Convex duality for Epstein–Zin stochastic differential utility

This paper introduces a dual problem to study a continuous‐time consumption and investment problem with incomplete markets and Epstein–Zin stochastic differential utilities. Duality between the primal and dual problems is established. Consequently, the optimal strategy of this consumption and investment problem is identified without assuming several technical conditions on market models, utility specifications, and agent's admissible strategies. Meanwhile, the minimizer of the dual problem is identified as the utility gradient of the primal value and is economically interpreted as the “least favorable” completion of the market.     

EXPLICIT SOLUTIONS OF CONSUMPTION-INVESTMENT PROBLEMS IN FINANCIAL MARKETS WITH REGIME SWITCHING

We consider a consumption and investment problem where the market presents different regimes. An investor taking decisions continuously in time selects a consumption–investment policy to maximize his expected total discounted utility of consumption. The market coefficients and the investor's utility of consumption are dependent on the regime of the financial market, which is modeled by an observable finite-state continuous-time Markov chain. We obtain explicit optimal consumption and investment policies for specific HARA utility functions. We show that the optimal policy depends on the regime. We also make an economic analysis of the solutions, and show that for every investor the optimal proportion to allocate in the risky asset is greater in a "bull market" than in a "bear market." This behavior is not affected by the investor's risk preferences. On the other hand, the optimal consumption to wealth ratio depends not only on the regime, but also on the investor's risk tolerance: high risk-averse investors will consume relatively more in a "bull market" than in a "bear market," and the opposite is true for low risk-averse investors.     

OPTIMAL INVESTMENT UNDER RELATIVE PERFORMANCE CONCERNS

We consider the problem of optimal investment when agents take into account their relative performance by comparison to their peers. Given N interacting agents, we consider the following optimization problem for agent i, : where is the utility function of agent i, his portfolio, his wealth, the average wealth of his peers, and is the parameter of relative interest for agent i. Together with some mild technical conditions, we assume that the portfolio of each agent i is restricted in some subset . We show existence and uniqueness of a Nash equilibrium in the following situations:

• ‐ unconstrained agents,
• ‐ constrained agents with exponential utilities and Black–Scholes financial market.

We also investigate the limit when the number of agents N goes to infinity. Finally, when the constraints sets are vector spaces, we study the impact of the s on the risk of the market.     

Backward SDEs for control with partial information

This paper considers a non‐Markov control problem arising in a financial market where asset returns depend on hidden factors. The problem is non‐Markov because nonlinear filtering is required to make inference on these factors, and hence the associated dynamic program effectively takes the filtering distribution as one of its state variables. This is of significant difficulty because the filtering distribution is a stochastic probability measure of infinite dimension, and therefore the dynamic program has a state that cannot be differentiated in the traditional sense. This lack of differentiability means that the problem cannot be solved using a Hamilton–Jacobi–Bellman equation. This paper will show how the problem can be analyzed and solved using backward stochastic differential equations, with a key tool being the problem's dual formulation.     

CASH SUBADDITIVE RISK MEASURES AND INTEREST RATE AMBIGUITY

A new class of risk measures called cash subadditive risk measures is introduced to assess the risk of future financial, nonfinancial, and insurance positions. The debated cash additive axiom is relaxed into the cash subadditive axiom to preserve the original difference between the numéraire of the current reserve amounts and future positions. Consequently, cash subadditive risk measures can model stochastic and/or ambiguous interest rates or defaultable contingent claims. Practical examples are presented, and in such contexts cash additive risk measures cannot be used. Several representations of the cash subadditive risk measures are provided. The new risk measures are characterized by penalty functions defined on a set of sublinear probability measures and can be represented using penalty functions associated with cash additive risk measures defined on some extended spaces. The issue of the optimal risk transfer is studied in the new framework using inf-convolution techniques. Examples of dynamic cash subadditive risk measures are provided via BSDEs where the generator can locally depend on the level of the cash subadditive risk measure.     

FROM SMILE ASYMPTOTICS TO MARKET RISK MEASURES

The left tail of the implied volatility skew, coming from quotes on out‐of‐the‐money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations, to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear partial differential equation and provide a small time‐to‐maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parameterized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.     

Arbitrage‐free XVA

We develop a framework for computing the total valuation adjustment (XVA) of a European claim accounting for funding costs, counterparty credit risk, and collateralization. Based on no‐arbitrage arguments, we derive backward stochastic differential equations associated with the replicating portfolios of long and short positions in the claim. This leads to the definition of buyer's and seller's XVA, which in turn identify a no‐arbitrage interval. In the case that borrowing and lending rates coincide, we provide a fully explicit expression for the unique XVA, expressed as a percentage of the price of the traded claim, and for the corresponding replication strategies. In the general case of asymmetric funding, repo, and collateral rates, we study the semilinear partial differential equations characterizing buyer's and seller's XVA and show the existence of a unique classical solution to it. To illustrate our results, we conduct a numerical study demonstrating how funding costs, repo rates, and counterparty risk contribute to determine the total valuation adjustment.     

The valuation of American options in a multidimensional exponential Lévy model

We consider the problem of valuation of American options written on dividend‐paying assets whose price dynamics follow a multidimensional exponential Lévy model. We carefully examine the relation between the option prices, related partial integro‐differential variational inequalities, and reflected backward stochastic differential equations. In particular, we prove regularity results for the value function and obtain the early exercise premium formula for a broad class of payoff functions.     

BILATERAL COUNTERPARTY RISK UNDER FUNDING CONSTRAINTS—PART II: CVA

The correction in value of an over‐the‐counter derivative contract due to counterparty risk under funding constraints is represented as the value of a dividend‐paying option on the value of the contract clean of counterparty risk and excess funding costs. This representation allows one to analyze the structure of this correction, the so‐called Credit Valuation Adjustment (CVA for short), in terms of replacement cost/benefits, credit cost/benefits, and funding cost/benefits. We develop a reduced‐form backward stochastic differential equations (BSDE) approach to the problem of pricing and hedging the CVA. In the Markov setup, explicit CVA pricing and hedging schemes are formulated in terms of semilinear partial differential equations.     

A MODEL OF OPTIMAL CONSUMPTION UNDER LIQUIDITY RISK WITH RANDOM TRADING TIMES

We consider a portfolio/consumption choice problem in a market model with liquidity risk. The main feature is that the investor can trade and observe stock prices only at exogenous Poisson arrival times. He may also consume continuously from his cash holdings, and his goal is to maximize his expected utility from consumption. This is a mixed discrete/continuous stochastic control problem, non‐standard in the literature. The dynamic programming principle leads to a coupled system of Integro‐Differential Equations (IDE), and we provide a convergent numerical algorithm for the resolution to this coupled system of IDE. Several numerical experiments illustrate the impact of the restricted liquidity trading opportunities, and we measure in particular the utility loss with respect to the classical Merton consumption problem.     

Calibrating a Diffusion Pricing Model with Uncertain Volatility: Regularization and Stability

A regularized (smoothed) version of the model calibration method of 1 ) is studied. We prove that the regularized formulation is solvable and that the solution depends continuously on the input data (observed derivative security prices). Associated issues of model credibility, stability, and robustness (insensitivity to model assumptions) are discussed. The Implicit Function Theorem for Banach spaces is used for the stability proof, and some numerical illustrations are included.     

Robust XVA

We introduce an arbitrage‐free framework for robust valuation adjustments. An investor trades a credit default swap portfolio with a risky counterparty, and hedges credit risk by taking a position in defaultable bonds. The investor does not know the exact return rate of her counterparty's bond, but she knows it lies within an uncertainty interval. We derive both upper and lower bounds for the XVA process of the portfolio, and show that these bounds may be recovered as solutions of nonlinear ordinary differential equations. The presence of collateralization and closeout payoffs leads to important differences with respect to classical credit risk valuation. The value of the super‐replicating portfolio cannot be directly obtained by plugging one of the extremes of the uncertainty interval in the valuation equation, but rather depends on the relation between the XVA replicating portfolio and the closeout value throughout the life of the transaction. Our comparative statics analysis indicates that credit contagion has a nonlinear effect on the replication strategies and on the XVA.     

Approximating Large Diversified Portfolios

This paper considers a financial market with asset price dynamics modeled by a system of lognormal stochastic differential equations. A one‐dimensional stochastic differential equation for the approximate evolution of a large diversified portfolio formed by these assets is derived. This identifies the asymptotic dynamics of the portfolio as being a lognormal diffusion. Consequentially an efficient way for computing probabilities, derivative prices, and other quantities for the portfolio are obtained. Additionally, the asymptotic strong and weak orders of convergence with respect to the number of assets in the portfolio are determined.     

Quantiles of the Euler Scheme for Diffusion Processes and Financial Applications

In this paper we briefly present the results obtained in our paper ( Talay and Zheng 2002a ) on the convergence rate of the approximation of quantiles of the law of one component of  ( X_t )  , where  ( X_t )  is a diffusion process, when one uses a Monte Carlo method combined with the Euler discretization scheme. We consider the case where  ( X_t )  is uniformly hypoelliptic (in the sense of Condition (UH) below), or the inverse of the Malliavin covariance of the component under consideration satisfies the condition (M) below. We then show that Condition (M) seems widely satisfied in applied contexts. We particularly study financial applications: the computation of quantiles of models with stochastic volatility, the computation of the VaR of a portfolio, and the computation of a model risk measurement for the profit and loss of a misspecified hedging strategy.     

Classical and Impulse Stochastic Control of the Exchange Rate Using Interest Rates and Reserves

We consider the problem of a Central Bank that wants the exchange rate to be as close as possible to a given target, and in order to do that uses both the interest rate level and interventions in the foreign exchange market. We model this as a mixed classical‐impulse stochastic control problem, and provide for the first time a solution to that kind of problem. We give examples of solutions that allow us to perform an interesting economic analysis of the optimal strategy of the Central Bank.     

MULTIDIMENSIONAL DYNAMIC RISK MEASURE VIA CONDITIONAL g‐EXPECTATION

This paper deals with multidimensional dynamic risk measures induced by conditional g‐expectations. A notion of multidimensional g‐expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem, and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, and translatability properties of multidimensional conditional g‐expectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic g‐risk measure is nonincreasingly convex if and only if the generator g satisfies a quasi‐monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex g‐risk measure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures. As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacting subsidiaries; optimal risk sharing for ‐tolerant g‐risk measures, and risk contribution for coherent g‐risk measures are investigated. Insurance g‐risk measure and other ways to induce g‐risk measures are also studied at the end of the paper.     

基于随机增长模型的低碳经济发展政策研究

低碳经济是一个充满不确定性的经济形态,它对于传统意义上的经济增长与经济产出有着很多不确定性的影响,因此,应用传统确定性条件下的经济增长模型研究低碳经济对于经济增长的影响存在很大局限性。运用随机微分对完全抽象的经济增长随机过程进行分析,其结论对于制定低碳经济条件下的宏观经济政策有着非常重要的意义。     

Credit risk pricing in a consumption-based equilibrium framework with incomplete accounting information

We present a consumption-based equilibrium framework for credit risk pricing based on the Epstein–Zin (EZ) preferences where the default time is modeled as the first hitting time of a default boundary and bond investors have imperfect/partial information about the firm value. The imperfect information is generated by the underlying observed state variables and a noisy observation process of the firm value. In addition, the consumption, the volatility, and the firm value process are modeled to follow affine diffusion processes. Using the EZ equilibrium solution as the pricing kernel, we provide an equivalent pricing measure to compute the prices of financial derivatives as discounted values of the future payoffs given the incomplete information. The price of a zero-coupon bond is represented in terms of the solutions of a stochastic partial differential equation (SPDE) and a deterministic PDE; the self-contained proofs are provided for both this representation and the well-posedness of the involved SPDE. Furthermore, this SPDE is numerically solved, which yields some insights into the relationship between the structure of the yield spreads and the model parameters.     

Size matters for OTC market makers: General results and dimensionality reduction techniques

In most over‐the‐counter (OTC) markets, a small number of market makers provide liquidity to other market participants. More precisely, for a list of assets, they set prices at which they agree to buy and sell. Market makers face therefore an interesting optimization problem: they need to choose bid and ask prices for making money while mitigating the risk associated with holding inventory in a volatile market. Many market‐making models have been proposed in the academic literature, most of them dealing with single‐asset market making whereas market makers are usually in charge of a long list of assets. The rare models tackling multiasset market making suffer however from the curse of dimensionality when it comes to the numerical approximation of the optimal quotes. The goal of this paper is to propose a dimensionality reduction technique to address multiasset market making by using a factor model. Moreover, we generalize existing market‐making models by the addition of an important feature: the existence of different transaction sizes and the possibility for the market makers in OTC markets to answer different prices to requests with different sizes.     

PRICING AND HEDGING OF DERIVATIVES BASED ON NONTRADABLE UNDERLYINGS

This paper is concerned with the study of insurance related derivatives on financial markets that are based on nontradable underlyings, but are correlated with tradable assets. We calculate exponential utility‐based indifference prices, and corresponding derivative hedges. We use the fact that they can be represented in terms of solutions of forward‐backward stochastic differential equations (FBSDE) with quadratic growth generators. We derive the Markov property of such FBSDE and generalize results on the differentiability relative to the initial value of their forward components. In this case the optimal hedge can be represented by the price gradient multiplied with the correlation coefficient. This way we obtain a generalization of the classical “delta hedge” in complete markets.