Market Viability and Martingale Measures under Partial Information

We consider a financial market model with a single risky asset whose price process evolves according to a general jump-diffusion with locally bounded coefficients and where market participants have only access to a partial information flow. For any utility function, we prove that the partial information financial market is locally viable, in the sense that the optimal portfolio problem has a solution up to a stopping time, if and only if the (normalised) marginal utility of the terminal wealth generates a partial information equivalent martingale measure (PIEMM). This equivalence result is proved in a constructive way by relying on maximum principles for stochastic control problems under partial information. We then characterize a global notion of market viability in terms of partial information local martingale deflators (PILMDs). We illustrate our results by means of a simple example.     

Asymptotic asset pricing and bubbles

We define the concept of asymptotic superreplication, and prove a duality principle of asset pricing for sequences of financial markets (e.g., weakly converging financial markets and large financial markets) based on contiguous sequences of equivalent local martingale measures. This provides a pricing mechanism to calculate the fundamental value of a financial asset in the asymptotic market. We introduce the notion of asymptotic bubbles by showing that this fundamental value can be strictly lower than the current price of the asset. In the case of weakly converging markets, we show that this fundamental value is equal to an expectation of the terminal value of the asset in the weak-limit market. From a practical perspective, we relate the asymptotic superreplication price to a limit of quantile-hedging prices. This shows that even when a price process is a true martingale, it can have properties similar to a bubble, up to a set of small probability. For practical applications, we give examples of weakly converging discrete-time models (e.g. some GARCH models) and large financial models that present bubbles.     

Optimal robust mean-variance hedging in incomplete financial markets

An optimal B-robust estimate is constructed for the multidimensional parameter in the drift coefficient of a diffusion-type process with a small noise. The optimal mean-variance robust (optimal V-robust) trading strategy is to hedge (in the mean-variance sense) the contingent claim in an incomplete financial market with an arbitrary information structure and a misspecified volatility of the asset price, which is modelled by a multidimensional continuous semimartingale. The obtained results are applied to the stochastic volatility model, where the model of the latent volatility process contains the unknown multidimensional parameter in the drift coefficient and a small parameter in the diffusion term. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 45, Martingale Theory and Its Application, 2007.     

Parameter estimation of an agent‐based stock price model

The influence of the behavior and strategies of traders on stock price formation has attracted much interest. It is assumed that there is a positive correlation between the total net demand and the price change. A buy order is expected to increase the price, whereas a sell order is assumed to decrease it. We perform data analysis based on a recently proposed stochastic model for stock prices. The model involves long‐range dependence, self‐similarity, and no arbitrage principle, as observed in real data. The arrival times of orders, their quantity, and their duration are created by a Poisson random measure. The aggregation of the effect of all orders based on these parameters yields the log‐price process. By scaling the parameters, a fractional Brownian motion or a stable Levy process can be obtained in the limit. In this paper, our aim is twofold; first, to devise statistical methodology to estimate the model parameters with an application on high‐frequency price data, and second, to validate the model by simulations with the estimated parameters. We find that the statistical properties of agent level behavior are reflected on the stock price, and can affect the entire process. Moreover, the price model is suitable for prediction through simulations when the parameters are estimated from real data. The methods developed in the present paper can be applied to frequently traded stocks in general. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust optimal investment and reinsurance problem for a general insurance company under Heston model

In this paper, we study a robust optimal investment and reinsurance problem for a general insurance company which contains an insurer and a reinsurer. Assume that the claim process described by a Brownian motion with drift, the insurer can purchase proportional reinsurance from the reinsurer. Both the insurer and the reinsurer can invest in a financial market consisting of one risk-free asset and one risky asset whose price process is described by the Heston model. Besides, the general insurance company’s manager will search for a robust optimal investment and reinsurance strategy, since the general insurance company faces model uncertainty and its manager is ambiguity-averse in our assumption. The optimal decision is to maximize the minimal expected exponential utility of the weighted sum of the insurer’s and the reinsurer’s surplus processes. By using techniques of stochastic control theory, we give sufficient conditions under which the closed-form expressions for the robust optimal investment and reinsurance strategies and the corresponding value function are obtained.     

Random times at which insiders can have free lunches

We consider models of time continuous financial markets with a regular trader and an insider who are able to invest into one risky asset. The insider's additional knowledge consists in his ability to stop at a random time which is inaccessible to the regular trader, such as the last passage of a certain level before maturity by some stock price process, or the time at which the stock price reaches its maximum during the trading interval. We show that under very mild assumptions on the coefficients of the diffusion process describing these price processes the information drift caused by the additional knowledge of the insider cannot be eliminated by an equivalent change of probability measure. As a consequence, all our models allow the insider to have free lunches with vanishing risk, or even to exercise arbitrage.     

Multivariate asset‐pricing model based on subordinated stable processes

In this paper, we consider a multidimensional time‐changed stochastic process in the context of asset‐pricing modeling. The proposed model is constructed from stable processes, and its construction is based on two popular concepts: multivariate subordination and Lévy copulas. From a theoretical point of view, our main result is Theorem 1, which yields a simulation method from the considered class of processes. Our empirical study shows that the model represents the correlation between asset returns quite well. Moreover, we provide some evidence that this model is more appropriate for describing stock prices than classical time‐changed Brownian motion, at least if the cumulative amount of transactions is used for a stochastic time change.     

带负债的连续时间最优资产组合选择

构造了一个带外生负债的连续时间均值-方差最优投资组合选择模型.假定风险资产价格的演变服从几何布朗运动,累积负债服从带漂移的布朗运动,并且市场系数恒为常数,借助随机LQ控制方法得到相应的均值-方差优化问题的最优策略和有效边界.     

One‐Dimensional Birth‐Death Process and Delbrück‐Gillespie Theory of Mesoscopic Nonlinear Chemical Reactions

As a mathematical theory for the stochastic, nonlinear dynamics of individuals within a population, Delbrück‐Gillespie process (DGP) is a birth–death system with state‐dependent rates which contain the system size V as a natural parameter. For large V , it is intimately related to an autonomous, nonlinear ODE as well as a diffusion process. For nonlinear dynamical systems with multiple attractors, the quasi‐stationary and stationary behavior of such a birth–death process can be understood in terms of a separation of time scales by a T*～e^αV (α > 0) : a relatively fast, intra‐basin diffusion for t?T* and a much slower inter‐basin Markov jump process for t?T* . In this paper for one‐dimensional systems, we study both stationary behavior (t=∞ ) in terms of invariant distribution , and finite time dynamics in terms of the mean first passsage time (MFPT) . We obtain an asymptotic expression of MFPT in terms of the “stochastic potential”. We show in general no continuous diffusion process can provide asymptotically accurate representations for both the MFPT and the for a DGP. When n₁ and n₂ belong to two different basins of attraction, the MFPT yields the T*(V) in terms of Φ (x, V) ≈φ₀(x) + (1/V)φ₁(x) . For systems with saddle‐node bifurcations and catastrophe, discontinuous “phase transition” emerges, which can be characterized by Φ (x, V) in the limit of . In terms of timescale separation, the relation between deterministic local nonlinear bifurcations, and stochastic global phase transition is discussed. The one‐dimensional theory is a pedagogic first step toward a general theory of DGP.     

Uniform asymptotic estimates for ruin probabilities of renewal risk models with exponential Lévy process investment returns and dependent claims

This paper investigates the ruin probabilities of a renewal risk model with stochastic investment returns and dependent claim sizes. The investment is described as a portfolio of one risk‐free asset and one risky asset whose price process is an exponential Lévy process. The claim sizes are assumed to follow a one‐sided linear process with independent and identically distributed step sizes. When the step‐size distribution is heavy tailed, we establish some uniform asymptotic estimates for the ruin probabilities of this renewal risk model. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal investment–reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets

In this paper, we investigate the optimal time-consistent investment–reinsurance strategies for an insurer with state dependent risk aversion and Value-at-Risk (VaR) constraints. The insurer can purchase proportional reinsurance to reduce its insurance risks and invest its wealth in a financial market consisting of one risk-free asset and one risky asset, whose price process follows a geometric Brownian motion. The surplus process of the insurer is approximated by a Brownian motion with drift. The two Brownian motions in the insurer’s surplus process and the risky asset’s price process are correlated, which describe the correlation or dependence between the insurance market and the financial market. We introduce the VaR control levels for the insurer to control its loss in investment–reinsurance strategies, which also represent the requirement of regulators on the insurer’s investment behavior. Under the mean–variance criterion, we formulate the optimal investment–reinsurance problem within a game theoretic framework. By using the technique of stochastic control theory and solving the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations, we derive the closed-form expressions of the optimal investment–reinsurance strategies. In addition, we illustrate the optimal investment–reinsurance strategies by numerical examples and discuss the impact of the risk aversion, the correlation between the insurance market and the financial market, and the VaR control levels on the optimal strategies.     

Pricing Asian options of discretely monitored geometric average in the regime‐switching model

This paper provides analytic pricing formulas of discretely monitored geometric Asian options under the regime‐switching model. We derive the joint Laplace transform of the discount factor, the log return of the underlying asset price at maturity, and the logarithm of the geometric mean of the asset price. Then using the change of measures and the inversion of the transform, the prices and deltas of a fixed‐strike and a floating‐strike geometric Asian option are obtained. As the numerical results, we calculate the price of a fixed‐strike and a floating‐strike discrete geometric Asian call option using our formulas and compare with the results of the Monte Carlo simulation. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal reinsurance–investment problem in a constant elasticity of variance stock market for jump‐diffusion risk model

In this paper, we consider the jump‐diffusion risk model with proportional reinsurance and stock price process following the constant elasticity of variance model. Compared with the geometric Brownian motion model, the advantage of the constant elasticity of variance model is that the volatility has correlation with the risky asset price, and thus, it can explain the empirical bias exhibited by the Black and Scholes model, such as volatility smile. Here, we study the optimal investment–reinsurance problem of maximizing the expected exponential utility of terminal wealth. By using techniques of stochastic control theory, we are able to derive the explicit expressions for the optimal strategy and value function. Numerical examples are presented to show the impact of model parameters on the optimal strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

Backward uniqueness of 3D MHD‐α model driven by linear multiplicative Gaussian noise

In this paper, we devote to proving the backward uniqueness property of the solution to three‐dimensional stochastic magnetohydrodynamic‐α model (3D stochastic MHD‐α model) driven by linear multiplicative Gaussian noise, which involves not only the study of multiplicative noise but also the challenging nonlinear drift terms.     

Stochastic equity volatility related to the leverage effect II: valuation of European equity options and warrants

We propose a general framework to assess the value of the financial claims issued by the firm, European equity options and warrantsin terms of the stock price. In our framework, the firm's asset is assumed to follow a standard stationary lognormal process with constant volatility. However, it is not the case for equity volatility. The stochastic nature of equity volatility is endogenous, and comes from the impact of a change in the value of the firm's assets on the financial leverage. In a previous paper we studied the stochastic process for equity volatility, and proposed analytic approximations for different capital structures. In this companion paper we derive analytic approximations for the value of European equity options and warrants for a firm financed by equity, debt and warrants. We first present the basic model, which is an extension of the Black-Scholes model, to value corporate securities either as a function of the stock price, or as a function of the firm's total assets. Since stock prices are observable, then for practical purposes, traders prefer to use the stock as the underlying instrument, we concentrate on valuation models in terms of the stock price. Second, we derive an exact solution for the valuation in terms of the stock price of (i) a European call option on the stock of a levered firm, i.e. a European compound call option on the total assets of the firm, (ii) an equity warrant for an all-equity firm, and (iii) an equity warrant for a firm financed by equity and debt. Unfortunately, to compute these solutions we need to specify the function of the stock price in terms of the firm's assets value. In general we are unable to specify this expression, but we propose tight bounds for the value of these options which can be easily computed as a function of the stock price. Our results provide useful extensions of the Black-Scholes model.     

Simultaneity and non‐linear variability in financial markets: simulation and forecasting

Non‐linear variability in financial markets can emerge from several mechanisms, including simultaneity and time‐varying coefficients. In simultaneous equation systems, the reduced‐form coefficients that determine the behaviour of jointly dependent variables are products and ratios of the original structural coefficients. If the coefficients are stochastic, the resulting multiplicative interactions will result in high degrees of non‐linearity. Processes generated in this way will scale as fractals: they will exhibit intermittent outliers and scaling symmetries, i.e. proportionality relationships between fluctuations at different separation distances. A model is specified in which both the exchange rate itself and the exchange rate residual exhibit simultaneity. The exchange rate depends on other exchange rates, while the residual depends on the other residuals. The model is then simulated using embedding noise from a t‐distribution. The simulations replicate the observed properties of exchange rates, heavy‐tailed distributions and long memory in the variance. A forecasting algorithm is specified in two stages. The first stage is a model for the actual process. In the second stage the residuals are modelled as a function of the predicted rate of change. The first and second stage models are then combined. This algorithm exploits the scaling symmetry: the residual is proportional to the predicted rate of change at separation distances corresponding to the forecast horizon. The procedure is tested empirically on three exchange rates. At a daily frequency and a 1‐day forecast horizon, two‐stage models reduce the forecast error by one fourth. At a 5‐day horizon, the improvement is 10–15 percent. At a weekly frequency, the improvement at the 1‐week horizon is on the order of 30–40 percent. Copyright © 2006 John Wiley & Sons, Ltd.     

Mean-variance portfolio selection for a non-life insurance company

We consider a collective insurance risk model with a compound Cox claim process, in which the evolution of a claim intensity is described by a stochastic differential equation driven by a Brownian motion. The insurer operates in a financial market consisting of a risk-free asset with a constant force of interest and a risky asset which price is driven by a Lévy noise. We investigate two optimization problems. The first one is the classical mean-variance portfolio selection. In this case the efficient frontier is derived. The second optimization problem, except the mean-variance terminal objective, includes also a running cost penalizing deviations of the insurer’s wealth from a specified profit-solvency target which is a random process. In order to find optimal strategies we apply techniques from the stochastic control theory.     

Variance swaps under the threshold Ornstein–Uhlenbeck model

Variance swap is a typical financial tool for managing volatility risk. In this paper, we evaluate different types of variance swaps under a threshold Ornstein–Uhlenbeck model, which exhibits both mean reversion and regime switching features in the underlying asset price. We derive the analytical solution for the joint moment generating function of log‐asset prices at two distinct time points. This enables us to price various types of variance swaps analytically. Copyright © 2017 John Wiley & Sons, Ltd.     

Optimal investment,consumption and proportional reinsurance under model uncertainty

This paper considers the optimal investment, consumption and proportional reinsurance strategies for an insurer under model uncertainty. The surplus process of the insurer before investment and consumption is assumed to be a general jump–diffusion process. The financial market consists of one risk-free asset and one risky asset whose price process is also a general jump–diffusion process. We transform the problem equivalently into a two-person zero-sum forward–backward stochastic differential game driven by two-dimensional Lévy noises. The maximum principles for a general form of this game are established to solve our problem. Some special interesting cases are studied by using Malliavin calculus so as to give explicit expressions of the optimal strategies.     

Fast-slow stochastic dynamical system with singular coefficients

This paper aims to study the asymptotic behavior of a fast-slow stochastic dynamical system with singular coefficients, where the fast motion is given by a continuous diffusion process while the slow component is driven by an α-stable noise with α ∈ [1, 2). Using Zvonkin’s transformation and the technique of the Poisson equation, we have that both the strong and weak convergences in the averaging principle are established, which can be viewed as a functional law of large numbers. Then we study the small fluctuations between the original system around its average. We show that the normalized difference converges weakly to an Ornstein-Uhlenbeck type Gaussian process, which is a form of the functional central limit theorem. Furthermore, sharp rates for the above convergences are also obtained, and these convergences are shown to not depend on the regularities of the coefficients with respect to the fast variable, which reflect the effects of noises on the multi-scale systems.