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.     

European‐Type Contingent Claims in an Incomplete Market with Constrained Wealth and Portfolio

This paper considers the problem of hedgeability and replicability of European‐type contingent claims in an incomplete market with the wealth and the portfolio possibly being constrained. For the case of no constraint, using the idea of a Four Step Scheme (Ma, Protter, and Yong 1994), we prove the replicability of a class of contingent claims (including European call and put options) without assuming ad hoc technical conditions. For the case with the wealth and portfolio being constrained, several positive and negative results concerning hedgeability and replicability are presented.     

证券投资者保护基金的收支系统研究——基于倒向随机微分方程的系统模型

证券投资者保护基金是投资者权益保护的重要工具,而对于证券投资者保护基金的收入和支出的定量分析研究更是其中的核心问题之一。本文基于倒向随机微分方程对证券投资者保护基金的收支系统进行探索式研究,将相关的现实内容进行模型化处理与数理表达。根据模型的数理结果,证券投资者保护基金向证券公司收费金额是预期赔偿金额的增函数,与市场风险资产的波动正向关联;但与无风险资产的收益率及风险资产的预期收益率均负向相关。本文据此提出了有针对性的政策建议。     

Fair bilateral pricing under funding costs and exogenous collateralization

Bielecki and Rutkowski introduced and studied a generic nonlinear market model, which includes several risky assets, multiple funding accounts, and margin accounts. In this paper, we examine the pricing and hedging of contract from the perspective of both the hedger and the counterparty with arbitrary initial endowments. We derive inequalities for unilateral prices and we study the range of fair bilateral prices. We also examine the positive homogeneity and monotonicity of unilateral prices with respect to the initial endowments. Our study hinges on results from Nie and Rutkowski for backward stochastic differential equations (BSDEs) driven by continuous martingales, but we also derive the pricing partial differential equations (PDEs) for path‐independent contingent claims of a European style in a Markovian framework.     

A PRIMAL–DUAL ALGORITHM FOR BSDES

We generalize the primal–dual methodology, which is popular in the pricing of early‐exercise options, to a backward dynamic programming equation associated with time discretization schemes of (reflected) backward stochastic differential equations (BSDEs). Taking as an input some approximate solution of the backward dynamic program, which was precomputed, e.g., by least‐squares Monte Carlo, this methodology enables us to construct a confidence interval for the unknown true solution of the time‐discretized (reflected) BSDE at time 0. We numerically demonstrate the practical applicability of our method in two 5‐dimensional nonlinear pricing problems where tight price bounds were previously unavailable.     

基于均值-方差准则下的套期保值问题研究

当有重大信息出现时,股票价格会呈现不连续的跳跃,在股票价格服从跳-扩散过程时,研究了均值-方差准则下的套期保值问题。运用倒向随机微分方程及随机控制理论得到了均值-方差准则下的最优套期保值策略。     

LINKED RECURSIVE PREFERENCES AND OPTIMALITY

We study a class of optimization problems involving linked recursive preferences in a continuous‐time Brownian setting. Such links can arise when preferences depend directly on the level or volatility of wealth, in principal–agent (optimal compensation) problems with moral hazard, and when the impact of social influences on preferences is modeled via utility (and utility diffusion) externalities. We characterize the necessary first‐order conditions, which are also sufficient under additional conditions ensuring concavity. We also examine applications to optimal consumption and portfolio choice, and applications to Pareto optimal allocations.     

OPTIMALITY AND STATE PRICING IN CONSTRAINED FINANCIAL MARKETS WITH RECURSIVE UTILITY UNDER CONTINUOUS AND DISCONTINUOUS INFORMATION

We study marginal pricing and optimality conditions for an agent maximizing generalized recursive utility in a financial market with information generated by Brownian motion and marked point processes. The setting allows for convex trading constraints, non-tradable income, and non-linear wealth dynamics. We show that the FBSDE system of the general optimality conditions reduces to a single BSDE under translation or scale invariance assumptions, and we identify tractable applications based on quadratic BSDEs. An appendix relates the main optimality conditions to duality.     

Optimal hedging for fund and insurance managers with partially observable investment flows

All financial practitioners are working in incomplete markets full of unhedgeable risk factors. Making the situation worse, they are only equipped with imperfect information on the relevant processes. In addition to the market risk, fund and insurance managers have to be prepared for sudden and possibly contagious changes in the investment flows from their clients so that they can avoid the over- as well as under-hedging. In this work, the prices of securities, the occurrences of insured events and (possibly a network of) investment flows are used to infer their drifts and intensities by a stochastic filtering technique. We utilize the inferred information to provide the optimal hedging strategy based on the mean-variance (or quadratic) risk criterion. A BSDE approach allows a systematic derivation of the optimal strategy, which is shown to be implementable by a set of simple ODEs and standard Monte Carlo simulation. The presented framework may also be useful for manufacturers and energy firms to install an efficient overlay of dynamic hedging by financial derivatives to minimize the costs.