Abstract: We study the optimal design of a menu of funds by a manager who is required to use linear pricing and does not observe the beliefs of investors regarding one of the risky assets. The optimal menu involves bundling of assets and can be constructed from the solution to a calculus of variations problem that optimizes over the indirect utility that each type receives. We provide a complete characterization of the optimal menu and show that the need to maintain incentive compatibility leads the manager to offer funds that are inefficiently tilted towards the asset that is not subject to the information friction.

Publication: Mathematical Finance Vol.: 32 No.: 2 ISSN: 0960-1627

ID: CaltechAUTHORS:20211207-155923022

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Abstract: We consider the problem of eliciting truthful responses to a survey question when the respondents share a common prior that the survey planner is agnostic about. The planner would therefore like to have a "universal” mechanism, which would induce honest answers for all possible priors. If the planner also requires a locality condition that ensures that the mechanism payoffs are determined by the respondents' posterior probabilities of the true state of nature, we prove that, under additional smoothness and sensitivity conditions, the payoff in the truth-telling equilibrium must be a logarithmic function of those posterior probabilities. Moreover, the respondents are necessarily ranked according to those probabilities. Finally, we discuss implementation issues.

Publication: Theory of Probability & Its Applications Vol.: 65 No.: 2 ISSN: 0040-585X

ID: CaltechAUTHORS:20201001-101339197

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Abstract: We consider a stochastic tournament game in which each player is rewarded based on her rank in terms of the completion time of her own task and is subject to cost of effort. When players are homogeneous and the rewards are purely rank dependent, the equilibrium has a surprisingly explicit characterization, which allows us to conduct comparative statics and obtain explicit solution to several optimal reward design problems. In the general case when the players are heterogenous and payoffs are not purely rank dependent, we prove the existence, uniqueness and stability of the Nash equilibrium of the associated mean field game, and the existence of an approximate Nash equilibrium of the finite-player game.

Publication: Annals of Applied Probability Vol.: 29 No.: 6 ISSN: 1050-5164

ID: CaltechAUTHORS:20200123-081310030

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Abstract: This paper aims to develop the insights into Bayesian truth serum (BTS) mechanism by postulating a sequence of seven natural conditions reminiscent of axioms in information theory. The condition that reduces a larger family of mechanisms to BTS is additivity, akin to the axiomatic development of entropy. The seven conditions identify BTS as the unique scoring rule for ranking respondents in situations in which respondents are asked to choose an alternative from a finite set and provide predictions of their peers’ propensities to choose, for finite or infinite sets of respondents.

Publication: IEEE Transactions on Information Theory Vol.: 65 No.: 1 ISSN: 0018-9448

ID: CaltechAUTHORS:20180906-140959870

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Abstract: We consider the problem of finding equilibrium asset prices in a financial market in which a portfolio manager (Agent) invests on behalf of an investor (Principal), who compensates the manager with an optimal contract. We extend a model from Buffa, Vayanos and Woolley (2014) by allowing general contracts, and by allowing the portfolio manager to invest privately in individual risky assets or the index. To alleviate the effect of moral hazard, Agent is optimally compensated by benchmarking to the index, which, however, may incentivize him to be too much of a “closet indexer”. To counter those incentives, the optimal contract rewards Agent for taking specific risk of individual assets in excess of the systematic risk of the index, by rewarding the deviation between the portfolio return and the return of an index portfolio, and the deviation's quadratic variation.

Publication: Journal of Economic Theory Vol.: 173ISSN: 0022-0531

ID: CaltechAUTHORS:20171114-155621178

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Abstract: We consider a general formulation of the principal–agent problem with a lump-sum payment on a finite horizon, providing a systematic method for solving such problems. Our approach is the following. We first find the contract that is optimal among those for which the agent’s value process allows a dynamic programming representation, in which case the agent’s optimal effort is straightforward to find. We then show that the optimization over this restricted family of contracts represents no loss of generality. As a consequence, we have reduced a non-zero-sum stochastic differential game to a stochastic control problem which may be addressed by standard tools of control theory. Our proofs rely on the backward stochastic differential equations approach to non-Markovian stochastic control, and more specifically on the recent extensions to the second order case.

Publication: Finance and Stochastics Vol.: 22 No.: 1 ISSN: 0949-2984

ID: CaltechAUTHORS:20171027-093026885

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Abstract: We consider a contracting problem in which a principal hires an agent to manage a risky project. When the agent chooses volatility components of the output process and the principal observes the output continuously, the principal can compute the quadratic variation of the output, but not the individual components. This leads to moral hazard with respect to the risk choices of the agent. To find the optimal contract, we develop a novel approach to solving principal–agent problems: first, we identify a family of admissible contracts for which the optimal agent’s action is explicitly characterized; then, we show that we do not lose on generality when finding the optimal contract inside this family, up to integrability conditions. To do this, we use the recent theory of singular changes of measures for Itô processes. We solve the problem in the case of CARA preferences and show that the optimal contract is linear in these factors: the contractible sources of risk, including the output, the quadratic variation of the output and the cross-variations between the output and the contractible risk sources. Thus, like sample Sharpe ratios used in practice, path-dependent contracts naturally arise when there is moral hazard with respect to risk management. In a numerical example, we show that the loss of efficiency can be significant if the principal does not use the quadratic variation component of the optimal contract.

Publication: Management Science Vol.: 63 No.: 10 ISSN: 0025-1909

ID: CaltechAUTHORS:20171117-085646065

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Abstract: We analyze a game in which a group of agents exerts costly effort over time to make progress on a project. The project is completed once the cumulative effort reaches a prespecified threshold, at which point it generates a lump-sum payoff. We characterize a budget-balanced mechanism that induces each agent to exert the first-best effort level as the outcome of a Markov perfect equilibrium, thus eliminating the free-rider problem. We also show how our mechanism can be adapted to other dynamic games with externalities, such as strategic experimentation and the dynamic extraction of a common resource.

Publication: American Economic Journal: Microeconomics Vol.: 8 No.: 4 ISSN: 1945-7669

ID: CaltechAUTHORS:20161206-105045384

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Abstract: We consider a continuous-time principal–agent model in which the agent's effort cannot be contracted upon, and both the principal and the agent may have non-standard, cumulative prospect theory type preferences. We find that the optimal contracts are likely to be “more nonlinear” than in the standard case with concave utility preferences. In the special case when the principal is risk-neutral, we show that she will offer a contract which effectively makes the agent less risk averse in the gain domain and less risk seeking in the loss domain, in order to align the agent's risk preference better with the principal's.

Publication: Journal of Mathematical Analysis and Applications Vol.: 428 No.: 2 ISSN: 0022-247X

ID: CaltechAUTHORS:20160108-095142845

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Abstract: We explore theoretically and experimentally the general equilibrium price and allocation implications of delegated portfolio management when the investor–manager relationship is nonexclusive. Our theory predicts that competition forces managers to promise portfolios that mimic Arrow–Debreu (AD) securities, which investors then combine to fit their preferences. A weak version of the capital asset pricing model (CAPM) obtains, where state prices (relative to state probabilities) implicit in prices of traded securities will be inversely ranked to aggregate wealth across states. Our experiment broadly corroborates the price and choice predictions of the theory. However, price quality deteriorates when only a few managers attract most of the available wealth. Wealth concentration increases because funds flow toward managers who offer portfolios closer to replicating AD securities (as in the theory), but also because funds flow to managers who had better performance in the immediate past (an observation unrelated to the theory).

Publication: Management Science Vol.: 61 No.: 8 ISSN: 0025-1909

ID: CaltechAUTHORS:20150828-090251047

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Abstract: We consider a market in which traders arrive at random times, with random private values for the single-traded asset. A trader's optimal trading decision is formulated in terms of exercising the option to trade one unit of the asset at the optimal stopping time. We solve the optimal stopping problem under the assumption that the market price follows a mean-reverting diffusion process. the model is calibrated to experimental data taken from Alton and Plott (Principles of continuous price determination in the experimental environment with flows of random arrivals and departures. Working paper, Caltech, 2010), resulting in a very good fit. In particular, the estimated long-term mean of the traded prices is close to the theoretical long-term mean at which the expected number of buys is equal to the expected number of sells. We call that value long-term competitive equilibrium, extending the concept of flow competitive equilibrium of Alton and Plott (Principles of continuous price determination in an experimental environment with flows of random arrivals and departures. Working paper, Caltech, 2010).

Publication: Decisions in Economics and Finance Vol.: 38 No.: 1 ISSN: 1593-8883

ID: CaltechAUTHORS:20140128-101150546

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Abstract: We consider a continuous time principal-agent model where the principal/firm compensates an agent/manager who controls the output’s exposure to risk and its expected return. Both the firm and the manager have exponential utility and can trade in a frictionless market. When the firm observes the manager’s choice of effort and volatility, there is an optimal contract that induces the manager to not hedge. In a two factor specification of the model where an index and a bond are traded, the optimal contract is linear in output and the log return of the index. We also consider a manager who receives exogenous share or option compensation and illustrate how risk taking depends on the relative size of the systematic and firm-specific risk premia of the output and index. Whilst in most cases, options induce greater risk taking than shares, we find that there are also situations under which the hedging manager may take less risk than the non-hedging manager.

Publication: Mathematics and Financial Economics Vol.: 8 No.: 4 ISSN: 1862-9679

ID: CaltechAUTHORS:20170616-103207801

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Abstract: We provide a representation for the nonmyopic optimal portfolio of an agent consuming only at the terminal horizon when the single state variable follows a general diffusion process and the market consists of one risky asset and a risk-free asset. The key term of our representation is a new object that we call the “rate of macroeconomic fluctuation” whose properties are fundamental for the portfolio dynamics. We show that, under natural cyclicality conditions, (i) the agent’s hedging demand is positive (negative) when the product of his prudence and risk tolerance is below (above) two and (ii) the portfolio weights decrease in risk aversion. We apply our results to study a general continuous-time capital asset pricing model and show that under the same cyclicality conditions, the market price of risk is countercyclical and the price of the risky asset exhibits excess volatility.

Publication: Mathematics and Financial Economics Vol.: 8 No.: 1 ISSN: 1862-9679

ID: CaltechAUTHORS:20170622-142956786

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Abstract: We study consequences of regulatory interventions in limit order markets that aim at stabilizing the market after an occurrence of a “flash crash”. We use a simulation platform that creates random arrivals of trade orders, that allows us to analyze subtle theoretical features of liquidity and price variability under various market structures. The simulations are performed under continuous double-auction microstructure, and under alternatives, including imposing minimum resting times, shutting off trading for a period of time, and switching to call auction mechanisms. We find that the latter is the most effective in restoring the liquidity of the book and recovery of the price level. However, one has to be cautious about possible consequences of the intervention on the traders’ strategies, including an undesirable slowdown of a convergence to a new equilibrium after a change in fundamentals.

Publication: Journal of Applied Economics Vol.: 16 No.: 2 ISSN: 1514-0326

ID: CaltechAUTHORS:20140103-132748356

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Abstract: In recent years there has been a significant increase of interest in continuous-time Principal-Agent models, or contract theory, and their applications. Continuous-time models provide a powerful and elegant framework for solving stochastic optimization problems of finding the optimal contracts between two parties, under various assumptions on the information they have access to, and the effect they have on the underlying "profit/loss" values. This monograph surveys recent results of the theory in a systematic way, using the approach of the so-called Stochastic Maximum Principle, in models driven by Brownian Motion. Optimal contracts are characterized via a system of Forward-Backward Stochastic Differential Equations. In a number of interesting special cases these can be solved explicitly, enabling derivation of many qualitative economic conclusions.

ID: CaltechAUTHORS:20200203-133006886

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Abstract: We study consequences of regulatory interventions in limit order markets that aim at stabilizing the market after an occurrence of a "flash crash". We use a simulation platform that creates random arrivals of trade orders, that allows us to analyze subtle features of liquidity and price variability under various market structures. The simulations are performed under continuous double-auction microstructure, and under alternatives, including imposing minimum resting times, shutting off trading for a period of time, and switching to call auction mechanisms. We find that the latter is the most effective in restoring the liquidity of the book and recovery of the price level. However, one has to be cautious about possible long term consequences of the intervention on the traders’ strategies.

No.: 1365
ID: CaltechAUTHORS:20170727-091614199

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Abstract: We consider a model of correlated defaults in which the default times of multiple entities depend not only on common and specific factors, but also on the extent of past defaults in the market, via the average loss process, including the average number of defaults as a special case. The paper characterizes the average loss process when the number of entities becomes large, showing that under some monotonicity conditions the limiting average loss process can be determined by a fixed point problem.We also show that the Law of Large Numbers holds under certain compatibility conditions.

Publication: Stochastic Processes and their Applications Vol.: 122 No.: 8 ISSN: 0304-4149

ID: CaltechAUTHORS:20120813-111328071

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Abstract: We find the optimal time for entering a joint venture by two firms, and the optimal linear contract for sharing the profits. We consider risk-sharing, timing-incentive and asymmetric decisions contract designs. If the firms are risk-neutral and the cash payments are allowed, all three designs are equivalent. With risk aversion, the optimal contract parameters may vary significantly across the three designs and across varying levels of risk aversion. We also analyze a dataset of joint biomedical ventures, in which, in agreement with our theoretical predictions, both royalty and cash payments are mostly increasing in the smaller firm's experience, and the time of entry happens sooner for more experienced small firms.

Publication: Journal of Economic Dynamics and Control Vol.: 35 No.: 10 ISSN: 0165-1889

ID: CaltechAUTHORS:20120228-151158196

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Abstract: Standard optimal portfolio selection models take no account of the special information that active investors believe they possess. For example, active investors who believe they can place bounds on the price of a security will want to use that information when assessing risk and expected return in order to construct an optimal portfolio. In this paper, we use two continuous-time models to analyze how placing boundaries on the price of a stock affects assessed risk, expected returns, and the optimal holdings of an active investor, and how those vary as a function of the relation between the stock price and the boundaries. In particular, the optimal strategy takes significant long/short positions as the price nears its lower/upper boundary.

Publication: Annals of Finance Vol.: 6ISSN: 1614-2446

ID: CaltechAUTHORS:20190821-103126134

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Abstract: In all the existing literature on survival in heterogeneous economies, the rate at which an agent vanishes in the long run relative to another agent can be characterized by the difference of the so-called survival indices, where each survival index only depends on the preferences of the corresponding agent and the properties of the aggregate endowment. In particular, one agent experiences extinction relative to another (that is, the wealth ratio of the two agents goes to zero) if and only if she has a smaller survival index. We consider a simple complete market model and show that the survival index is more complex if there are more than two agents in the economy. In fact, the following phenomenon may take place: even if agent one experiences extinction relative to agent two, adding a third agent to the economy may reverse the situation and force the agent two to experience extinction relative to agent one. We also calculate the rates of convergence.

Publication: B. E. Journal of Theoretical Economics Vol.: 10 No.: 1 ISSN: 1935-1704

ID: CaltechAUTHORS:20100712-090750198

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Abstract: We propose a structural model for the valuation of defaultable securities of a firm which models the effect of deliberate misreporting done by insiders in the firm and unobserved by others. We derive exact formulas for equity and bond prices and approximate expressions for the conditional default probability, recovery rate, and credit spread under the proposed credit risk framework. We propose a novel estimation approach to structural model estimation which accounts for noisy observed asset values. We apply the proposed method to calibrate a simple version of our model to the case of Parmalat and show that the model is able to recover a certain amount of misreporting during the years of accounting irregularities.

Publication: International Journal of Theoretical and Applied Finance Vol.: 12 No.: 1 ISSN: 0219-0249

ID: CaltechAUTHORS:20090820-142436958

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Abstract: We consider a problem of finding optimal contracts in continuous time, when the agent’s actions are unobservable by the principal, who pays the agent with a one-time payoff at the end of the contract. We fully solve the case of quadratic cost and separable utility, for general utility functions. The optimal contract is, in general, a nonlinear function of the final outcome only, while in the previously solved cases, for exponential and linear utility functions, the optimal contract is linear in the final output value. In a specific example we compute, the first-best principal’s utility is infinite, while it becomes finite with hidden action, which is increasing in value of the output. In the second part of the paper we formulate a general mathematical theory for the problem. We apply the stochastic maximum principle to give necessary conditions for optimal contracts. Sufficient conditions are hard to establish, but we suggest a way to check sufficiency using non-convex optimization.

Publication: Applied Mathematics and Optiumization Vol.: 59 No.: 1 ISSN: 0095-4616

ID: CaltechAUTHORS:CVIamo09

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Abstract: We consider the problem of when to deliver the contract payoff, in a continuous-time principal-agent setting, in which the agent's effort is unobservable. The principal can design contracts of a simple form that induce the agent to ask for the payoff at the time of the principal's choosing. The optimal time of payment depends on the agent's and the principal's outside options. We develop a theory for general utility functions, while with CARA utilities we are able to specify conditions under which the optimal payment time is not random. However, in general, the optimal payment time is typically random. One illustrative application is the case when the agent can be fired, after having been paid a severance payment, and then replaced by another agent. The methodology we use is the stochastic maximum principle and its link to Forward-Backward Stochastic Differential Equations.

Publication: B.E. Journal of Theoretical Economics Vol.: 8 No.: 1 ISSN: 1935-1704

ID: CaltechAUTHORS:CVIbejte08

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Abstract: We model the risky asset as driven by a pure jump process, with non-trivial and tractable higher moments. We compute the optimal portfolio strategy of an investor with CRRA utility and study the sensitivity of the investment in the risky asset to the higher moments, as well as the resulting wealth loss from ignoring higher moments. We find that ignoring higher moments can lead to significant overinvestment in risky securities, especially when volatility is high.

Publication: Annals of Finance Vol.: 4 No.: 1 ISSN: 1614-2446

ID: CaltechAUTHORS:20190826-124741137

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Abstract: We consider continuous-time models in which the agent is paid at the end of the time horizon by the principal, who does not know the agent’s type. The agent dynamically affects either the drift of the underlying output process, or its volatility. The principal’s problem reduces to a calculus of variation problem for the agent’s level of utility. The optimal ratio of marginal utilities is random, via dependence on the underlying output process. When the agent affects the drift only, in the risk- neutral case lower volatility corresponds to the more incentive optimal contract for the smaller range of agents who get rent above the reservation utility. If only the volatility is affected, the optimal contract is necessarily non-incentive, unlike in the first-best case. We also suggest a procedure for finding simple and reasonable contracts, which, however, are not necessarily optimal.

Publication: Mathematics and Financial Economics Vol.: 1 No.: 1 ISSN: 1862-9679

ID: CaltechAUTHORS:20190828-102317415

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Abstract: We consider first-best risk-sharing problems in which “the agent” can control both the drift (effort choice) and the volatility of the underlying process (project selection). In a model of delegated portfolio management, it is optimal to compensate the manager with an option-type payoff, where the functional form of the option is obtained as a solution to an ordinary differential equation. In the general case, the optimal contract is a fixed point of a functional that connects the agent's and the principal's maximization problems. We apply martingale/duality methods familiar from optimal consumption-investment problems.

Publication: Journal of Economic Theory Vol.: 133 No.: 1 ISSN: 0022-0531

ID: CaltechAUTHORS:20100909-143417799

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Abstract: This paper is concerned with nonlinear filtering of the coefficients in asset price models with stochastic volatility. More specifically, we assume that the asset price process S=(St)t≥0 is given by dSt=m(θt)St dt+v(θt)St dBt, where B=(Bt)t≥0 is a Brownian motion, v is a positive function and θ=(θt)t≥0 is a cádlág strong Markov process. The random process θ is unobservable. We assume also that the asset price St is observed only at random times 0<τ1<τ2< ... . This is an appropriate assumption when modeling high frequency financial data (e.g., tick-by-tick stock prices). In the above setting the problem of estimation of θ can be approached as a special nonlinear filtering problem with measurements generated by a multivariate point process (τk, log Sτk). While quite natural, this problem does not fit into the “standard” diffusion or simple point process filtering frameworks and requires more technical tools. We derive a closed form optimal recursive Bayesian filter for θt, based on the observations of (τk, log Sτk)k≥1. It turns out that the filter is given by a recursive system that involves only deterministic Kolmogorov-type equations, which should make the numerical implementation relatively easy.

Publication: Annals of Applied Probability Vol.: 16 No.: 3 ISSN: 1050-5164

ID: CaltechAUTHORS:CVIaap06

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Abstract: We present a unified approach to solving contracting problems with full information in models driven by Brownian motion. We apply the stochastic maximum principle to give necessary and sufficient conditions for contracts that implement the so-called first-best solution. The optimal contract is proportional to the difference between the underlying process controlled by the agent and a stochastic, state-contingent benchmark. Our methodology covers a number of frameworks considered in the existing literature. The main finance applications of this theory are optimal compensation of company executives and of portfolio managers.

Publication: Journal of Applied Mathematics and Stochastic Analysis Vol.: 2006ISSN: 1048-9533

ID: CaltechAUTHORS:CVIjamsa06

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Abstract: This paper aims to open a door to Monte-Carlo methods for numerically solving Forward-Backward SDEs, without computing over all Cartesian grids as usually done in the literature. We transform the FBSDE to a control problem and propose the steepest descent method to solve the latter one. We show that the original (coupled) FBSDE can be approximated by {it decoupled} FBSDEs, which further comes down to computing a sequence of conditional expectations. The rate of convergence is obtained, and the key to its proof is a new well-posedness result for FBSDEs. However, the approximating decoupled FBSDEs are non-Markovian. Some Markovian type of modification is needed in order to make the algorithm efficiently implementable.

Publication: Electronic Journal of Probability Vol.: 10 No.: 45 ISSN: 1083-6489

ID: CaltechAUTHORS:CVIejp05

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Abstract: We introduce a method that relies exclusively on Monte Carlo simulation in order to compute numerically optimal portfolio values for utility maximization problems. Our method is quite general and only requires complete markets and knowledge of the dynamics of the security processes. It can be applied regardless of the number of factors and of whether the agent derives utility from intertemporal consumption, terminal wealth or both. We also perform some comparative statics analysis. Our comparative statics show that risk aversion has by far the greatest influence on the value of the optimal portfolio.

Publication: Journal of Economic Dynamics and Control Vol.: 27 No.: 6 ISSN: 0165-1889

ID: CaltechAUTHORS:20111010-091503480

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Abstract: We consider the problem of computing hedging portfolios for options that may have discontinuous payoffs, in the framework of diffusion models in which the number of factors may be larger than the number of Brownian motions driving the model. Extending the work of Fournié et al. (1999), as well as Ma and Zhang (2000), using integration by parts of Malliavin calculus, we find two representations of the hedging portfolio in terms of expected values of random variables that do not involve differentiating the payoff function. Once this has been accomplished, the hedging portfolio can be computed by simple Monte Carlo. We find the theoretical bound for the error of the two methods. We also perform numerical experiments in order to compare these methods to two existing methods, and find that no method is clearly superior to others.

Publication: Mathematical Finance Vol.: 13 No.: 1 ISSN: 0960-1627

ID: CaltechAUTHORS:20111007-134050070

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Abstract: This paper solves the following problem of mathematical finance: to find a solution to the problem of maximizing utility from terminal wealth of an agent with a random endowment process, in the general, semimartingale model for incomplete markets, and to characterize it via the associated dual problem. We show that this is possible if the dual problem and its domain are carefully defined. More precisely, we show that the optimal terminal wealth is equal to the inverse of marginal utility evaluated at the solution to the dual problem, which is in the form of the regular part of an element of (L^∞)∗ (the dual space of L^∞).

Publication: Finance and Stochastics Vol.: 5 No.: 2 ISSN: 0949-2984

ID: CaltechAUTHORS:20170614-102418542

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Abstract: In this paper we study a class of forward-backward stochastic differential equations with reflecting boundary conditions (FBSDER for short). More precisely, we consider the case in which the forward component of the FBSDER is restricted to a fixed, convex region, and the backward component will stay, at each fixed time, in a convex region that may depend on time and is possibly random. The solvability of such FBSDER is studied in a fairly general way. We also prove that if the coefficients are all deterministic and the backward equation is one-dimensional, then the adapted solution of such FBSDER will give the viscosity solution of a quasilinear variational inequality (obstacle problem) with a Neumann boundary condition. As an application, we study how the solvability of FBSDERs is related to the solvability of an American game option.

Publication: Journal of Applied Mathematics and Stochastic Analysis Vol.: 14 No.: 2 ISSN: 1048-9533

ID: CaltechAUTHORS:MAJjamsa01

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Abstract: In the framework of continuous-time, Itô processes models for financial markets, we study the problem of maximizing the probability of an agent's wealth at time T being no less than the value C of a contingent claim with expiration time T. The solution to the problem has been known in the context of complete markets and recently also for incomplete markets; we rederive the complete markets solution using a powerful and simple duality method, developed in utility maximization literature. We then show how to modify this approach to solve the problem in a market with partial information, the one in which we have only a prior distribution on the vector of return rates of the risky assets. Finally, the same problem is solved in markets in which the wealth process of the agent has a nonlinear drift. These include the case of different borrowing and lending rates, as well as "large investor" models. We also provide a number of explicitly solved examples.

Publication: Annals of Applied Probability Vol.: 9 No.: 4 ISSN: 1050-5164

ID: CaltechAUTHORS:SPIaap99

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Abstract: We study a financial market with incompleteness arising from two sources: stochastic volatility and portfolio constraints. The latter are given in terms of bounds imposed on the borrowing and short-selling of a `hedger' in this market, and can be described by a closed convex set K. We find explicit characterizations of the minimal price needed to super-replicate European-type contingent claims in this framework. The results depend on whether the volatility is bounded away from zero and/or infinity, and also, on if we have linear dynamics for the stock price process, and whether volatility process depends on the stock price. We use a previously known representation of the minimal price as a supremum of the prices in the corresponding shadow markets, and we derive a PDE characterization of that representation.

Publication: Journal of Applied Probability Vol.: 36 No.: 2 ISSN: 0021-9002

ID: CaltechAUTHORS:CVIjap99

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Abstract: We consider backward stochastic differential equations with convex constraints on the gains (or intensity-of-noise) process. Existence and uniqueness of a minimal solution are established in the case of a drift coefficient which is Lipschitz continuous in the state and gains processes and convex in the gains process. It is also shown that the minimal solution can be characterized as the unique solution of a functional stochastic control-type equation. This representation is related to the penalization method for constructing solutions of stochastic differential equations, involves change of measure techniques, and employs notions and results from convex analysis, such as the support function of the convex set of constraints and its various properties.

Publication: Annals of Probability Vol.: 26 No.: 4 ISSN: 0091-1798

ID: CaltechAUTHORS:20111220-131043977

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Abstract: We establish existence and uniqueness results for adapted solutions of backward stochastic differential equations (BSDE's) with two reflecting barriers, generalizing the work of El Karoui, Kapoudjian, Pardoux, Peng and Quenez. Existence is proved first by solving a related pair of coupled optimal stopping problems, and then, under different conditions, via a penalization method. It is also shown that the solution coincides with the value of a certain Dynkin game, a stochastic game of optimal stopping. Moreover, the connection with the backward SDE enables us to provide a pathwise (deterministic) approach to the game.

Publication: Annals of Probability Vol.: 24 No.: 4 ISSN: 0091-1798

ID: CaltechAUTHORS:20120201-094412791

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Abstract: In the classical continuous-time financial market model, stock prices have been understood as solutions to linear stochastic differential equations, and an important problem to solve is the problem of hedging options (functions of the stock price values at the expiration date). In this paper we consider the hedging problem not only with a price model that is nonlinear, but also with coefficients of the price equations that can depend on the portfolio strategy and the wealth process of the hedger. In mathematical terminology, the problem translates to solving a forward-backward stochastic differential equation with the forward diffusion part being degenerate. We show that, under reasonable conditions, the four step scheme of Ma, Protter and Yong for solving forward-backward SDE's still works in this case, and we extend the classical results of hedging contingent claims to this new model. Included in the examples is the case of the stock volatility increase caused by overpricing the option, as well as the case of different interest rates for borrowing and lending.

Publication: Annals of Applied Probability Vol.: 6 No.: 2 ISSN: 1050-5164

ID: CaltechAUTHORS:20120209-085609546

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Abstract: Conventional wisdom holds that since continuous-time, Black-Scholes hedging is infinitely expensive in a model with proportional transaction costs, there is no continuous-time strategy which hedges a European call option perfectly. Of course, if one is attempting to dominate the European call rather than replicate it, then one can use the trivial strategy of buying one share of the underlying stock and holding to maturity. In this paper we prove that this is, in fact, the least expensive method of dominating a European call in a Black-Scholes model with proportional transaction costs.

Publication: Annals of Applied Probability Vol.: 5 No.: 2 ISSN: 1050-5164

ID: CaltechAUTHORS:SONaap95

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Abstract: We employ a stochastic control approach to study the question of hedging contingent claims by portfolios constrained to take values in a given closed, convex subset of R^d. In the framework of our earlier work for utility maximization with constrained portfolios, we extend results of El Karoui and Quenez on incomplete markets and treat the case of different interest rates for borrowing and lending.

Publication: Annals of Applied Probability Vol.: 3 No.: 3 ISSN: 1050-5164

ID: CaltechAUTHORS:20120309-140207905

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Abstract: We study the stochastic control problem of maximizing expected utility from terminal wealth and/or consumption, when the portfolio is constrained to take values in a given closed, convex subset of R^d. The setting is that of a continuous-time, Itô process model for the underlying asset prices. General existence results are established for optimal portfolio/consumption strategies, by suitably embedding the constrained problem in an appropriate family of unconstrained ones, and finding a member of this family for which the corresponding optimal policy obeys the constraints. Equivalent conditions for optimality are obtained, and explicit solutions leading to feedback formulae are derived for special utility functions and for deterministic coefficients. Results on incomplete markets, on short-selling constraints and on different interest rates for borrowing and lending are covered as special cases. The mathematical tools are those of continuous-time martingales, convex analysis and duality theory.

Publication: Annals of Applied Probability Vol.: 2 No.: 4 ISSN: 1050-5164

ID: CaltechAUTHORS:20120316-103944783

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