Martingale Measure Research Articles

VALUATION OF GENERAL CONTINGENT CLAIMS WITH SHORT SELLING BANS: AN EQUAL-RISK PRICING APPROACH

This paper studies the valuation of general contingent claims with short selling bans under the equal-risk pricing (ERP) framework proposed in I. Guo & S.-P. Zhu (2017) [Journal of Economic Dynamics and Control 76, 136–151]. In existing literature, analytical pricing formulae were derived in the special case, where the payoff function is monotonic under risk-neutral measures. In this paper, we establish a unified framework for this new pricing approach so that its range of application can be significantly expanded. The results of I. Guo & S.-P. Zhu (2017) [Journal of Economic Dynamics and Control 76, 136–151] are extended to the case of non-monotonic payoffs (such as a butterfly spread option) under risk-neutral measures. We also provide numerical schemes for computing equal-risk prices under other measures such as the original physical measure. Furthermore, we demonstrate how short selling bans can affect the valuation of contingent claims by comparing equal-risk prices with Black–Scholes prices.

Equilibrium in Complete Market with Jump-Diffusion Processes

This paper establishes the existence of a solution to the optimization problem. Supposing that risk assets pay continuous dividend regarded as the function of time. It is established that the behaviour model of the stock pricing process is jump-diffusion driven by a count process. We give a characterization of the optimal portfolio by means of the value function and the equivalent martingale measure defined by the utility function. The unique equivalent martingale measure, the unique optimal consumption and portfolio pair and the corresponding wealth process are deduced. We provide a simple characterization of an equilibrium market and discuss existence and uniqueness of equilibrium in the economy.

White-noise driven conditional McKean–Vlasov limits for systems of particles with simultaneous and random jumps

We study the convergence of N-particle systems described by SDEs driven by Brownian motion and Poisson random measure, where the coefficients depend on the empirical measure of the system. Every particle jumps with a jump rate depending on its position and on the empirical measure of the system. Jumps are simultaneous, that is, at each jump time, all particles of the system are affected by this jump and receive a random jump height that is centred and scaled in \(N^{-1/2}\). This particular scaling implies that the limit of the empirical measures of the system is random, describing the conditional distribution of one particle in the limit system. We call such limits conditional McKean–Vlasov limits. The conditioning in the limit measure reflects the dependencies between coexisting particles in the limit system such that we are dealing with a conditional propagation of chaos property. As a consequence of the scaling in \(N^{-1/2}\) and of the fact that the limit of the empirical measures is not deterministic the limit system turns out to be solution of a non-linear SDE, where not independent martingale measures and white noises appear having an intensity that depends on the conditional law of the process.

Fundamental theorem of asset pricing: a strengthened version and а martingale measures search area

In the article a strengthened version of the ’Fundamental Theorem of Asset Pricing’ for a one-period market model is proven. The principal role in this result is played by total and nonanihilating cones.

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.

A reusable discounting framework under jump-diffusion process

Due to the restriction of duality structure, existing discounting methods in Lévy models have not met the requirements from psychologists and economists. To address this weakness, our paper invents a stochastic heterogeneous quasi-hyperbolic (SHQH) clock. The SHQH clock has the following strengths: (i) being invisible in the duality structure; (ii) capturing multi-period, multi-goods and multi-agent preferences; (iii) nesting many discount functions as its special cases. Using a self-driven approach, this paper applies the SHQH clock to an insurance policy-holder who is both a farsighted planner and a myopic doer. The time-consistent insurance strategy is explicitly solved via equivalent martingale measure.

Valuing the Future and Discounting in Random Environments: A Review.

We address the process of discounting in random environments, which allows valuation of the future in economic terms. We review several approaches to the problem regarding different well-established stochastic market dynamics in the continuous-time context and include the Feynman–Kac approach. We also review the relation between bond-pricing theory and discounting and introduce both the market price of risk and the risk neutral measure from an intuitive point of view devoid of excessive formalism. We provide the discount for each economic model and discuss their key results. We finally present a summary of our previous empirical studies for several countries on the long-run discount problem.

Risk-hedging a European option with a convex risk measure and without no-arbitrage condition

In this article, we revisit the discrete-time problem of pricing a contingent claim with respect to a dynamic risk measure defined by its acceptance sets. Without any no-arbitrage condition, we show that it is possible to characterize the prices of a European claim. Our analysis reveals a natural weak no-arbitrage condition that we study. This is a condition formulated in terms of the (risk) hedging prices instead of the attainable claims. Our approach is not based on a robust representation of the risk measure and we do not suppose the existence of a risk-neutral probability measure.

The risk-neutral non-additive probability with market frictions

The fundamental theory of asset pricing has been developed under the two main assumptions that markets are frictionless and have no arbitrage opportunities. In this case the market enforces that replicable assets are valued by a linear function of their payoffs, or as the discounted expectation with respect to the so-called risk-neutral probability. Important evidence of the presence of frictions in financial markets has led to study market pricing rules in such a framework. Recently, Cerreia-Vioglio et al. (J Econ Theory 157:730–762, 2015) have extended the Fundamental Theorem of Finance by showing that, with markets frictions, requiring the put–call parity to hold, together with the mild assumption of translation invariance, is equivalent to the market pricing rule being represented as a discounted Choquet expectation with respect to a risk-neutral nonadditive probability. This paper continues this study by characterizing important properties of the (unique) risk-neutral nonadditive probability v_f associated with a Choquet pricing rule f, when it is not assumed to be subadditive. First, we show that the observed violation of the call–put parity, a condition considered by Chateauneuf et al. (Math Financ 6:323–330, 1996) similar to the put–call parity in Cerreia-Vioglio et al. (2015), is consistent with the existence of bid-ask spreads. Second, the balancedness of v_f—or equivalently the non-vacuity of its core—is characterized by an arbitrage-free condition that eliminates all the arbitrage opportunities that can be obtained by splitting payoffs in parts; moreover the (nonempty) core of v_f consists of additive probabilities below v_f whose associated (standard) expectations are all below the Choquet pricing rule f. Third, by strengthening again the previous arbitrage-free condition, we show the existence of a strictly positive risk-neutral probability below v_f, which allows to recover the standard formulation of the Fundamental Theorem of Finance for frictionless markets.

Dynamic importance allocated nested simulation for variable annuity risk measurement

AbstractEstimating tail risk measures for portfolios of complex variable annuities is an important enterprise risk management task which usually requires nested simulation. In the nested simulation, the outer simulation stage involves projecting scenarios of key risk factors under the real-world measure, while the inner simulations are used to value pay-offs under guarantees of varying complexity, under a risk-neutral measure. In this paper, we propose and analyse an efficient simulation approach that dynamically allocates the inner simulations to the specific outer scenarios that are most likely to generate larger losses. These scenarios are identified using a proxy calculation that is used only to rank the outer scenarios, not to estimate the tail risk measure directly. As the proxy ranking will not generally provide a perfect match to the true ranking of outer scenarios, we calculate a measure based on the concomitant of order statistics to test whether further tail scenarios are required to ensure, with given confidence, that the true tail scenarios are captured. This procedure, which we call the dynamic importance allocated nested simulation approach, automatically adjusts for the relationship between the proxy calculations and the true valuations and also signals when the proxy is not sufficiently accurate.

Estimating actual probability of default from structural models

In this paper, we study the probability of default, the credit default swap (CDS) implied probability of default, and the estimated actual probability of default from the structural models. We first show that the conditional risk-neutral probability of default is totally different from the CDS implied probability of default, and the CDS implied probability of default provides biased estimation of a corporation’s actual probability of default. Hence, the common belief that the CDS implied probability of default gives unbiased (or reliable) estimations of the firm’s probability of default is false. Furthermore, we use the credit spread risk premium as the difference of the physical measure and the risk-neutral measure of the credit spread to estimate the actual bond spread, and relate the credit spread risk premium in terms of the market price of risk and the credit spread volatility. Using the credit spread volatility and the risk-neutral credit spread and the market price of risk in hybrid HPD model, we obtain the estimated actual probability of default in a structural model. The estimated actual probability of default would be useful for credit rating agencies, investors, credit risk managers and regulators.

Submodular financial markets with frictions

This paper studies arbitrage-free financial markets with bid-ask spreads whose super-hedging prices are submodular. The submodular assumption on the super-hedging price, or the supermodularity usually assumed on utility functions, is the formal expression of perfect complementarity, which dates back to Fisher, Pareto, and Edgeworth, according to Samuelson (J Econ Lit 12:1255–1289, 1974). Our main contribution provides several characterizations of financial markets with frictions that are submodular as a consequence of a more general study of submodular pricing rules. First, a market is submodular if and only if its super-hedging price is a Choquet integral and if and only if its set of risk-neutral probabilities is representable as the core of a submodular non-additive probability that is uniquely defined, called risk-neutral capacity. Second, a market is representable by its risk neutral capacity if and only if it is equivalent to a market, only composed of bid-ask event securities.

Valuation of Annuity Guarantees Under a Self-Exciting Switching Jump Model

This article investigates the valuation of annuity guarantees under a regime-switching model when the dynamics of the underlying stock price follow a self-exciting switching jump-diffusion process. In this framework, we add a jump component to a regime-switching geometric Brownian for large shocks on the stock price. The intensity of shock arrivals is a Hawkes process modulated by a continuous time hidden Markov chain with a finite number of states. The interest rate used for discounting is stochastic and correlated to the stock market. In an incomplete market, we define an equivalent martingale measure to price a variable annuity contract that guarantees a minimum living or death benefit. Under this equivalent martingale measure, we propose closed-form approximation formulas using the inverse Fourier transform technique. A numerical implementation highlights the impact of self-exciting jumps and economic regimes on the valuation of guarantees.

Real options, risk aversion and markets: A corporate finance perspective

We analyze how the presence of financial markets affects the optimal exercise of real options for a risk averse agent. Extending the results of Shackleton and Sodal (2005), we characterize the optimal exercise rule in terms of a benchmark portfolio, even for the case of an incomplete market, facilitating the minimal martingale measure. We unambiguously characterize the effect of idiosyncratic risk on the speed of exercise of the option. We further show that systematic risk can accelerate execution and reduce the value of a call-type option, in contrast with the standard view that both value and execution threshold are increasing in volatility.

Stochastic model for determination of arbitrariness and completeness of information system

When evaluating information system reliability indicators, the necessary initial data for a priori probability calculations are often missing, and statistical evaluation is complicated by a large amount of tests, which can only be used to determine the estimates of the moments of the determining random variables of the process of functioning of the information system or its constituent parts (mathematical expectation and variance of failure time processing, recovery time, backup time, etc.). However, in this situation, it is necessary to justify some characteristics of the information system, for example, time reserve, guaranteed exact limits of the probability of fault-free operation of the system and availability coefficient. In this case, «justification» means the construction of precise upper and lower limits of the change of functionals, which characterize the efficiency of the functioning of the information system, in the conditions of incomplete a priori information about the distribution functions of the determining random variables. In this paper, an N-periodic set of information states with one risky and one risk-free process is considered, the arbitrariness and completeness of this set is investigated, and the entropy for the Radon-Nikodym derivative of the corresponding equivalent martingale measures is found. Conditions for the signal loss distribution function were also found, under which the set of signal states is arbitragefree and complete. It is shown that the use of the proposed signal change function made it possible to effectively investigate a set of information states for impartiality and completeness. The application of the obtained results to the determination of the fair price for the аsian arithmetic option with the initially specified payment function is considered.

The weak functional representation of historical martingales

A weak extension of the Dupire derivative is derived, which turns out to be the adjoint operator of the integral with respect to the martingale measure associated with the historical Brownian motion a benchmark example of a measure valued process. This extension yields the explicit form of the martingale representation of historical functionals, which we compare to a classical result on the representation of historical functionals derived in [7].

Asset price bubbles: Invariance theorems

<p style='text-indent:20px;'>This paper provides invariance theorems that facilitate testing for the existence of an asset price bubble in a market where the price evolves as a Markov diffusion process. The test involves only the properties of the price process' quadratic variation under the statistical probability. It does not require an estimate of either the equivalent local martingale measure or the asset's drift. To augment its use, a new family of stochastic volatility price processes is also provided where the processes' strict local martingale behavior can be characterized.</p>

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.

A mean-field stochastic linear-quadratic optimal control problem with jumps under partial information

In this article, the stochastic linear-quadratic optimal control problem of mean-field type with jumps under partial information is discussed. The state equation which contains affine terms is a SDE with jumps driven by a multidimensional Brownian motion and a Poisson stochastic martingale measure, and the quadratic cost function contains cross terms. In addition, the state and the control as well as their expectations are contained both in the state equation and the cost functional. This is the so-called optimal control problem of mean-field type. Firstly, the existence and uniqueness of the optimal control is proved. Secondly, the adjoint processes of the state equation is introduced, and by using the duality technique, the optimal control is characterized by the stochastic Hamiltonian system. Thirdly, by applying a decoupling technology, we deduce two integro-differential Riccati equations and get the feedback representation of the optimal control under partial information. Fourthly, the existence and uniqueness of the solutions of two Riccati equations are proved. Finally, we discuss a special case, and establish the corresponding feedback representation of the optimal control by means of filtering technique.

Lévy Interest Rate Models with a Long Memory

This article proposes an interest rate model ruled by mean reverting Lévy processes with a sub-exponential memory of their sample path. This feature is achieved by considering an Ornstein–Uhlenbeck process in which the exponential decaying kernel is replaced by a Mittag–Leffler function. Based on a representation in term of an infinite dimensional Markov processes, we present the main characteristics of bonds and short-term rates in this setting. Their dynamics under risk neutral and forward measures are studied. Finally, bond options are valued with a discretization scheme and a discrete Fourier’s transform.