Nonlinear Martingale Research Articles

Option returns

Nonlinear martingale theory is used to form lower and upper price processes straddling a martingale. The upper and lower expected returns under the martingale measure are the seen to reflect risk charges associated with the nonlinear martingales. The move to physically expected returns is made via the usual covariation construction. There are however two covariation terms, one for the lower and the other for the upper return. The expected returns are then seen to incorporate in addition to covariation considerations that address the martingale, the risk charges that assess out of martingale issues. The theory is tested on prices of options for $ 10 $ underliers over $ 170 $ days covering $ 4 $ maturities and $ 21 $ strikes for a sample of around $ 140,000 $ assets. It is observed that the risk charges have positive significant coefficients while the upper covariation has a positive price with the lower covariation receives a negative price. The covariation prices are consistent with the view that the upper prices are entry prices related to the assumption of risk while the lower price is an exit price reflecting the shedding of risk.

A New Dawn

Dilip Madan argues that one has to discard the old clothes borrowed from statistics and economics and take up an understanding of risk anew. One has to move away from the theory of expectations and martingales towards the fast-developing theory of nonlinear expectations and nonlinear martingales.

Lower and upper pricing of financial assets

<p style='text-indent:20px;'>Modeling of uncertainty by probability errs by ignoring the uncertainty in probability. When financial valuation recognizes the uncertainty of probability, the best the market may offer is a two price framework of a lower and upper valuation. The martingale theory of asset prices is then replaced by the theory of nonlinear martingales. When dealing with pure jump compensators describing probability, the uncertainty in probability is captured by introducing parametric measure distortions. The two price framework then alters asset pricing theory by requiring two required return equations, one each for the lower upper valuation. Proxying lower and upper valuations by daily lows and highs, the paper delivers the first empirical study of nonlinear martingales via the modeling and simultaneous estimation of the two required return equations.</p>

Lower and Upper Pricing of Financial Assets

Modeling of uncertainty by probability errs by ignoring the uncertainty in probability. When financial valuation recognizes the uncertainty of probability the best the market may offer is a two price framework of a lower and upper valuation. The martingale theory of asset prices is then replaced by the theory of nonlinear martingales. When dealing with pure jump compensators describing probability the uncertainty in probability is captured by introducing parametric measure distortions. The two price framework then alters asset pricing theory by requiring two required return equations one each for the lower upper valuation. Proxying lower and upper valuations by daily lows and highs the paper delivers the first empirical study of nonlinear martingales via the modeling and simultaneous estimation of the two required return equations.

Theory, methods and meaning of nonlinear expectation theory

This is a survey on the research developments of nonlinear expectation theory. We first recall the basic definition of a space of nonlinear expectation, and then, through the representation theorem and some examples of nonlinear i.i.d. (independent and identically distributed), to explain why this new framework can be applied to calculate and quantitatively analyze probabilistic and distributional uncertainty hidden behind a real world (possibly high-dimensional) data sequence. Then we introduce two fundamentally important nonlinear normal distribution and maxima distribution and the corresponding nonlinear law of large numbers and nonlinear central limit theorem, which are crucial and fundamental breakthroughs of this new research domain. A typical application is a basic algorithm named ``$\\varphi$-max-mean. We also present a basic continuous-time stochastic process---nonlinear Brownian motion and its stochastic calculus, including stochastic integral, stochastic differential equations, and the corresponding nonlinear martingale theory. This new theoretical framework has generalized the axiomatical probability theory founded by Kolmogorov (1933). The key difference is the notion of nonlinear expectation $\\hat{\\mathbb{E}}$, whose special linear case corresponds a probability space $(\\Omega,{\\mathcal{F}},P)$. It is the nonlinearity that allows us to quantitatively measure the uncertainty of probabilities and probabilistic distributions inhabited in our real world.

Benchmarking in two price financial markets

No arbitrage for two price economies with no locally risk free asset implies that suitably benchmarked prices are nonlinear martingales. However, both the benchmarking asset and the measure change depend on the process being benchmarked. Further assumptions allow the nonlinear martingales in discrete time to become expectations with respect to a nonadditivity probability. Such nonlinear expectations are imminently reasonable given the lack of experience with tail events on both sides of the gain loss spectrum. Continuous time extensions employ measure distortions. The general valuation of economic activities and the leveraging of stability in benchmarked price processes is then addressed. Traditional asset pricing questions and investigations are then reopened for benchmarked prices. In particular, the analytics for benchmarked option pricing and the asset pricing theory for benchmarked prices in a limiting stationary state are developed.

Relativities in Financial Markets

No arbitrage for two price economies with no locally risk free asset implies that suitably deflated prices are nonlinear martingales. However, both the deflating process and the measure change depend on the process being deflated. Further assumptions allow the nonlinear martingales in discrete time to become expectations with respect to a nonadditive probability. Such nonlinear expectations that distort probabilities are imminently reasonable given the lack of experience with tail events on both sides of the gain loss spectrum. Continuous time extensions replace probability distortions with measure distortions that distort the arrival rates of jumps. The general valuation of economic activities and the leveraging of stability in deflated price processes is then addressed. Applications include the pricing of options on relativities and the asset pricing theory for relativities in a limiting stationary state.

Asset pricing theory for two price economies

We show that nonlinearly discounted nonlinear martingales are related to no arbitrage in two price economies as linearly discounted martingales were related to no arbitrage in economies satisfying the law of one price. Furthermore, assuming risk acceptability requires a positive physical expectation, we demonstrate that expected rates of return on ask prices should be dominated by expected rates of return on bid prices. A preliminary investigation conducted here, supports this hypothesis. In general we observe that asset pricing theory in two price economies leads to asset pricing inequalities. A model incorporating both nonlinear discounting and nonlinear martingales is developed for the valuation of contingent claims in two price economies. Examples illustrate the interactions present between the severity of measure changes and their associated discount rates. As a consequence arbitrage free two price economies can involve unique discount curves and measure changes that are however specific to both the product being priced and the trade direction. Furthermore the developed valuation operators call into question the current practice of Debt Valuation Adjustments.

Asset Pricing Theory for Two Price Economies

We show that nonlinearly discounted nonlinear martingales are related to no arbitrage in two price economies as linearly discounted martingales were related to no arbitrage in economies satisfying the law of one price. Furthermore, assuming risk acceptability requires a positive physical expectation, we demonstrate that expected rates of return on ask prices should be dominated by expected rates of return on bid prices. A preliminary investigation conducted here, supports this hypothesis. In general we observe that asset pricing theory in two price economies leads to asset pricing inequalities. A model incorporating both nonlinear discounting and nonlinear martingales is developed for the valuation of contingent claims in two price economies. Examples illustrate the interactions present between the severity of measure changes and their associated discount rates. As a consequence arbitrage free two price economies can involve unique discount curves and measure changes that are however specific to both the product being priced and the trade direction.

Superhedging and Dynamic Risk Measures under Volatility Uncertainty

We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a càdlàg nonlinear martingale which is also the value process of a superhedging problem. The superhedging strategy is obtained from a representation similar to the optional decomposition. Furthermore, we prove an optional sampling theorem for the nonlinear martingale and characterize it as the solution of a second order backward SDE. The uniqueness of dynamic extensions of static sublinear expectations is also studied.

Superhedging and Dynamic Risk Measures Under Volatility Uncertainty

We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a cadlag nonlinear martingale which is also the value process of a superhedging problem. The superhedging strategy is obtained from a representation similar to the optional decomposition. Furthermore, we prove an optional sampling theorem for the nonlinear martingale and characterize it as the solution of a second order backward SDE. The uniqueness of dynamic extensions of static sublinear expectations is also studied.

A mini-tutorial on measure-valued Markov processes and nonlinear martingale problems—As a reply to McCauley's comment

One goal of this mini-tutorial is to provide an introduction into the theory of measure-valued Markov processes and nonlinear martingales defined by strongly nonlinear Fokker–Planck equations and to discuss the physical relevance of the associated processes. Another goal is to reply to McCauley's comment on T.D. Frank [Physica A 331, 391 (2004)]. The tutorial addresses in detail two approaches found in physics and mathematics. The first approach exploits a mapping between linear and nonlinear Fokker–Planck equations. The second approach exploits martingale theory. Several examples of Markov processes and martingales in quantum mechanical, nonextensive, and self-organizing systems defined by nonlinear Fokker–Planck equations are discussed.

Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature

We develop a nonlinear martingale theory for time discrete processes $(Y_n)_{n\in \NN_0}$. These processes are defined on any probability space $(\O,\F,\F_n,\P)_{n\in\NN_0}$ and have values in a metric space (N,d) of nonpositive curvature (in the sense of A. D. Alexandrov). The defining martingale property for such processes is \[ \E(Y_{n+1}|\F_n)=Y_n, \qquad \P\mbox{-a.s.,} \] where the conditional expectation on the left-hand side is defined as the minimizer of the functional \[ Z\mapsto\E d^2(Z,Y_{n+1}) \] within the space of $\F_n$-measurable maps $Z\dvtx \O\to N$. We give equivalent characterization of N-valued martingales (using merely the usual linear conditional expectations) and derive fundamental properties of these martingales, for example, a martingale convergence theorem. Finally, we exploit the relation with harmonic maps. It turns out that a map $f\dvtx M\to N$ is harmonic w.r.t. a given Markov kernel p on M if and only if it maps Markov chains $(X_n)_{n\in\NN}$ (with transition kernel p) on M onto martingales $(f(X_n))_{n\in\NN}$ with values in $N$. The nonlinear heat flow $f\dvtx \N_0\times M\to N$ of a given initial map $f(0,\cdot)\dvtx M\to N$ at time n is obtained as the filtered expectation, \[ f(n,x) := \E_x [ f(X_n) |\!|\!| (\F_k)_{k\ge 0}] \] of the random map $f(X_n)$. Similarly, the unique solution to the Dirichlet problem for a given map $g\dvtx M\to N$ and a subset $D\subset M$ is obtained as \[ f(x) := \E_x [ g(X_{\tau(D)})|\!|\!| (\F_k)_{k\ge 0}]. \] In both cases, a crucial role is played by the notion of expectation $\E_x [\cdot |\!|\!| (\F_k)_{k\ge 0}]$ which will be discussed in detail. Moreover, we prove Jensen's inequality for expectations and expectations and we prove (weak and strong) laws of large numbers for sequences of i.i.d. random variables with values in N. Our theory is an extension of the classical linear martingale theory and of the nonlinear theory of martingales with values in manifolds as developed, for example, in Emery and Kendall. The goal is to extend the previous framework towards processes with values in metric spaces. This will lead to a stochastic approach to the theory of (generalized) harmonic maps with values in such singular spaces as developed by Jost and Korevaar and Schoen.

Filtration-consistent nonlinear expectations and related g-expectations

From a general definition of nonlinear expectations, viewed as operators preserving monotonicity and constants, we derive, under rather general assumptions, the notions of conditional nonlinear expectation and nonlinear martingale. We prove that any such nonlinear martingale can be represented as the solution of a backward stochastic equation, and in particular admits continuous paths. In other words, it is a g-martingale.

DISCRETE-TIME MARTINGALES WITH SPATIAL PARAMETERS

Our analysis of a certain stochastic difference equation driven by a martingale k↦M(x,k) that depends on a spatial parameter x∈R d requires some regularity properties of the underlying martingale be satisfied. Because of their independent interest, we present these regularity properties in this article. We study first the continuity and Lipschitz continuity properties under corresponding conditions on the quadratic covariation of the martingale. We follow this with differentiability and integrability properties. Our analysis of the stochastic difference equation requires a discrete-time version of Itô's formula. The discrete-time Itô formula we have derived involves a martingale transform term. The purpose of the final section is to introduce linear and nonlinear martingale transforms and analyze their properties.

A DECOMPOSITION THEOREM OF g-MARTINGALES *

In this paper, we first introduce the notions of nonlinear mathematical expectations and nonlinear martingales via backward stochastic differential equations(BSDEs) introduced by Duffie & Epstein and Skiadas. And then, we prove a general nonlinear decomposition theorem. Our decomposition theorem generalizes Doob-Meyer decomposition theorem.