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166
Optimal control of execution costs
 JOURNAL OF FINANCIAL MARKETS 1 (1998) 1—50
, 1998
"... We derive dynamic optimal trading strategies that minimize the expected cost of trading a large block of equity over a fixed time horizon. Specifically, given a fixed block SM of shares to be executed within a fixed finite number of periods , and given a priceimpact function that yields the executi ..."
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Cited by 193 (3 self)
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We derive dynamic optimal trading strategies that minimize the expected cost of trading a large block of equity over a fixed time horizon. Specifically, given a fixed block SM of shares to be executed within a fixed finite number of periods , and given a priceimpact function that yields the execution price of an individual trade as a function of the shares traded and market conditions, we obtain the optimal sequence of trades as a function of market conditions — closedform expressions in some cases — that minimizes the expected cost of executing SM within periods. Our analysis is extended to the portfolio case in which price impact across stocks can have an important effect on the total cost of trading a portfolio.
Transactions costs and portfolio choice in a discretecontinuoustime setting
 Journal of Economic Dynamics and Control
, 1990
"... Abstract: This paper makes the following observation concerning a new formulation of the consumption and portfolio choice model of Merton (1971), with transactions costs. Suppose an investor observes his or her current wealth only when making a transaction, that transactions are costly, and that de ..."
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Cited by 65 (3 self)
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Abstract: This paper makes the following observation concerning a new formulation of the consumption and portfolio choice model of Merton (1971), with transactions costs. Suppose an investor observes his or her current wealth only when making a transaction, that transactions are costly, and that decisions to transact can be made at any time based on all current information. If, at each transaction, the agent is charged a fixed fraction of current portfolio value, an optimal policy exists and the optimal interval of time between transactions is fixed, independent of time and current wealth. We thank Monique Pontier, Monique JeanblancPicqué, and a referee for many suggestions and comments. Correspondence should be directed to Darrell Duffie at
Continuoustime methods in finance: A review and an assessment
 Journal of Finance
, 2000
"... I survey and assess the development of continuoustime methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. ..."
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Cited by 52 (0 self)
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I survey and assess the development of continuoustime methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. During the period 1981 to 1999 the theory has been extended and modified to better explain empirical regularities in various subfields of finance. This latter subperiod has seen significant progress in econometric theory, computational and estimation methods to test and implement continuoustime models. Capital market frictions and bargaining issues are being increasingly incorporated in continuoustime theory. THE ROOTS OF MODERN CONTINUOUSTIME METHODS in finance can be traced back to the seminal contributions of Merton ~1969, 1971, 1973b! in the late 1960s and early 1970s. Merton ~1969! pioneered the use of continuoustime modeling in financial economics by formulating the intertemporal consumption and portfolio choice problem of an investor in a stochastic dynamic programming setting.
Optimal Replication of Contingent Claims Under Portfolio Constraints
 Rev. of Financial Studies
, 1998
"... We study the problem of determining the minimum cost of superreplicating a nonnegative contingent claim when there are convex constraints on the portfolio weights. It is shown that the optimal cost with constraints is equal to the price of a related claim without constraints. The related claim is a ..."
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Cited by 49 (3 self)
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We study the problem of determining the minimum cost of superreplicating a nonnegative contingent claim when there are convex constraints on the portfolio weights. It is shown that the optimal cost with constraints is equal to the price of a related claim without constraints. The related claim is a dominating claim, i.e., a claim whose payoffs are increased in an appropriate way relative to the original claim. The results hold for a wide variety of options, including standard European and American calls and puts, multiasset options, and some pathdependent options. We also provide a somewhat similar analysis when there are constraints on the gamma of the replicating portfolio. Constraints on portfolio amounts and constraints on number of shares of assets are also considered. Optimal Replication of Contingent Claims Under Portfolio Constraints 2 Since the pioneering option pricing work of Black and Scholes (1973) and Merton (1973), much research has focused on relaxing the assumptio...
A ClosedForm Solution to the Problem of SuperReplication Under Transaction Costs
, 1997
"... We study the problem of finding the minimal price needed to dominate Europeantype contingent claims under proportional transaction costs in a continuoustime diffusion model. The result we prove has already been known in special cases  the minimal superreplicating strategy is the least expensive b ..."
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Cited by 42 (2 self)
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We study the problem of finding the minimal price needed to dominate Europeantype contingent claims under proportional transaction costs in a continuoustime diffusion model. The result we prove has already been known in special cases  the minimal superreplicating strategy is the least expensive buyandhold strategy. Our contribution consists in showing that this result remains valid for general pathindependent claims, and in providing a shorter and more intuitive, financial mathematicstype proof. It is based on a previously known representation of the minimal price as a supremum of the prices in corresponding shadow markets, and on a PDE (viscosity) characterization of that representation. Key words: transaction costs, superreplicating strategies, viscosity solutions. JEL classification: G11, G12. AMS 1991 subject classifications: Primary 90A09, 93E20, 60H30; secondary 60G44, 90A16. Research of the first author partially supported by NSF grant #DMS9503582. 1 Introductio...
Numerical Convergence Properties of Option Pricing PDEs with Uncertain Volatility
 IMA Journal of Numerical Analysis
, 2003
"... The pricing equations derived from uncertain volatility models in finance are often cast in the form of nonlinear partial differential equations. Implicit timestepping leads to a set of nonlinear algebraic equations which must be solved at each timestep. To solve these equations, an iterative approa ..."
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Cited by 38 (16 self)
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The pricing equations derived from uncertain volatility models in finance are often cast in the form of nonlinear partial differential equations. Implicit timestepping leads to a set of nonlinear algebraic equations which must be solved at each timestep. To solve these equations, an iterative approach is employed. In this paper, we prove the convergence of a particular iterative scheme for one factor uncertain volatility models. We also demonstrate how nonmonotone discretization schemes (such as standard CrankNicolson timestepping) can converge to incorrect solutions, or lead to instability. Numerical examples are provided.
Rollover risk and credit risk
 Journal of Finance
, 2012
"... Our model shows that deterioration in debt market liquidity leads to an increase in not only the liquidity premium of corporate bonds but also credit risk. The latter effect originates from firms ’ debt rollover. When liquidity deterioration causes a firm to suffer losses in rolling over its maturin ..."
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Cited by 36 (7 self)
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Our model shows that deterioration in debt market liquidity leads to an increase in not only the liquidity premium of corporate bonds but also credit risk. The latter effect originates from firms ’ debt rollover. When liquidity deterioration causes a firm to suffer losses in rolling over its maturing debt, equity holders bear the losses while maturing debt holders are paid in full. This conflict leads the firm to default at a higher fundamental threshold. Our model demonstrates an intricate interaction between the liquidity premium and default premium and highlights the role of shortterm debt in exacerbating rollover risk. THE YIELD SPREAD OF a firm’s bond relative to the riskfree interest rate directly determines the firm’s debt financing cost, and is often referred to as its credit spread. It is widely recognized that the credit spread reflects not only a default premium determined by the firm’s credit risk but also a liquidity premium due to illiquidity of the secondary debt market (e.g., Longstaff, Mithal, and Neis (2005) and Chen, Lesmond, and Wei (2007)). However, academics and policy makers tend to treat both the default premium and the liquidity premium as independent, and thus ignore interactions between them. The financial crisis of 2007 to 2008 demonstrates the importance of such an interaction— deterioration in debt market liquidity caused severe financing difficulties for many financial firms, which in turn exacerbated their credit risk. In this paper, we develop a theoretical model to analyze the interaction between debt market liquidity and credit risk through socalled rollover risk: when debt market liquidity deteriorates, firms face rollover losses from issuing new bonds to replace maturing bonds. To avoid default, equity holders need to bear the rollover losses, while maturing debt holders are paid in full. This
THE FEEDBACK EFFECT OF HEDGING IN ILLIQUID MARKETS
, 2000
"... This paper analyzes the influence of dynamic trading strategies on the prices in financial markets. After a thorough discussion of the modeling issues involved we derive the modification of the stochastic process of the underlying asset that follows from the presence of dynamic trading strategies. ..."
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Cited by 35 (1 self)
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This paper analyzes the influence of dynamic trading strategies on the prices in financial markets. After a thorough discussion of the modeling issues involved we derive the modification of the stochastic process of the underlying asset that follows from the presence of dynamic trading strategies. We analyze the nonlinear effects and the feedback from prices to trading strategy. The pricing, hedging, and replication of options in the context of illiquid markets is discussed and a nonlinear partial differential equation for an option replication strategy is derived. Finally the effects of one of the most popular trading strategies—Putoption replication—on the price of the underlying asset are illustrated using numerical simulations.
The Price of Options Illiquidity
 The Journal of Finance
, 2001
"... suggestions. Special thanks to Ken Garbade for spending many hours reading and commenting on every draft of this paper. Many thanks to the Bank of Israel and the TelAviv Stock Exchange for providing the data and responding to our innumerable questions. Finally, we thank René Stulz and the referee o ..."
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Cited by 34 (3 self)
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suggestions. Special thanks to Ken Garbade for spending many hours reading and commenting on every draft of this paper. Many thanks to the Bank of Israel and the TelAviv Stock Exchange for providing the data and responding to our innumerable questions. Finally, we thank René Stulz and the referee of this paper for their many helpful comments and suggestions. THE PRICE OF OPTIONS ILLIQUIDITY