzing the dynamic of the stock markets.

Market efficiency and random number generators in Solvency II

Complex insurance policies are valued by Economic Scenario Generators (ESG), stochastic simulations developed via Monte Carlo techniques that span the evolution of risk factors important for insurers (financial and technical) consistent with market data. The either pseudo or quasi-random number generators used in the model turn out to be important for the precision of the valuation of Asset/Liability models. I shall compare five different random number generators in a G[Formula: see text] model in terms of market efficiency, by testing for the fulfilment of the martingale condition. The enhancement of existing methods is crucial in the field of computational insurance based on Monte Carlo techniques given that the number of simulations is perilously limited.

Semiparametric estimation of structural nested mean models with irregularly spaced longitudinal observations.

Structural nested mean models (SNMMs) are useful for causal inference of treatment effects in longitudinal observational studies. Most existing works assume that the data are collected at prefixed time points for all subjects, which, however, may be restrictive in practice. To deal with irregularly spaced observations, we assume a class of continuous-time SNMMs and a martingale condition of no unmeasured confounding (NUC) to identify the causal parameters. We develop the semiparametric efficiency theory and locally efficient estimators for continuous-time SNMMs. This task is nontrivial due to the restrictions from the NUC assumption imposed on the SNMM parameter. In the presence of ignorable censoring, we show that the complete-case estimator is optimal among a class of weighting estimators including the inverse probability of censoring weighting estimator, and it achieves a double robustness feature in that it is consistent if at least one of the models for the potential outcome mean function and the treatment process is correctly specified. The new framework allows us to conduct causal analysis respecting the underlying continuous-time nature of data processes. The simulation study shows that the proposed estimator outperforms existing approaches. We estimate the effect of time to initiate highly active antiretroviral therapy on the CD4 count at year 2 from the observational Acute Infection and Early Disease Research Program database.

Recover Dynamic Utility from Observable Process: Application to the Economic Equilibrium

Decision making under uncertainty is often viewed as an optimization problem under choice criterium, but its calibration raises the inverse problem to recover the criterium from the data. An example is the theory of preference by Samuelson in the 40s, \cite{samuelson1938}. The observable is a so-called increasing characteristic process $\mbX=(\scX_t(x))$ and the objective is to recover a dynamic stochastic utility $\bfU$ in the sense where $U(t,\scX_t(x))$ is a martingale. A linearized version is provided by the first order conditions $U_x(t,\scX_t(x))=Y_t(u_z(x)$, and the additional martingale conditions of the processes $\scX_x(t,x)Y_t(u_z(x))$ and $\scX_t(x)Y_t(u_z(x))$. When $\mbX$ and $\bfY$ are regular solutions of two SDEs with random coefficients, under strongly martingale condition, any revealed utility is solution of a non linear SPDE, and is the stochastic value function of some optimization problem. More interesting is the dynamic equilibrium problem as in He and Leland \cite{HL}, where $Y$ is coupled with $\scX$ so that the monotonicity of $Y_t(z,u_z(z)) $ is lost. Nevertheless, we solve the He \& Leland problem (in random environment), by characterizing all the equilibria: the adjoint process still linear in $y$ (GBM in Markovian case) and the conjugate utilities are a deterministic mixture of stochastic dual power utilities. Besides, the primal utility is the value function of an optimal wealth allocation in the Pareto problem.

Policy Evaluation and Temporal-Difference Learning in Continuous Time and Space: A Martingale Approach

We propose a unified framework to study policy evaluation (PE) and the associated temporal difference (TD) methods for reinforcement learning in continuous time and space. We show that PE is equivalent to maintaining the martingale condition of a process. From this perspective, we find that the mean--square TD error approximates the quadratic variation of the martingale and thus is not a suitable objective for PE. We present two methods to use the martingale characterization for designing PE algorithms. The first one minimizes a ``martingale loss function, whose solution is proved to be the best approximation of the true value function in the mean--square sense. This method interprets the classical gradient Monte-Carlo algorithm. The second method is based on a system of equations called the ``martingale orthogonality conditions with ``test functions''. Solving these equations in different ways recovers various classical TD algorithms, such as TD($\lambda$), LSTD, and GTD. Different choices of test functions determine in what sense the resulting solutions approximate the true value function. Moreover, we prove that any convergent time-discretized algorithm converges to its continuous-time counterpart as the mesh size goes to zero. We demonstrate the theoretical results and corresponding algorithms with numerical experiments and applications.

Spontaneous symmetry breaking in quantum finance

We analyze the phenomena of spontaneous symmetry breaking in quantum finance by using as a starting point the Black-Scholes (BS) and the Merton-Garman (MG) equations expressed in the Hamiltonian form. In this scenario the martingale condition (state) corresponds to the vacuum state which becomes degenerate when the symmetry of the system is spontaneously broken. We then analyze the broken symmetries of the system and we interpret from the perspective of financial markets the possible appearance of the Nambu-Goldstone bosons.

Learning, parameter drift, and the credibility revolution

This paper analyses extrapolation and inference using tax experiments in dynamic economies when shock processes are latent regime-shifting Markov chains. Belief revisions result in severe parameter drift: Response signs and magnitudes vary widely over time despite ideal exogeneity. Even with linear causal effects, shock responses are non-linear, preventing direct extrapolation. Analytical formulae are derived for extrapolating responses or inferring causal parameters. Extrapolation and inference hinges upon shock histories and correct assumptions regarding potential data generating processes. A martingale condition is necessary and sufficient for shock responses to directly recover comparative statics, but stochastic monotonicity is insufficient for correct sign inference.

Computational methods for martingale optimal transport problems

We develop computational methods for solving the martingale optimal transport (MOT) problem—a version of the classical optimal transport with an additional martingale constraint on the transport’s dynamics. We prove that a general, multi-step multi-dimensional, MOT problem can be approximated through a sequence of linear programming (LP) problems which result from a discretization of the marginal distributions combined with an appropriate relaxation of the martingale condition. Further, we establish two generic approaches for discretising probability distributions, suitable respectively for the cases when we can compute integrals against these distributions or when we can sample from them. These render our main result applicable and lead to an implementable numerical scheme for solving MOT problems. Finally, specialising to the one-step model on real line, we provide an estimate of the convergence rate which, to the best of our knowledge, is the first of its kind in the literature.

Semiparametric estimation of structural failure time models in continuous-time processes.

Structural failure time models are causal models for estimating the effect of time-varying treatments on a survival outcome. G-estimation and artificial censoring have been proposed for estimating the model parameters in the presence of time-dependent confounding and administrative censoring. However, most existing methods require manually pre-processing data into regularly spaced data, which may invalidate the subsequent causal analysis. Moreover, the computation and inference are challenging due to the nonsmoothness of artificial censoring. We propose a class of continuous-time structural failure time models that respects the continuous-time nature of the underlying data processes. Under a martingale condition of no unmeasured confounding, we show that the model parameters are identifiable from a potentially infinite number of estimating equations. Using the semiparametric efficiency theory, we derive the first semiparametric doubly robust estimators, which are consistent if the model for the treatment process or the failure time model, but not necessarily both, is correctly specified. Moreover, we propose using inverse probability of censoring weighting to deal with dependent censoring. In contrast to artificial censoring, our weighting strategy does not introduce nonsmoothness in estimation and ensures that resampling methods can be used for inference.

Merton’s equation and the quantum oscillator II: Option pricing

Merton has proposed a model for contingent claims on a firm as an option on the firms value, and is based on a generalization of the Black–Scholes stochastic equation (Merton, 1974). A special case of Merton’s model is proposed – based on the quantum oscillator – for pricing options. Two cases of the option price are obtained: both these cases yield possible candidates for the generalization of the Black–Scholes option pricing formula. However, one of the proposed option prices does not obey the martingale condition and the other does not yield the correct discounting of future cash flows. For these reasons, the option prices do not obey put–call parity. The options can, however, be used to approximately price market traded options. The oscillator model for the option price has an extra parameter that is absent for the Black–Scholes case. Similar to the model studied by Baaquie et al. (2014), which that does not obey put–call parity, the option’s price can be studied empirically and the extra parameter in the model could, in principal, generate implied volatility.

Reflected solutions of backward stochastic differential equations driven by G-Brownian motion

In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by $G$-Brownian motion. The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected $G$-BSDEs, we apply a “martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization. We then give some applications including a generalized Feynman-Kac formula of an obstacle problem for fully nonlinear partial differential equation and option pricing of American types under volatility uncertainty.

Optimal Portfolio for N-Risky and N-Riskless Irreversible Investments

When a firm decides to make irreversible investment expenditure, it neglects its right without obligation to invest but wait for new desirable information that might affect investment decision. Studies have shown irreversible investments with the underlying value process driven by stochastic differential equations. However, this study had no consideration for a portfolio of N-risky and Nriskless investments (N = 2; 3;;T <1) in relation to the set of: Variable Production Costs (VPC), Periodic Loan Repayment (PLR) with penalty for default, Investment Reserve (IR) and Equipment Value (EV) equations. This study was designed to consider a portfolio of N-risky and N-riskless investments including a set of processes describing VPC for the portfolio, interval PLR for the same investments with penalty processes for default in PLR. A set of instantaneous Expected Revenue (ER) from the risky investments were formulated using Geometric Brownian Motion (GBM) with drift term restricted to zero and volatility strictly positive (Martingale condition). The resulting equation was solved using Ito’s formula. The IR and EV equations were formulated with predictive adaptive processes to examine liquidity of business plan and Equipment Depreciation (ED) value respectively. Maintenance Process (MP) was introduced into the EV equation.

Pricing currency derivatives with Markov-modulated Lévy dynamics

Using a Lévy process we generalize formulas in Bo et al. (2010) for the Esscher transform parameters for the log-normal distribution which ensure that the martingale condition holds for the discounted foreign exchange rate. Using these values of the parameters we find a risk-neural measure and provide new formulas for the distribution of jumps, the mean jump size, and the Poisson process intensity with respect to this measure. The formulas for a European call foreign exchange option are also derived. We apply these formulas to the case of the log-double exponential distribution of jumps. We provide numerical simulations for the European call foreign exchange option prices with different parameters.

Robust hedging with proportional transaction costs

A duality for robust hedging with proportional transaction costs of path-dependent European options is obtained in a discrete-time financial market with one risky asset. The investor’s portfolio consists of a dynamically traded stock and a static position in vanilla options, which can be exercised at maturity. Trading of both options and stock is subject to proportional transaction costs. The main theorem is a duality between hedging and a Monge–Kantorovich-type optimization problem. In this dual transport problem, the optimization is over all probability measures that satisfy an approximate martingale condition related to consistent price systems, in addition to an approximate marginal constraint.

Currency Derivatives Pricing for Markov-Modulated Merton Jump-Diffusion Spot Forex Rate

We derive results similar to Bo et al. (2010), but in the case of dynamics of the FX rate driven by a general Merton jump-diffusion process. The main results of our paper are as follows: 1) formulas for the Esscher transform parameters which ensure that the martingale condition for the discounted foreign exchange rate is a martingale for a general Merton jump-diffusion process are derived; using the values of these parameters we proceed to a risk-neural measure and provide new formulas for the distribution of jumps, the mean jump size, and the Poisson Process intensity with respect to the measure; pricing formulas for European foreign exchange call options have been given as well; 2) obtained formulas are applied to the case of the exponential processes; 3) numerical simulations of European call foreign exchange option prices for different parameters are also provided.

Option volatility and the acceleration Lagrangian

This paper develops a volatility formula for option on an asset from an acceleration Lagrangian model and the formula is calibrated with market data. The Black–Scholes model is a simpler case that has a velocity dependent Lagrangian.The acceleration Lagrangian is defined, and the classical solution of the system in Euclidean time is solved by choosing proper boundary conditions. The conditional probability distribution of final position given the initial position is obtained from the transition amplitude. The volatility is the standard deviation of the conditional probability distribution. Using the conditional probability and the path integral method, the martingale condition is applied, and one of the parameters in the Lagrangian is fixed. The call option price is obtained using the conditional probability and the path integral method.

Testing for financial crashes using the Log Periodic Power Law model

Many papers claim that a Log Periodic Power Law (LPPL) model fitted to financial market bubbles that precede large market falls or ‘crashes’, contains parameters that are confined within certain ranges. Further, it is claimed that the underlying model is based on influence percolation and a martingale condition. This paper examines these claims and their validity for capturing large price falls in the Hang Seng stock market index over the period 1970 to 2008. The fitted LPPLs have parameter values within the ranges specified post hoc by Johansen and Sornette (2001) for only seven of these 11 crashes. Interestingly, the LPPL fit could have predicted the substantial fall in the Hang Seng index during the recent global downturn. Overall, the mechanism posited as underlying the LPPL model does not do so, and the data used to support the fit of the LPPL model to bubbles does so only partially.