Real Probability Measure Research Articles

Shadow martingales – a stochastic mass transport approach to the peacock problem

Given a family of real probability measures $(\mu_t)_{t\geq 0}$ increasing in convex order (a peacock) we describe a systematic method to create a martingale exactly fitting the marginals at any time. The key object for our approach is the obstructed shadow of a measure in a peacock, a generalization of the (obstructed) shadow introduced in \cite{BeJu16,NuStTa17}. As input data we take an increasing family of measures $(\nu^\alpha)_{\alpha \in [0,1]}$ with $\nu^\alpha(\mathbb{R})=\alpha$ that are submeasures of $\mu _0$, called a parametrization of $\mu_0$. Then, for any $\alpha$ we define an evolution $(\eta^\alpha_t)_{t\geq 0}$ of the measure $\nu^\alpha=\eta^\alpha_0$ across our peacock by setting $\eta^\alpha_t$ equal to the obstructed shadow of $\nu^\alpha$ in $(\mu _s)_{s \in [0,t]}$. We identify conditions on the parametrization $(\nu^\alpha)_{\alpha\in [0,1]}$ such that this construction leads to a unique martingale measure $\pi$, the shadow martingale, without any assumptions on the peacock. In the case of the left-curtain parametrization $(\nu_{\text{lc}}^\alpha)_{\alpha \in [0,1]}$ we identify the shadow martingale as the unique solution to a continuous-time version of the martingale optimal transport problem. Furthermore, our method enriches the knowledge on the Predictable Representation Property (PRP) since any shadow martingale comes with a canonical Choquet representation in extremal Markov martingales.

Correcting for risk premium on Extended Generalised Leland Models: an empirical study on Dow Jones Industrial Average (DJIA) index options

The relative option pricing performance of Extended Generalised Leland models is examined in this study. We generalise the extended Leland models based on the implied adjusted volatility introduced in the models. Non-parametric framework is fitted into parametric option-pricing framework based on the Leland models to achieve a more realistic option pricing. To reflect the real probability measure, the implied adjusted information is corrected in terms of risk premium. This study concentrates mainly in examining the option-implied information produced by the models after correcting for risk-premium. Data extracted from DJIA index options are employed in this study, which covers the period from January 2009 until the end of 2015. We discovered that the option-implied volatility, which is priced using the Extended Generalised Leland models, especially after being corrected for risk-premium factor improves the option valuation accuracy significantly.

Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk

For a G-inhomogeneous semi-Markov chain and G-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theorem that the forward probability measure does not alter the semi Markov structure. In addition, foundation of a G-inhohomogeneous Markov renewal process is done and a theorem is provided where it is proved that the Markov renewal process is maintained under the forward probability measure. We show that for an inhomogeneous semi-Markov there are martingales that characterize it. We show that the same is true for a Markov renewal processes. We discuss in depth the calibration of the G-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds.

Effect of risk preference on the valuation of time charter contracts with extension options

ABSTRACTThe literature on valuation of time charter contracts and real options in shipping generally relies on the complete markets hypothesis and the risk-neutrality of agents. However, these assumptions fail completely in some shipping market segments. This study proposes a numerical approach—based on discounting the certainty equivalent payoff at the risk-free rate—which incorporates the agent’s risk preferences through an exponential utility function. The method comprises an iterative Monte Carlo nested simulation with the real probability measure. This method is applied to a case of Suezmax tankers. The stochastic evolution of the time charter rates is modelled as a geometric mean-reverting process. The case study supports the applicability of the proposed method and evidences that the effect of risk preference may be significant, mainly for more risk-averse agents. Although the method involves intensive computation, it has the benefits of theoretical ease and flexibility, which could encourage utilisation by practitioners.

Δ-Cumulants in terms of moments

The [Formula: see text]-convolution of real probability measures, introduced by Bożejko, generalizes both free and Boolean convolutions. It is linearized by the [Formula: see text]-cumulants, and Yoshida gave a combinatorial formula for moments in terms of [Formula: see text]-cumulants, that implicitly defines the latter. It relies on the definition of an appropriate weight on noncrossing partitions. We give here two different expressions for the [Formula: see text]-cumulants: the first one is a simple variant of Lagrange inversion formula, and the second one is a combinatorial inversion of Yoshida’s formula involving Schröder trees.

Martingales associated to peacocks using the curtain coupling

We consider right-continuous peacocks, that is, families of real probability measures $(\mu _t)_{t\in [0,1]}$ that are increasing in convex order. Given a sequence of time partitions we associate the sequence of martingales characterised by the fact that they are Markovian, constant on the partition intervals $[t_k,t_{k+1}[$, and such that the transition kernels at times $t_{k+1}$ are the curtain couplings of marginals $\mu _{t_k}$ and $\mu _{t_{k+1}}$. We study the limit curtain processes obtained when the mesh of the partition tends to zero and study existence, uniqueness and relevancy with respect to the original data. For any right-continuous peacock we show there exist sequences of partitions such that a limit process exists (for the finite-dimensional convergence). Under certain additional regularity assumptions, we prove that there is a unique limit curtain process and that it is a Markovian martingale. We first study by elementary methods peacocks whose marginals correspond to uniform distributions in convex order. In this case, the results and techniques complete the results and techniques used in a parallel work by Henry-Labordere, Tan and Touzi [9]. We obtain the same type of results for all limit curtain processes associated to a class of analytic discrete peacocks, i.e., the measures $\mu _t$ are finitely supported and vary analytically in $t$. Finally, we give examples of peacocks and sequences of partitions such that the limit curtain process is a non-Markovian martingale.

Stability of the shadow projection and the left-curtain coupling

Le couplage rideau (gauche), introduit par Beiglböck et Juillet est un point extrémal de l’ensemble des couplages « martingale » entre deux mesures de probabilité réelles prises dans l’ordre convexe. Ce couplage possède des propriétés remarquables quant aux relations d’ordre entre mesures d’une part, et par rapport à un problème de minimisation issu de la théorie du transport optimal d’autre part. Une représentation explicite a été récemment mise en évidence par Henry-Labordère et Touzi. Nous démontrons dans cet article que le couplage rideau dépend continument des mesures marginales prescrites et nous quantifions cette dépendance à l’aide d’inégalités lipschitziennes.

Semi-Markov migration process in a stochastic market in credit risk

In the present the idea of stochastic Market environment comes into play to express the changes in general economy, which affects any industry in small or great amounts of turbulence. We model the evolution of the Market among its possible ν-states as an F-inhomogeneous semi-Markov process. This idea leads us to modeling the migration process of defaultable bonds as ν different F-inhomogeneous semi-Markov process. The survival probabilities of a defaultable bond in every credit grade are found. The asymptotic behaviour of the survival probabilities is established under certain conditions. Also, it is proved under what conditions the convergence is geometrically fast. The stochastic foundation of the general stochastic discrete-time Market is provided, by proving that the market is viable, if and only if, there exists an equivalent martingale measure, from which we construct the forward probability measure and under which the discounted default free bond price process for all possible states of the Market is a martingale. The term structure of credit spread and the change of real-world probability measure to forward probability measure are studied. In the form of a theorem it is proved that under certain conditions, changing the real probability measure to a forward probability measure, does not affect the inhomogeneous semi Markov process modeling the migration of defaultable bonds. That is, it is proved that it only changes the basic parameters and we provide a relation among the transition probabilities under the two measures. Finally, parameter estimation and calibration of the inhomogeneous semi-Markov chain in stochastic environment is being provided.

Parameter Estimation in Credit Models Under Incomplete Information

We consider the filtering model of Frey and Schmidt (2012) stated under the real probability measure and develop a method for estimating the parameters in this framework by using time-series data of CDS index spreads and classical maximum-likelihood algorithms. The estimation-approach incorporates the Kushner-Stratonovich SDE for the dynamics of the filtering probabilities. The convenient formula for the survival probability is a prerequisite for our estimation algorithm. We apply the developed maximum-likelihood algorithms on market data for historical CDS index spreads (iTraxx Europe Main Series) in order to estimate the parameters in the nonlinear filtering model for an exchangeable credit portfolio. Several such estimations are performed as well as accompanying statistical and numerical computations.

Parameter Estimation in Credit Models Under Incomplete Information

We consider the filtering model of Frey & Schmidt (2012) stated under the real probability measure and develop a method for estimating the parameters in this framework by using time-series data of CDS index spreads and classical maximum-likelihood algorithms. The estimation-approach incorporates the Kushner-Stratonovich SDE for the dynamics of the filtering probabilities. The convenient formula for the survival probability is a prerequisite for our estimation algorithm. We apply the developed maximum-likelihood algorithms on market data for historical CDS index spreads (iTraxx Europe Main Series) in order to estimate the parameters in the nonlinear filtering model for an exchangeable credit portfolio. Several such estimations are performed as well as accompanying statistical and numerical computations.

Fuzzy semi-Markov migration process in credit risk

We explore the rating system used by credit agencies with a focus on problems that justify the use of fuzzy set theory. We prove that a fuzzy market is viable if and only if an equivalent martingale measure exists, from which we construct the forward probability measure and under which the discounted price of a default-free bond is a martingale. We model the evolution of credit migration of a defaultable bond as an inhomogeneous semi-Markov process with fuzzy states. We study the effects of changing the real probability measure to a forward probability measure. In addition, we investigate the asymptotic behaviour of the survival probability in each fuzzy state given in the absence of default. Finally, we discuss parameter estimation and calibration of the inhomogeneous Markov chain with fuzzy states.

Free Bessel Laws

Abstract We introduce and study a remarkable family of real probability measures πst that we call free Bessel laws. These are related to the free Poisson law π via the formulae πs1 = π⊠s and π1t=π⊠t. Our study includes definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.

Default Risk and its Greeks under an Objective Probability Measure

This paper investigates default probabilities and their comparative statics (default Greeks) in the Black and Scholes, Merton framework, using the objective or real probability measure. First we show how a risk neutral default probability can be transformed into an objective probability. The difference between the two is analysed numerically. Risk neutral probabilities appear to be accurate only when they are not useful, and to be too inaccurate for normal credit risk management. Several sensitivity measures for default probability are derived. Default probability is shown to decrease with the value and the drift of assets and to increase with asset volatility, while debt maturity has an ambigous effect. The approach in this paper facilitates the use of the BSM framework for risk management purposes and provides a theoretical basis for empirical analyses of default probability.

Relativistic Feynman-type integrals

It is shown that within the framework of the Kershaw stochastic model generalized by the author to the relativistic case a Feynman-type process may be constructed which can formally be understood as a diffusion phenomenon in Euclidean space. This makes it possible to introduce a real probability measure in the scheme of quantum mechanics proposed by Feynman.