Optimal Portfolio Choice and the Valuation of Illiquid Securities
TL;DR: In this article, the authors analyze a model where investors are restricted to trading strategies that are of bounded variation, and show that large price discounts can be sustained in a rational model.
Abstract: Traditional models of portfolio choice assume that investors can continuously trade unlimited amounts of securities. In reality, investors face liquidity constraints. I analyze a model where investors are restricted to trading strategies that are of bounded variation. An investor facing this type of illiquidity behaves very differently from an unconstrained investor. A liquidity-constrained investor endogenously acts as if facing borrowing and short-selling constraints, and one may take riskier positions than in liquid markets. I solve for the shadow cost of illiquidity and show that large price discounts can be sustained in a rational model. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.
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TL;DR: In this paper, the authors review the theories on how liquidity affects the required returns of capital assets and the empirical studies that test these theories and find the effects of liquidity on asset prices to be statistically significant and economically important, controlling for traditional risk measures and asset characteristics.
Abstract: We review the theories on how liquidity affects the required returns of capital assets and the empirical studies that test these theories. The theory predicts that both the level of liquidity and liquidity risk are priced, and empirical studies find the effects of liquidity on asset prices to be statistically significant and economically important, controlling for traditional risk measures and asset characteristics. Liquidity-based asset pricing empirically helps explain (1) the cross-section of stock returns, (2) how a reduction in stock liquidity result in a reduction in stock prices and an increase in expected stock returns, (3) the yield differential between on- and off-the-run Treasuries, (4) the yield spreads on corporate bonds, (5) the returns on hedge funds, (6) the valuation of closed-end funds, and (7) the low price of certain hard-to-trade securities relative to more liquid counterparts with identical cash flows, such as restricted stocks or illiquid derivatives. Liquidity can thus play a role in resolving a number of asset pricing puzzles such as the small-firm effect, the equity premium puzzle, and the risk-free rate puzzle.
625 citations
Cites background from "Optimal Portfolio Choice and the Va..."
...Longstaff (2001) considers a continuous-time model in which an investor must limit his trading intensity, thus capturing the idea that investors cannot unwind a position immediately....
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...Longstaff (2004) provides another test of the effect of liquidity on bond yields by comparing yields on U....
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Book•
01 Dec 2005TL;DR: In this paper, the authors review the theories on how liquidity affects the required returns of capital assets and the empirical studies that test these theories and find the effects of liquidity on asset prices to be statistically significant and economically important, controlling for traditional risk measures and asset characteristics.
Abstract: We review the theories on how liquidity affects the required returns of capital assets and the empirical studies that test these theories. The theory predicts that both the level of liquidity and liquidity risk are priced, and empirical studies find the effects of liquidity on asset prices to be statistically significant and economically important, controlling for traditional risk measures and asset characteristics. Liquidity-based asset pricing empirically helps explain (1) the cross-section of stock returns, (2) how a reduction in stock liquidity result in a reduction in stock prices and an increase in expected stock returns, (3) the yield differential between on- and off-the-run Treasuries, (4) the yield spreads on corporate bonds, (5) the returns on hedge funds, (6) the valuation of closed-end funds, and (7) the low price of certain hard-to-trade securities relative to more liquid counterparts with identical cash flows, such as restricted stocks or illiquid derivatives. Liquidity can thus play a role in resolving a number of asset pricing puzzles such as the small-firm effect, the equity premium puzzle, and the risk-free rate puzzle.
572 citations
TL;DR: In this article, the authors present evidence on the relation between hedge fund returns and restrictions imposed by funds that limit the liquidity of fund investors, and they find that the excess returns of funds with lockup restrictions are approximately 4-7% per year higher than those of non-lockup funds.
519 citations
TL;DR: In this article, the authors study the implications of jumps in prices and volatility on investment strategies and provide an analytical solution to the optimal portfolio problem, finding that event risk dramatically affects the optimal strategy.
Abstract: An inherent risk facing investors in financial markets is that a major event may trigger a large abrupt change in stock prices and market volatility. This paper studies the implications of jumps in prices and volatility on investment strategies. Using the event-risk framework of Duffie, Pan, and Singleton, we provide an analytical solution to the optimal portfolio problem. We find that event risk dramatically affects the optimal strategy. An investor facing event risk is less willing to take leveraged or short positions. In addition, the investor acts as if some portion of his wealth may become illiquid and the optimal strategy blends elements of both dynamic and buy-and-hold portfolio strategies. Jumps in prices and volatility both have an important influence on the optimal strategy.
387 citations
TL;DR: This paper examined whether there is a flight-to-liquidity premium in Treasury bond prices by comparing them with prices of bonds issued by Refcorp, a U.S. Government agency.
Abstract: We examine whether there is a flight-to-liquidity premium in Treasury bond prices by comparing them with prices of bonds issued by Refcorp, a U.S. Government agency. Since Refcorp bonds are, in effect, guaranteed by the Treasury, they have the same credit as Treasury bonds. We find a large liquidity premium in Treasury bonds, which can be more than fifteen percent of the value of some Treasury bonds. We find strong evidence that this liquidity premium is related to changes in consumer con- fidence, flows into equity and money market mutual funds, and changes in foreign ownership of Treasury debt. This suggests that the popularity of Treasury bonds directly affects their value
364 citations
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TL;DR: In this paper, a theoretical valuation formula for options is derived, based on the assumption that options are correctly priced in the market and it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks.
Abstract: If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks. Using this principle, a theoretical valuation formula for options is derived. Since almost all corporate liabilities can be viewed as combinations of options, the formula and the analysis that led to it are also applicable to corporate liabilities such as common stock, corporate bonds, and warrants. In particular, the formula can be used to derive the discount that should be applied to a corporate bond because of the possibility of default.
28,434 citations
"Optimal Portfolio Choice and the Va..." refers background in this paper
...This assumption also underlies standard option pricing theory, such as Black and Scholes (1973), Merton (1973b), Harrison and Kreps (1979), and Harrison and Pliska (1981) where the number of shares of stock needed to replicate an option generally follows a stochastic process of unbounded variation, implying that infinite amounts of the stock must be traded....
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Book•
12 Sep 2011
TL;DR: In this paper, the authors deduced a set of restrictions on option pricing formulas from the assumption that investors prefer more to less, which are necessary conditions for a formula to be consistent with a rational pricing theory.
Abstract: The long history of the theory of option pricing began in 1900 when the French mathematician Louis Bachelier deduced an option pricing formula based on the assumption that stock prices follow a Brownian motion with zero drift. Since that time, numerous researchers have contributed to the theory. The present paper begins by deducing a set of restrictions on option pricing formulas from the assumption that investors prefer more to less. These restrictions are necessary conditions for a formula to be consistent with a rational pricing theory. Attention is given to the problems created when dividends are paid on the underlying common stock and when the terms of the option contract can be changed explicitly by a change in exercise price or implicitly by a shift in the investment or capital structure policy of the firm. Since the deduced restrictions are not sufficient to uniquely determine an option pricing formula, additional assumptions are introduced to examine and extend the seminal Black-Scholes theory of option pricing. Explicit formulas for pricing both call and put options as well as for warrants and the new "down-and-out" option are derived. The effects of dividends and call provisions on the warrant price are examined. The possibilities for further extension of the theory to the pricing of corporate liabilities are discussed.
9,635 citations
Book•
01 Jan 1987
TL;DR: In this paper, the authors present a characterization of continuous local martingales with respect to Brownian motion in terms of Markov properties, including the strong Markov property, and a generalized version of the Ito rule.
Abstract: 1 Martingales, Stopping Times, and Filtrations.- 1.1. Stochastic Processes and ?-Fields.- 1.2. Stopping Times.- 1.3. Continuous-Time Martingales.- A. Fundamental inequalities.- B. Convergence results.- C. The optional sampling theorem.- 1.4. The Doob-Meyer Decomposition.- 1.5. Continuous, Square-Integrable Martingales.- 1.6. Solutions to Selected Problems.- 1.7. Notes.- 2 Brownian Motion.- 2.1. Introduction.- 2.2. First Construction of Brownian Motion.- A. The consistency theorem.- B. The Kolmogorov-?entsov theorem.- 2.3. Second Construction of Brownian Motion.- 2.4. The SpaceC[0, ?), Weak Convergence, and Wiener Measure.- A. Weak convergence.- B. Tightness.- C. Convergence of finite-dimensional distributions.- D. The invariance principle and the Wiener measure.- 2.5. The Markov Property.- A. Brownian motion in several dimensions.- B. Markov processes and Markov families.- C. Equivalent formulations of the Markov property.- 2.6. The Strong Markov Property and the Reflection Principle.- A. The reflection principle.- B. Strong Markov processes and families.- C. The strong Markov property for Brownian motion.- 2.7. Brownian Filtrations.- A. Right-continuity of the augmented filtration for a strong Markov process.- B. A "universal" filtration.- C. The Blumenthal zero-one law.- 2.8. Computations Based on Passage Times.- A. Brownian motion and its running maximum.- B. Brownian motion on a half-line.- C. Brownian motion on a finite interval.- D. Distributions involving last exit times.- 2.9. The Brownian Sample Paths.- A. Elementary properties.- B. The zero set and the quadratic variation.- C. Local maxima and points of increase.- D. Nowhere differentiability.- E. Law of the iterated logarithm.- F. Modulus of continuity.- 2.10. Solutions to Selected Problems.- 2.11. Notes.- 3 Stochastic Integration.- 3.1. Introduction.- 3.2. Construction of the Stochastic Integral.- A. Simple processes and approximations.- B. Construction and elementary properties of the integral.- C. A characterization of the integral.- D. Integration with respect to continuous, local martingales.- 3.3. The Change-of-Variable Formula.- A. The Ito rule.- B. Martingale characterization of Brownian motion.- C. Bessel processes, questions of recurrence.- D. Martingale moment inequalities.- E. Supplementary exercises.- 3.4. Representations of Continuous Martingales in Terms of Brownian Motion.- A. Continuous local martingales as stochastic integrals with respect to Brownian motion.- B. Continuous local martingales as time-changed Brownian motions.- C. A theorem of F. B. Knight.- D. Brownian martingales as stochastic integrals.- E. Brownian functionals as stochastic integrals.- 3.5. The Girsanov Theorem.- A. The basic result.- B. Proof and ramifications.- C. Brownian motion with drift.- D. The Novikov condition.- 3.6. Local Time and a Generalized Ito Rule for Brownian Motion.- A. Definition of local time and the Tanaka formula.- B. The Trotter existence theorem.- C. Reflected Brownian motion and the Skorohod equation.- D. A generalized Ito rule for convex functions.- E. The Engelbert-Schmidt zero-one law.- 3.7. Local Time for Continuous Semimartingales.- 3.8. Solutions to Selected Problems.- 3.9. Notes.- 4 Brownian Motion and Partial Differential Equations.- 4.1. Introduction.- 4.2. Harmonic Functions and the Dirichlet Problem.- A. The mean-value property.- B. The Dirichlet problem.- C. Conditions for regularity.- D. Integral formulas of Poisson.- E. Supplementary exercises.- 4.3. The One-Dimensional Heat Equation.- A. The Tychonoff uniqueness theorem.- B. Nonnegative solutions of the heat equation.- C. Boundary crossing probabilities for Brownian motion.- D. Mixed initial/boundary value problems.- 4.4. The Formulas of Feynman and Kac.- A. The multidimensional formula.- B. The one-dimensional formula.- 4.5. Solutions to selected problems.- 4.6. Notes.- 5 Stochastic Differential Equations.- 5.1. Introduction.- 5.2. Strong Solutions.- A. Definitions.- B. The Ito theory.- C. Comparison results and other refinements.- D. Approximations of stochastic differential equations.- E. Supplementary exercises.- 5.3. Weak Solutions.- A. Two notions of uniqueness.- B. Weak solutions by means of the Girsanov theorem.- C. A digression on regular conditional probabilities.- D. Results of Yamada and Watanabe on weak and strong solutions.- 5.4. The Martingale Problem of Stroock and Varadhan.- A. Some fundamental martingales.- B. Weak solutions and martingale problems.- C. Well-posedness and the strong Markov property.- D. Questions of existence.- E. Questions of uniqueness.- F. Supplementary exercises.- 5.5. A Study of the One-Dimensional Case.- A. The method of time change.- B. The method of removal of drift.- C. Feller's test for explosions.- D. Supplementary exercises.- 5.6. Linear Equations.- A. Gauss-Markov processes.- B. Brownian bridge.- C. The general, one-dimensional, linear equation.- D. Supplementary exercises.- 5.7. Connections with Partial Differential Equations.- A. The Dirichlet problem.- B. The Cauchy problem and a Feynman-Kac representation.- C. Supplementary exercises.- 5.8. Applications to Economics.- A. Portfolio and consumption processes.- B. Option pricing.- C. Optimal consumption and investment (general theory).- D. Optimal consumption and investment (constant coefficients).- 5.9. Solutions to Selected Problems.- 5.10. Notes.- 6 P. Levy's Theory of Brownian Local Time.- 6.1. Introduction.- 6.2. Alternate Representations of Brownian Local Time.- A. The process of passage times.- B. Poisson random measures.- C. Subordinators.- D. The process of passage times revisited.- E. The excursion and downcrossing representations of local time.- 6.3. Two Independent Reflected Brownian Motions.- A. The positive and negative parts of a Brownian motion.- B. The first formula of D. Williams.- C. The joint density of (W(t), L(t), ? +(t)).- 6.4. Elastic Brownian Motion.- A. The Feynman-Kac formulas for elastic Brownian motion.- B. The Ray-Knight description of local time.- C. The second formula of D. Williams.- 6.5. An Application: Transition Probabilities of Brownian Motion with Two-Valued Drift.- 6.6. Solutions to Selected Problems.- 6.7. Notes.
8,639 citations
TL;DR: In this paper, a closed-form solution for the price of a European call option on an asset with stochastic volatility is derived based on characteristi c functions and can be applied to other problems.
Abstract: I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spotasset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset’s price is important for explaining return skewness and strike-price biases in the BlackScholes (1973) model. The solution technique is based on characteristi c functions and can be applied to other problems.
7,867 citations
TL;DR: In this paper, the authors use an intertemporal general equilibrium asset pricing model to study the term structure of interest rates and find that anticipations, risk aversion, investment alternatives, and preferences about the timing of consumption all play a role in determining bond prices.
Abstract: This paper uses an intertemporal general equilibrium asset pricing model to study the term structure of interest rates. In this model, anticipations, risk aversion, investment alternatives, and preferences about the timing of consumption all play a role in determining bond prices. Many of the factors traditionally mentioned as influencing the term structure are thus included in a way which is fully consistent with maximizing behavior and rational expectations. The model leads to specific formulas for bond prices which are well suited for empirical testing. 1. INTRODUCTION THE TERM STRUCTURE of interest rates measures the relationship among the yields on default-free securities that differ only in their term to maturity. The determinants of this relationship have long been a topic of concern for economists. By offering a complete schedule of interest rates across time, the term structure embodies the market's anticipations of future events. An explanation of the term structure gives us a way to extract this information and to predict how changes in the underlying variables will affect the yield curve. In a world of certainty, equilibrium forward rates must coincide with future spot rates, but when uncertainty about future rates is introduced the analysis becomes much more complex. By and large, previous theories of the term structure have taken the certainty model as their starting point and have proceeded by examining stochastic generalizations of the certainty equilibrium relationships. The literature in the area is voluminous, and a comprehensive survey would warrant a paper in itself. It is common, however, to identify much of the previous work in the area as belonging to one of four strands of thought. First, there are various versions of the expectations hypothesis. These place predominant emphasis on the expected values of future spot rates or holdingperiod returns. In its simplest form, the expectations hypothesis postulates that bonds are priced so that the implied forward rates are equal to the expected spot rates. Generally, this approach is characterized by the following propositions: (a) the return on holding a long-term bond to maturity is equal to the expected return on repeated investment in a series of the short-term bonds, or (b) the expected rate of return over the next holding period is the same for bonds of all maturities. The liquidity preference hypothesis, advanced by Hicks [16], concurs with the importance of expected future spot rates, but places more weight on the effects of the risk preferences of market participants. It asserts that risk aversion will cause forward rates to be systematically greater than expected spot rates, usually
7,014 citations