Order Flow, Transaction Clock, and Normality of Asset Returns

The goal of this paper is to show that normality of asset returns can be recovered through a stochastic time change. Clark (1973) addressed this issue by representing the price process as a subordinated process with volume as the lognormally distributed subordinator. We extend Clark's results and find the following: (i) stochastic time changes are mathematically much less constraining than subordinators; (ii) the cumulative number of trades is a better stochastic clock than the volume for generating virtually perfect normality in returns; (iii) this clock can be modeled nonparametrically, allowing both the time-change and price processes to take the form of jump diffusions.     

On a symmetrization of diffusion processes

The latter author, together with collaborators, proposed a numerical scheme to calculate the price of barrier options. The scheme is based on a symmetrization of diffusion processes. The present paper aims to give a basis to the use of the numerical scheme for Heston and SABR-type stochastic volatility models. This will be done by showing a fairly general result on the symmetrization (in multi-dimension/multi-reflections). Further applications (to time-inhomogeneous diffusions/ to time-dependent boundaries/to curved boundaries) are also discussed.     

Pricing Options under Generalized GARCH and Stochastic Volatility Processes

In this paper, we develop an efficient lattice algorithm to price European and American options under discrete time GARCH processes. We show that this algorithm is easily extended to price options under generalized GARCH processes, with many of the existing stochastic volatility bivariate diffusion models appearing as limiting cases. We establish one unifying algorithm that can price options under almost all existing GARCH specifications as well as under a large family of bivariate diffusions in which volatility follows its own, perhaps correlated, process.     

An Approximation Scheme for Diffusion Processes Based on an Antisymmetric Calculus over Wiener Space

In this paper, we show that every antisymmetric multiple stochastic (Ito’s) integral has a polynomial form of single and double ones. As an application, a new approximating scheme for the solution to a stochastic differential equation is proposed.     

Optimal liquidation under stochastic liquidity

We solve explicitly a two-dimensional singular control problem of finite fuel type for an infinite time horizon. The problem stems from the optimal liquidation of an asset position in a financial market with multiplicative and transient price impact. Liquidity is stochastic in that the volume effect process, which determines the intertemporal resilience of the market in the spirit of Predoiu et al. (SIAM J. Financ. Math. 2:183–212, 2011), is taken to be stochastic, being driven by its own random noise. The optimal control is obtained as the local time of a diffusion process reflected at a non-constant free boundary. To solve the HJB variational inequality and prove optimality, we need a combination of probabilistic arguments and calculus of variations methods, involving Laplace transforms of inverse local times for diffusions reflected at elastic boundaries.     

Contingent capital conversion under dual asset and equity jump–diffusions

We model contingent capital with market trigger under dual jump–diffusion processes in asset values and equity prices. Under the dual jump–diffusions, we show that the conversion ratio is no longer deterministic under the jump–diffusion. The conversion ratio becomes a stochastic process related to the jump process of the underlying equity and the conditional expectation of the contingent capital at the conversion time. Thus, making the implementation of contingent capital impossible. The best we can hope to practically implement this conversion design, is to give the minimal conversion ratio (at least the portion required to convert) to conform with Basel III.     

Comparison of Option Prices in Semimartingale Models

In this paper we generalize the recent comparison results of El Karoui et al. (Math Finance 8:93–126, 1998), Bellamy and Jeanblanc (Finance Stoch 4:209–222, 2000) and Gushchin and Mordecki (Proc Steklov Inst Math 237:73–113, 2002) to d-dimensional exponential semimartingales. Our main result gives sufficient conditions for the comparison of European options with respect to martingale pricing measures. The comparison is with respect to convex and also with respect to directionally convex functions. Sufficient conditions for these orderings are formulated in terms of the predictable characteristics of the stochastic logarithm of the stock price processes. As examples we discuss the comparison of exponential semimartingales to multivariate diffusion processes, to stochastic volatility models, to Lévy processes, and to diffusions with jumps. We obtain extensions of several recent results on nontrivial price intervals. A crucial property in this approach is the propagation of convexity property. We develop a new approach to establish this property for several further examples of univariate and multivariate processes.     

Displaced lognormal volatility skews: analysis and applications to stochastic volatility simulations

We analyze the implied volatility skews generated by displaced lognormal diffusions. In particular, we prove the global monotonicity of implied volatility, and an at-the-money bound on the steepness of downward volatility skews, under displaced lognormal dynamics, which therefore cannot reproduce some features observed in equity markets. A variant, the displaced anti-lognormal, overcomes the steepness constraint, but its state space is bounded above and unbounded below. In light of these limitations on what features the displaced (anti-)lognormal (DL) can model, we exploit the DL, not as a model, but as a control variate, to reduce variance in Monte Carlo simulations of the CEV and SABR local/stochastic volatility models. For either use—as model, or as control variate—the DL’s parameters require estimation. We find an explicit formula for the DL’s short-expiry limiting volatility skew, which allows direct calibration of its parameters to volatility skews implied by market data or by other models.     

Performance Evaluation Of U.K. Unit Trusts Within The Stochastic Discount Factor Framework

We examine the performance of U.K. unit trusts between January 1982 and December 1996 within the stochastic discount factor approach across a wide class of models. No one model dominates the others in correctly pricing passive portfolios or detecting superior performance for hypothetical trading strategies. We find no evidence of significant superior performance by the unit trusts for any model of the stochastic discount factor. Also, the charges of the trust have a mixed effect on trust performance.     

Additive subordination and its applications in finance

This paper studies additive subordination, which we show is a useful technique for constructing time-inhomogeneous Markov processes with analytical tractability. This technique is a natural generalization of Bochner’s subordination that has proved to be extremely useful in financial modeling. Probabilistically, Bochner’s subordination corresponds to a stochastic time change with respect to an independent Lévy subordinator, while in additive subordination, the Lévy subordinator is replaced by an additive one. We generalize the classical Phillips theorem for Bochner’s subordination to the additive subordination case, based on which we provide Markov and semimartingale characterizations for a rich class of jump-diffusions and pure jump processes obtained from diffusions through additive subordination, and obtain spectral decomposition for them. To illustrate the usefulness of additive subordination, we develop an analytically tractable cross-commodity model for spread option valuation that is able to calibrate the implied volatility surface of each commodity. Moreover, our model can generate implied correlation patterns that are consistent with market observations and economic intuitions.     

Time reversal and last passage time of diffusions with applications to credit risk management

Finance and Stochastics - We study time reversal, last passage time and $h$ -transform of linear diffusions. For general diffusions with killing, we obtain the probability density of the last...     

行为如何影响资产定价——理论模型与实证分析

本文基于金融经济学中状态价格与随机折现因子等理论的分析,认为资产定价会受到行为因素的影响.在此基础上本文提出了状态价格函数,并建立了行为影响资产定价的多项式模型.     

An SPDE model for systemic risk with endogenous contagion

Finance and Stochastics - We propose a dynamic mean-field model for ‘systemic risk’ in large financial systems, derived from a system of interacting diffusions on the positive half-line...     

Pricing participating policies under the Meixner process and stochastic volatility

We propose a model for the valuation of participating life insurance products under the Meixner process, which belongs to the family of semi-heavy tailed processes. This particular model assumption is extremely desirable as it captures the stylised features of the return distribution, with existing moment generating functions. The highlight of the paper is the analytical solution derived for minimising the relative entropy between the historical and risk-neutral measures, when driving a pricing kernel. Further, we capture the stochastic volatility effect using an accurate polynomial approximation technique. Finally, to highlight the practical applications, we conduct a simulation experiment.     

Second order multiscale stochastic volatility asymptotics: stochastic terminal layer analysis and calibration

Multiscale stochastic volatility models have been developed as an efficient way to capture the principal effects on derivative pricing and portfolio optimization of randomly varying volatility. The recent book by Fouque et al. (Multiscale Stochastic Volatility for Equity, Interest-Rate and Credit Derivatives, 2011) analyzes models in which the volatility of the underlying is driven by two diffusions – one fast mean-reverting and one slowly varying – and provides a first order approximation for European option prices and for the implied volatility surface, which is calibrated to market data. Here, we present the full second order asymptotics, which are considerably more complicated due to a terminal layer near the option expiration time. We find that to second order, the implied volatility approximation depends quadratically on log-moneyness, capturing the convexity of the implied volatility curve seen in data. We introduce a new probabilistic approach to the terminal layer analysis needed for the derivation of the second order singular perturbation term, and calibrate to S&P 500 options data.     

Bayesian testing volatility persistence in stochastic volatility models with jumps

Whether or not there is a unit root persistence in volatility of financial assets has been a long-standing topic of interest to financial econometricians and empirical economists. The purpose of this article is to provide a Bayesian approach for testing the volatility persistence in the context of stochastic volatility with Merton jump and correlated Merton jump. The Shanghai Composite Index daily return data is used for empirical illustration. The result of Bayesian hypothesis testing strongly indicates that the volatility process doesn’t have unit root volatility persistence in this stock market.     

Nonparametric estimation and testing of stochastic discount factor

This paper attempts to estimate stochastic discount factor (SDF) proxies nonparametrically using the conditional Hansen–Jagannathan distance. Nonparametric estimation can not only avoid misspecification when dealing with nonlinearity in the model but also provide more precise information about the local properties of the estimators. Empirical studies show that our method performs better than the alternative parametric polynomial models, and furthermore, we find that the return on aggregate wealth can sufficiently explain the SDF proxies when one deals with nonlinearity appropriately.     

Common Stochastic Trends in a System of Exchange Rates

Univariate tests reveal strong evidence for the presence of a unit root in the univariate time-series representation for seven daily spot and forward exchange rate series. Furthermore, all seven spot and forward rates appear to be cointegrated; that is, the forward premiums are stationary, and one common unit root, or stochastic trend, is detectable in the multivariate time-series models for the seven spot and forward rates, respectively. This is consistent with the hypothesis that the seven exchange rates possess one long-run relationship and that the disequilibrium error around that relationship partly accounts for subsequent movements in the exchange rates.     

Information variables for monetary policy in an estimated structural model of the euro area

A small scale new keynesian model for the euro area is estimated with maximum likelihood under the assumptions of imperfect information and discretionary monetary policy. The estimated parametrization of this widely used dynamic stochastic model unveils the monetary authorities’ objectives and the information content of two indicator variables: monetary aggregates and real unit labour costs. The results highlight a significant policy concern about interest-rate smoothing and inflation; almost no concern for output gap stabilization emerges. Regarding indicator variables, unit labour costs provide information on potential output that is helpful for stabilization policy; no useful information role emerges for monetary aggregates.