An Asymptotic Expansion for Forward–Backward SDEs: A Malliavin Calculus Approach

This paper proposes a new analytical approximation scheme for the representation of the forward–backward stochastic differential equations (FBSDEs) of Ma and Zhang (Ann Appl Probab, 2002). In particular, we obtain an error estimate for the scheme applying Malliavin calculus method for the forward SDEs combined with the Picard iteration scheme for the BSDEs. We also show numerical examples for pricing option with counterparty risk under local and stochastic volatility models, where the credit value adjustment is taken into account.     

Optimal risk transfer and investment policies based upon stochastic differential utilities

This paper addresses the stochastic differential utility (SDU) version of the issue raised by Barrieu and El Karoui (Quantitative Finance, 2:181–188, 2002a) in which optimal risk transfer from a bank to an investor, realized by transacting well-designed derivatives written on relevant illiquid assets, was␣mainly studied in two cases with and without an available financial market. From a stochastic maximum principle as described in Yong and Zhou (Stochastic controls: Hamiltonian systems and HJB equations. Springer-Verlag, New York, 1999) we shall derive necessary and sufficient conditions for optimality in several SDU-based maximization problems. It is also shown that the optimal risk transfer, consumptions, investment policies of both agents are characterized by a forward–backward stochastic differential equation (FBSDE) system.     

Consumption–investment optimization with Epstein–Zin utility in incomplete markets

In a market with stochastic investment opportunities, we study an optimal consumption–investment problem for an agent with recursive utility of Epstein–Zin type. Focusing on the empirically relevant specification where both risk aversion and elasticity of intertemporal substitution are in excess of one, we characterize optimal consumption and investment strategies via backward stochastic differential equations. The superdifferential of indirect utility is also obtained, meeting demands from applications in which Epstein–Zin utilities were used to resolve several asset pricing puzzles. The empirically relevant utility specification introduces difficulties to the optimization problem due to the fact that the Epstein–Zin aggregator is neither Lipschitz nor jointly concave in all its variables.     

Dynamic Investment Strategies to Reaction–Diffusion Systems Based upon Stochastic Differential Utilities

We study the dynamic investment strategies in continuous-time settings based upon stochastic differential utilities of Duffie and Epstein (Econometrica 60:353–394, 1992). We assume that the asset prices follow interacting Itô-Poisson processes, which are known to be the so-called reaction–diffusion systems. Stochastic maximum principle for stochastic control problems described by some backward-stochastic differential equations that are driven by Poisson jump processes allows us to derive the optimal investment strategies as well as optimal consumption. We shall furthermore propose a numerical procedure for solving the associated nested quasi-linear partial differential equations.     

Can Asset Pricing Models Price Idiosyncratic Risk in U.K. Stock Returns?

I examine how well different linear factor models and consumption‐based asset pricing models price idiosyncratic risk in U.K. stock returns. Correctly pricing idiosyncratic risk is a significant challenge for many of the models I consider. For some consumption‐based models, there is a clear tradeoff in the performance of the models between correctly pricing systematic risk and idiosyncratic risk. Linear factor models do a better job in most cases in pricing systematic risk than consumption‐based models but the reverse is true for idiosyncratic risk.     

How Do Investors' Expectations Drive Asset Prices?

Based on an extension of the process of investors' expectations to stochastic volatility we derive asset price processes in a general continuous time pricing kernel framework. Our analysis suggests that stochastic volatility of asset price processes results from the fact that investors do not know the risk of an asset and therefore the volatility of the process of their expectations is stochastic, too. Furthermore, our model is consistent with empirical studies reporting negative correlation between asset prices and their volatility as well as significant variations in the Sharpe ratio.     

Optimal pricing barriers in a regulated market using reflected diffusion processes

We consider a class of one-dimensional (1D) reflected stochastic differential equations (SDEs). Such reflected SDE models arise as the key approximating processes in a regulated financial market system, and our main goal is to determine the set of optimal pricing barriers. We consider the running cost associated with the deviation of the process from the desired target level, and also the control cost from the interventions in an effort to keep the process inside the boundaries. Both a long-time average (ergodic) cost criterion and an infinite horizon discount cost criterion, where the discount factor is allowed to vary from one period to another, are studied, with numerical examples illustrating our main results.     

Examination of Conditional Asset Pricing in UK Stock Returns

I examine the empirical performance of various specifications of the capital asset pricing model (CAPM) in UK stock returns, using the stochastic discount framework. When the proxy for the market portfolio includes a proxy for labor income growth in addition to the stock market index, the performance of the CAPM improves. The improvement in performance shows in the magnitude and significance of the pricing errors and in the reduced impact of asset characteristics and other factors in the pricing of assets. There is further improvement when I use conditional versions of the models.     

A reduced PDE method for European option pricing under multi-scale,multi-factor stochastic volatility

The number of tailor-made hybrid structured products has risen more prominently to fit each investor’s preferences and requirements as they become more diversified. The structured products entail synthetic derivatives such as combinations of bonds and/or stocks conditional on how they are backed up by underlying securities, stochastic volatility, stochastic interest rates or exchanges rates. The complexity of these multi-asset structures yields lots of difficulties of pricing the products. Because of the complexity, Monte-Carlo simulation is a possible choice to price them but it may not produce stable Greeks leading to a trouble in hedging against risks. In this light, it is desirable to use partial differential equations with relevant analytic and numerical techniques. Even if the partial differential equation method would generate stable security prices and Greeks for single asset options, however, it may result in the curse of dimensionality when pricing multi-asset derivatives. In this study, we make the best use of multi-scale nature of stochastic volatility to lift the curse of dimensionality for up to three asset cases. Also, we present a transformation formula by which the pricing group parameters required for the multi-asset options in illiquid market can be calculated from the underlying market parameters.     

Pricing pension buy-outs under stochastic interest and mortality rates

Pension buy-out is a special financial asset issued to offload the pension liabilities holistically in exchange for an upfront premium. In this paper, we concentrate on the pricing of pension buy-outs under dependence between interest and mortality rates risks with an explicit correlation structure in a continuous time framework. Change of measure technique is invoked to simplify the valuation. We also present how to obtain the buy-out price for a hypothetical benefit pension scheme using stochastic models to govern the dynamics of interest and mortality rates. Besides employing a non-mean reverting specification of the Ornstein–Uhlenbeck process and a continuous version of Lee–Carter setting for modeling mortality rates, we prefer Vasicek and Cox–Ingersoll–Ross models for short rates. We provide numerical results under various scenarios along with the confidence intervals using Monte Carlo simulations.     

An Empirical Investigation of the Campbell-Cochrane Habit Utility Model

Abstract:  This paper tests whether the Campbell and Cochrane (1999) habit utility model generates a valid stochastic discount factor for the 25 Fama-French size/book-to-market and size/momentum sorted portfolios. Campbell and Cochrane (1999) derive a consumption based habit utility asset pricing model and calibrate it to aggregate US stock market data. However, they do not test whether their model is consistent with a larger cross section of asset returns. We test their model using the methodology of Hansen and Jagannathan (1991) and Burnside (1994) . In contrast to previous studies, we find that for reasonable parameter values, the model's stochastic discount factor is inside the Hansen-Jagannathan bounds and therefore satisfies the necessary conditions for a valid stochastic discount factor. We trace the difference between our results and previous studies to the method used to estimate the model's parameters and the parameter values themselves.     

Optimal investment-consumption-insurance with random parameters

This paper discusses an optimal investment, consumption, and life insurance purchase problem for a wage earner in a complete market with Brownian information. Specifically, we assume that the parameters governing the market model and the wage earner, including the interest rate, appreciation rate, volatility, force of mortality, premium-insurance ratio, income and discount rate, are all random processes adapted to the Brownian motion filtration. Our modeling framework is very general, which allows these random parameters to be unbounded, non-Markovian functionals of the underlying Brownian motion. Suppose that the wage earner’s preference is described by a power utility. The wage earner’s problem is then to choose an optimal investment-consumption-insurance strategy so as to maximize the expected, discounted utilities from intertemporal consumption, legacy and terminal wealth over an uncertain lifetime horizon. We use a novel approach, which combines the Hamilton–Jacobi–Bellman equation and backward stochastic differential equation (BSDE) to solve this problem. In general, we give explicit expressions for the optimal investment-consumption-insurance strategy and the value function in terms of the solutions to two BSDEs. To illustrate our results, we provide closed-form solutions to the problem with stochastic income, stochastic mortality, and stochastic appreciation rate, respectively.     

Exchange Rates and the Conversion of Currency‐Specific Risk Premia

How do the risk factors that drive asset prices influence exchange rates? Are the parameters of asset price processes relevant for specifying exchange rate processes? Most international asset pricing models focus on the analysis of asset returns given exchange rate processes. Little work has been done on the analysis of exchange rates dependent on asset returns. This paper uses an international stochastic discount factor (SDF) framework to analyse the interplay between asset prices and exchange rates. So far, this approach has only been implemented in international term structure models. We find that exchange rates serve to convert currency‐specific discount factors and currency‐specific prices of risk – a result linked to the international arbitrage pricing theory (IAPT). Our empirical investigation of exchange rates and stock markets of four countries presents evidence for the conversion of currency‐specific risk premia by exchange rates.     

Recursive marginal quantization of higher-order schemes

Quantization techniques have been applied in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and efficient calibration of large derivative books. Recursive marginal quantization (RMQ) of the Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This method involves recursively quantizing the conditional marginals of the discrete-time Euler approximation of the underlying process. By generalizing this approach, we show that it is possible to perform RMQ for two higher-order schemes: the Milstein scheme and a simplified weak order 2.0 scheme. We further extend the applicability of RMQ by showing how absorption and reflection at the zero boundary may be incorporated, when necessary. To illustrate the improved accuracy of the higher-order schemes, various computations are performed using geometric Brownian motion and the constant elasticity of variance model. For both models, we provide evidence of improved weak order convergence and computational efficiency. By pricing European, Bermudan and barrier options, further evidence of improved accuracy of the higher-order schemes is demonstrated.     

Pricing average options under time-changed Lévy processes

This paper presents an approximate formula for pricing average options when the underlying asset price is driven by time-changed Lévy processes. Time-changed Lévy processes are attractive to use for a driving factor of underlying prices because the processes provide a flexible framework for generating jumps, capturing stochastic volatility as the random time change, and introducing the leverage effect. There have been very few studies dealing with pricing problems of exotic derivatives on time-changed Lévy processes in contrast to standard European derivatives. Our pricing formula is based on the Gram–Charlier expansion and the key of the formula is to find analytic treatments for computing the moments of the normalized average asset price. In numerical examples, we demonstrate that our formula give accurate values of average call options when adopting Heston’s stochastic volatility model, VG-CIR, and NIG-CIR models.     

Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem

In this paper, we study a class of quadratic backward stochastic differential equations (BSDEs), which arises naturally in the utility maximization problem with portfolio constraints. We first establish the existence and uniqueness of solutions for such BSDEs and then give applications to the utility maximization problem. Three cases of utility functions, the exponential, power, and logarithmic ones, are discussed. This study is part of my PhD thesis supervised by Professor Ying Hu and defended at the University of Rennes 1 (in France) in October 2007.     

Set-valued risk measures as backward stochastic difference inclusions and equations

Scalar dynamic risk measures for univariate positions in continuous time are commonly represented via backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensions of scalar backward stochastic differential equations for the set-valued framework: (1) backward stochastic differential inclusions, which evaluate the risk dynamics on the selectors of acceptable capital allocations; or (2) set-valued backward stochastic differential equations, which evaluate the risk dynamics on the full set of acceptable capital allocations as a singular object. In this work, the discrete-time setting is investigated with difference inclusions and difference equations in order to provide insights for such differential representations for set-valued dynamic risk measures in continuous time.

     

动态资本资产定价理论评述

本文主要讨论了动态资产定价理论的产生和发展.默顿和布里登使用贝尔曼开创的动态规划方法和伊藤随机分析技术,重新考察在由随机过程驱动的不确定环境下,个人如何连续地做出消费/投资决策,使得终身效用最大化.无须单期框架中的严格假定,他们也获得了连续时间跨期资源配置的一般均衡模型--时际资产定价模型(ICAPM)以及消费资产定价模型(CCAPM).这些工作开启了连续时间金融方法论的新时代.     

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.