A Theory of the Term Structure of Interest Rates.
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
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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: This paper presented a consumption-based model that explains a wide variety of dynamic asset pricing phenomena, including the procyclical variation of stock prices, the long-horizon predictability of excess stock returns, and the countercyclical variations of stock market volatility.
Abstract: We present a consumption-based model that explains a wide variety of dynamic asset pricing phenomena, including the procyclical variation of stock prices, the long-horizon predictability of excess stock returns, and the countercyclical variation of stock market volatility The model captures much of the history of stock prices from consumption data It explains the short- and long-run equity premium puzzles despite a low and constant risk-free rate The results are essentially the same whether we model stocks as a claim to the consumption stream or as a claim to volatile dividends poorly correlated with consumption The model is driven by an independently and identically distributed consumption growth process and adds a slow-moving external habit to the standard power utility function These features generate slow countercyclical variation in risk premia The model posits a fundamentally novel description of risk premia: Investors fear stocks primarily because they do poorly in recessions unrelated to the risks of long-run average consumption growth
3,623 citations
Book•
01 Jan 1992
TL;DR: The "Dynamic Asset Pricing Theory" (DAT) as discussed by the authors is a textbook for doctoral students and researchers on the theory of asset pricing and portfolio selection in multi-period settings under uncertainty.
Abstract: "Dynamic Asset Pricing Theory" is a textbook for doctoral students and researchers on the theory of asset pricing and portfolio selection in multiperiod settings under uncertainty. The asset pricing results are based on the three increasingly restrictive assumptions: absence of arbitrage, single-agent optimaltiy, and equilibrium. These results are unified with two key concepts, state prices and martingales. Technicalities are given relatively little emphasis so as to draw connections between these concepts and to make plain the similarities between discrete and continuous-time models. For simplicity, all continuous-time models are based on Brownian motion. Applications include term structure models, derivative valuation and hedging methods, and dynamic programming algorithms for portfolio choice and optimal exercise of American options. Numerical methods covered include Monte Carlo simulation and finite-difference solvers for partial differential equations. Each chapter provides extensive problem exercises and notes to the literature. This second edition is substantially longer, while still retaining the consciseness for which the first edition was praised. All chapters from the first edition have been revised. Two new chapters have been added on term structure modeling and on derivative securities. References have been updated throughout. With this new edition, "Dynamic Asset Pricing Theory" remains the definitive textbook in the field.
2,857 citations
TL;DR: In this article, an option pricing model that allows volatility, interest rates and jumps to be stochastic is presented. But it is not known whether and by how much each generalization improves option pricing and hedging.
Abstract: Substantial progress has been made in developing more realistic option pricing models. Empirically, however, it is not known whether and by how much each generalization improves option pricing and hedging. We fill this gap by first deriving an option model that allows volatility, interest rates and jumps to be stochastic. Using S&P 500 options, we examine several alternative models from three perspectives: (1) internal consistency of implied parameters/volatility with relevant timeseries data, (2) out-of-sample pricing, and (3) hedging. Overall, incorporating stochastic volatility and jumps is important for pricing and internal consistency. But for hedging, modeling stochastic volatility alone yields the best performance. IN THE LAST TWO DECADES, option pricing has witnessed an explosion of new
2,777 citations
Cites methods from "A Theory of the Term Structure of I..."
...Next, to ensure proper discounting of future cash flows, we adopt a singlefactor term structure model of the Cox, Ingersoll, and Ross (1985) type as it requires the estimation of only three structural parameters: dR(t) = [OR - KRR(t)]dt + R R(t0dWR(t), (5) 3 See, for example, Bates (1996a,c),…...
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TL;DR: In this paper, the authors derived a single-beta asset pricing model in a multi-good, continuous-time model with uncertain consumption-goods prices and uncertain investment opportunities.
2,667 citations
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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
TL;DR: In this article, the authors derived a general form of the term structure of interest rates and showed that the expected rate of return on any bond in excess of the spot rate is proportional to its standard deviation.
6,160 citations
"A Theory of the Term Structure of I..." refers methods in this paper
...An arbitrage approach to bond pricing was developed in a series of papers by Brennan and Schwartz [3], Dothan [10], Garman [14], Richard [28], and Vasicek [37]....
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TL;DR: In this paper, the authors considered the continuous-time consumption-portfolio problem for an individual whose income is generated by capital gains on investments in assets with prices assumed to satisfy the geometric Brownian motion hypothesis, which implies that asset prices are stationary and lognormally distributed.
4,952 citations
TL;DR: In this paper, the optimal consumption-investment problem for an investor whose utility for consumption over time is a discounted sum of single-period utilities, with the latter being constant over time and exhibiting constant relative risk aversion (power-law functions or logarithmic functions), is discussed.
Abstract: Publisher Summary This chapter reviews the optimal consumption-investment problem for an investor whose utility for consumption over time is a discounted sum of single-period utilities, with the latter being constant over time and exhibiting constant relative risk aversion (power-law functions or logarithmic functions). It presents a generalization of Phelps' model to include portfolio choice and consumption. The explicit form of the optimal solution is derived for the special case of utility functions having constant relative risk aversion. The optimal portfolio decision is independent of time, wealth, and the consumption decision at each stage. Most analyses of portfolio selection, whether they are of the Markowitz–Tobin mean-variance or of more general type, maximize over one period. The chapter only discusses special and easy cases that suffice to illustrate the general principles involved and presents the lifetime model that reveals that investing for many periods does not itself introduce extra tolerance for riskiness at early or any stages of life.
2,369 citations
Additional excerpts
...4This type of separability has been shown in other contexts by Hakansson [15], Merton [22], and Samuelson [32]....
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TL;DR: In this paper, a continuous time general equilibrium model of a simple but complete economy is developed to examine the behavior of asset prices and their stochastic properties are determined endogenously, and the model is fully consistent with rational expectations and maximizing behavior on the part of all agents.
Abstract: This paper develops a continuous time general equilibrium model of a simple but complete economy and uses it to examine the behavior of asset prices. In this model, asset prices and their stochastic properties are determined endogenously. One principal result is a partial differential equation which asset prices must satisfy. The solution of this equation gives the equilibrium price of any asset in terms of the underlying real variables in the economy. IN THIS PAPER, we develop a general equilibrium asset pricing model for use in applied research. An important feature of the model is its integration of real and financial markets. Among other things, the model endogenously determines the stochastic process followed by the equilibrium price of any financial asset and shows how this process depends on the underlying real variables. The model is fully consistent with rational expectations and maximizing behavior on the part of all agents. Our framework is general enough to include many of the fundamental forces affecting asset markets, yet it is tractable enough to be specialized easily to produce specific testable results. Furthermore, the model can be extended in a number of straightforward ways. Consequently, it is well suited to a wide variety of applications. For example, in a companion paper, Cox, Ingersoll, and Ross [7], we use the model to develop a theory of the term structure of interest rates. Many studies have been concerned with various aspects of asset pricing under uncertainty. The most relevant to our work are the important papers on intertemporal asset pricing by Merton [19] and Lucas [16]. Working in a continuous time framework, Merton derives a relationship among the equilibrium expected rates of return on assets. He shows that when investment opportunities are changing randomly over time this relationship will include effects which have no analogue in a static one period model. Lucas considers an economy with homogeneous individuals and a single consumption good which is produced by a number of processes. The random output of these processes is exogenously determined and perishable. Assets are defined as claims to all or a part of the output of a process, and the equilibrium determines the asset prices. Our theory draws on some elements of both of these papers. Like Merton, we formulate our model in continuous time and make full use of the analytical tractability that this affords. The economic structure of our model is somewhat similar to that of Lucas. However, we include both endogenous production and
1,999 citations
"A Theory of the Term Structure of I..." refers background or methods in this paper
...In Section 3, we specialized the general equilibrium framework of Cox, Ingersoll, and Ross [6] to develop a complete model of bond pricing....
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...In this section, we briefly review and specialize the general equilibrium model of Cox, Ingersoll, and Ross [6]....
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...Section 2 summarizes the equilibrium model developed in Cox, Ingersoll, and Ross [6] and specializes it for studying the term structure....
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...We now cite two principal results from [6] which we will need frequently in this paper....
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...Arguments similar to those employed in the proof of Theorem 2 of Cox, Ingersoll, and Ross [6] are used to show that if there are no arbitrage opportunities, Y must have the form...
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