stochastic convenience yield and the pricing of oil contingent claims this paper develops and empirically tests a two factor model for pricing financial and real assets contingent on the price of oil the factors are the spot price of oil and the instantaneous convenience yield the parameters of the model are estimated using weekly oil futures contract prices from january 1984 to november 1988 and the model s performance is assessed out of sample by valuing futures contracts over the period november 1988 to may 1989 finally the model is applied to determine the present values of one barrel of oil deliverable in one to ten years time

bond pricing and the term structure of interest rates a new methodology for contingent claims valuation this paper presents a unifying theory for valuing contingent claims under a stochastic term structure of interest rates the methodology based on the equivalent martingale measure technique takes as given an initial forward rate curve and a family of potential stochastic processee for its subsequent movements a no arbitrage condition restricts this family of processes yielding valuation formulae for interest rate sensitive contingent claims which do not explicitly depend on the market prices of risk examples are provided to illustrate the key results in relation to the term structure of interest rates arbitrage pricing theory has two purposes the first is to price all zero coupon default free bonds of varying maturities from a finite number of economic fundamentals called state variables the second is to price all interest rate sensitive contingent claims taking as given the prices of the zero coupon bonds this paper presents a general theory and a unifying framework for understanding arbitrage pricing theory in this context of which all existing arbitrage pricing models are special cases in particular vasicek 1977 brennan and schwartz 1979 langetieg 1980 ball and torous 1983 ho and lee 1986 schaefer and schwartz 1987 and artzner and delbaen 1988 the primary contribution of this paper however is a new methodology for solving the second problem i e the pricing of interest rate sensitive contingent claims given the prices of all zero coupon bonds the methodology is new because i it imposes its stochastic structure directly on the evolution of the forward rate curve ii it does not require an inversion of the term structure to eliminate the market prices of risk from contingent claim values and iii it has a stochastic spot rate process with multiple stochastic factors influencing the term structure the model can be used to consistently price and hedge all contingent claims american or european on the term structure and it is derived from necessary and more importantly sufficient conditions for the absence of arbitrage the arbitrage pricing models of vasicek 1977 brennan and schwartz 1979 langetieg 1980 and artzner and delbaen 1988 all require an iformerly titled bond pricing and the term structure of interest rates a new methodology

equilibrium price dynamics of emission permits this article presents a stochastic equilibrium model for environmental markets that allows us to study the characteristic properties of emission permit prices induced by the design of today s cap and trade systems we characterize emission permits as highly nonlinear contingent claims on economy wide emissions and reveal their hybrid nature between investment and consumption assets our model makes predictions about the dynamics and volatility structure of emission permit prices the forward price curve and the implications for option pricing in this market empirical evidence from existing emissions markets shows that the model explains the stylized facts of emission permit prices and related derivatives

should investors include commodities in their portfolios after all new evidence abstract this paper investigates whether an investor is made better off by including commodities in a portfolio that consists of traditional asset classes first we revisit the posed question within an in sample setting by employing mean variance and non mean variance spanning tests then we form optimal portfolios by taking into account the higher order moments of the portfolio returns distribution and evaluate their out of sample performance under the in sample setting we find that commodities are beneficial only to non mean variance investors however these benefits are not preserved out of sample our findings challenge the alleged diversification benefits of commodities and are robust across a number of performance evaluation measures utility functions and datasets the results hold even when transaction costs are considered and across various sub periods not surprisingly the only exception appears over the 2005 2008 unprecedented commodity boom period

oil futures prices in a production economy with investment constraints we document a new stylized fact that the relationship between the volatility of oil futures prices and the slope of the forward curve is nonmonotone and has a v shape this pattern cannot be generated by standard models that emphasize storage we develop an equilibrium model of oil production in which investment is irreversible and capacity constrained investment constraints affect firms investment decisions and imply that the supply elasticity changes over time since demand shocks must be absorbed by changes in prices or changes in supply time varying supply elasticity results in time varying volatility of futures prices estimating this model we show it is quantitatively consistent with the v shape relationship between the volatility of futures prices and the slope of the forward curve in recent years commodity markets have experienced dramatic growth in trad

computing the market price of volatility risk in the energy commodity markets abstract in this paper we demonstrate the need for a negative market price of volatility risk to recover the difference between black scholes black f scholes m 1973 the pricing of options and corporate liabilities journal of political economy 81 637 654 black black f 1976 studies of stock price volatility changes in proceedings of the 1976 meetings of the business and economics statistics section american statistical association pp 177 181 implied volatility and realized term volatility initially using quasi monte carlo simulation we demonstrate numerically that a negative market price of volatility risk is the key risk premium in explaining the disparity between risk neutral and statistical volatility in both equity and commodity energy markets this is robust to multiple specifications that also incorporate jumps next using futures and options data from natural gas heating oil and crude oil contracts over a 10 year period we estimate the volatility risk premium and demonstrate that the premium is negative and significant for all three commodities additionally there appear distinct seasonality patterns for natural gas and heating oil where winter withdrawal months have higher volatility risk premiums computing such a negative market price of volatility risk highlights the importance of volatility risk in understanding priced volatility in these financial markets

short term variations and long term dynamics in commodity prices in this article we develop a two factor model of commodity prices that allows meanreversion in short term prices and uncertainty in the equilibrium level to which prices revert although these two factors are not directly observable they may be estimated from spot and futures prices intuitively movements in prices for long maturity futures contracts provide information about the equilibrium price level and differences between the prices for the short and long term contracts provide information about short term variations in prices we show that although this model does not explicitly consider changes in convenience yields over time this short term long term model is equivalent to the stochastic convenience yield model developed in gibson and schwartz 1990 we estimate the parameters of the model using prices for oil futures contracts and apply the model to some hypothetical oil linked assets to demonstrate its use and some of its advantages over the gibson schwartz model

unspanned stochastic volatility and fixed income derivatives pricing we propose a parsimonious unspanned stochastic volatility model of the term structure and study its implications for fixed income option prices the drift and quadratic variation of the short rate are affine in three state variables the short rate its long term mean and variance which follow a joint markov vector process yet bond prices are exponential affine functions of only two state variables independent of the current interest rate volatility level because this result holds for an arbitrary volatility process such a process can be calibrated to match fixed income derivative prices furthermore this model can be extended by relaxing the time homogeneity to fit any arbitrary term structure in its hjm form this model nests the analogous stochastic equity volatility model of heston 190 heston s l 1993 a closed form solution for options with stochastic volatility review of financial studies 6 327 343 in particular if the volatility process is specified to be affine closed form solutions for interest rate options obtain we propose an efficient algorithm to compute these prices an application using data on caps and floors shows that the model can capture very well the implied black spot volatility surface while simultaneously fitting the observed term structure c 2005 elsevier b v all rights reserved

vertical mergers and product differentiation this paper presents an equilibrium theory of vertical mergers that incorporates strategic behaviors in the hotelling type location model this enables one to consider the relationship between downstream firms strategies for product differentiation and vertical integration i show that vertical integration enhances the degree of product differentiation of the integrated firm under some conditions partial integration arises in equilibrium which may increase the profit of the nonintegrated downstream firm the paper also discusses the welfare implications of vertical integration

the dynamics of the forward interest rate curve a formulation with state variables the objective of this paper is twofold first the paper develops a class of models of the term structure of interest rates in the heath jarrow and morton 1992 framework with dynamics characterized by the evolution of a small set of state variables second the paper exploits this characterization of the dynamics of the term structure in an estimation exercise that makes use of both the time series and cross section of bond prices in this way our class of models bridges the gap between traditional models such as cox ingersoll and ross 1985 and vasicek 1977 that emphasize the dynamics of interest rates and the models of heath jarrow and morton 1992 that stress fitting the cross section of bond prices

valuation of commodity futures and options under stochastic convenience yields interest rates and jump diffusions in the spot this paper investigates the effects of stochastic convenience yields stochastic interest rates and jumps in the spot price on the pricing of commodity futures forwards and futures options numerical examples show that one factor prices differ materially from the stochastic convenience yield two factor prices when convenience yield is considerably above its long term average the extension to a three factor model with stochastic interest rates leads to a different futures price but the forward price is unchanged the extension ofthe three factor model to include jumps in the spot price process does not affect forward or futures prices but it can have an impact on options prices the model is applied to price the present value of future cash flows from a real asset

do bonds span the fixed income markets theory and evidence for unspanned stochastic volatility most term structure models assume bond markets are complete that is that all fixed income derivatives can be perfectly replicated using solely bonds however we find that in practice swap rates have limited explanatory power for returns on at the money straddles portfolios mainly exposed to volatility risk we term this empirical feature unspanned stochastic volatility usv while usv can be captured within an hjm framework we demonstrate that bivariate models cannot exhibit usv we determine necessary and sufficient conditions for trivariate markov affine systems to exhibit usv for such usv models bonds alone may not be sufficient to identify all parameters rather derivatives are needed

equilibrium exhaustible resource price dynamics we develop equilibrium models of exhaustible resource markets with endogenous extraction choices and prices our analysis demonstrates how adjustment costs can generate oil and gas forward price dynamics with two factors consistent with the behavior these commodities exhibit in the schwartz and smith 2000 calibration our two factor model predicts that stochastic volatility will arise in these markets as a natural consequence of production adjustments however and we provide supporting empirical evidence differences between endogenous price processes from our general equilibrium model and exogenous processes in earlier papers can generate significant differences in both financial and real option values

unspanned stochastic volatility and the pricing of commodity derivatives commodity derivatives are becoming an increasingly important part of the global derivatives market here we develop a tractable stochastic volatility model for pricing commodity derivatives the model features unspanned stochastic volatility quasi analytical prices of options on futures contracts and dynamics of the futures curve in terms of a low dimensional affine state vector we estimate the model on nymex crude oil derivatives using an extensive panel data set of 45 517 futures prices and 233 104 option prices spanning 4082 business days we find strong evidence for two predominantly unspanned volatility factors jel g13

pricing of options on commodity futures with stochastic term structures of convenience yields and interest rates we develop a model to value options on commodity futures in the presence of stochastic interest rates as well as stochastic convenience yields in the development of the model we distinguish between forward and future convenience yields a distinction that has not been recognized in the literature assuming normality of continuously compounded forward interest rates and convenience yields and log normality of the spot price of the underlying commodity we obtain closed form solutions generalizing the black scholes merton s formulas we provide numerical examples with realistic parameter values showing that both the effect of introducing stochastic convenience yields into the model and the effect of having a short time lag between the maturity of a european call option and the underlying futures contract have significant impact on the option prices

a theory of commodity price fluctuations this paper studies the price fluctuations of storable commodities that are traded in open markets and are subject to random shocks to demand or more particularly to supply it relaxes the common assumption that the shocks are identically and independently distributed in favor of temporally dependent and periodic disturbances the existence of a unique stationary rational expectations equilibrium is demonstrated for each of the models analyzed and testable implications of the models are derived an illustrative empirical investigation is then undertaken for the model with periodic disturbances using monthly time series observations for seven commodities over the period 1960 93

valuation of vix derivatives abstract we conduct an extensive empirical analysis of vix derivative valuation models before during and after the 2008 2009 financial crisis since the restrictive mean reversion and heteroskedasticity features of existing models yield large distortions during the crisis we propose generalisations with a time varying central tendency jumps and stochastic volatility analyse their pricing performance and implications for term structures of vix futures and volatility skews we find that a process for the log of the observed vix combining central tendency and stochastic volatility reliably prices vix derivatives we also uncover a significant risk premium that shifts the long run volatility level

pricing of long dated commodity derivatives do stochastic interest rates matter abstract does modelling stochastic interest rates beyond stochastic volatility improve pricing performance on long dated commodity derivatives to answer this question we consider futures price models for commodity derivatives that allow for stochastic volatility and stochastic interest rates and a correlation structure between the underlying variables we examine the empirical pricing performance of these models on pricing long dated crude oil derivatives estimating the model parameters from historical crude oil futures prices and option prices we find that stochastic interest rate models improve pricing performance on long dated crude oil derivatives when the interest rate volatility is relatively high furthermore increasing the model dimensionality does not tend to improve the pricing performance on long dated crude oil option prices but it matters for long dated futures prices we also find empirical evidence for a negative correlation between crude oil futures prices and interest rates that contributes to improving fit to long dated crude oil option prices

mean reversion in equilibrium asset prices evidence from the futures term structure we use the term structure of futures prices to test whether investors anticipate mean reversion in spot asset prices the empirical results indicate mean reversion in each market we examine for agricultural commodities and crude oil the magnitude of the estimated mean reversion is large for example point estimates indicate that 44 percent of a typical spot oil price shock is expected to be reversed over the subsequent eight months for metals the degree of mean reversion is substantially less but still statistically significant we detect only weak evidence of mean reversion in financial asset prices in this study we provide evidence of mean reversion in the prices of several real and financial assets rather than examining evidence of ex post reversion using time series of asset prices we use price data from futures contracts with varying delivery horizons to test whether investors expect asset prices to revert this approach offers two advantages first since futures prices are readily available for many markets our procedure can be implemented for many assets including those for which reliable spot price data is elusive second there is little ambiguity as to the source of any mean reversion detected using our method subject only to the maintained assumption that the no arbitrage cost of carry condition holds our test detects mean reversion that is expected to occur in equilibrium but has no power to detect mean reversion resulting from noise or inefficiencies our methodology focuses on relations between price levels and the slope of the futures term structure defined as the change across delivery dates in the futures prices observed on a given trading date an inverse relation between prices and the futures term slope constitutes evidence that investors expect mean reversion in spot prices to illustrate initially assume there are no

competitive storage and commodity price dynamics by buying cheap and selling dear risk neutral commodity speculators can smooth commodity prices and induce serial dependence in price even when none would exist under a simple process of supply and demand commodity prices are variable and strongly positively correlated from one year to the next the variability is often explained by supply factors and the autocorrelation by the activities of speculators we show that this explanation is not consistent with the evidence speculation can substantially increase autocorrelation for prices that are weakly autocorrelated in its absence but not to the high levels that are observed in the data

commodity derivatives pricing with cointegration and stochastic covariances abstract empirically cointegration and stochastic covariances including stochastic volatilities are statistically significant for commodity prices and energy products to capture such market phenomena we develop a continuous time dynamics of cointegrated assets with a stochastic covariance matrix and derive the joint characteristic function of asset returns in closed form the proposed model offers an endogenous explanation for the stochastic mean reverting convenience yield the time series of spot and futures prices of wti crude oil and gasoline shows cointegration relationship under both physical and risk neutral measures the proposed model also allows us to fit the observed term structure of futures prices and calibrate the market implied cointegration relationship we apply it to value options on a single commodity and on multiple commodities

unspanned stochastic volatility in affine models evidence from eurodollar futures and options unspanned stochastic volatility usv refers to the inability of bonds to replicate volatility sensitive derivative securities affine term structure models require special restrictions on the parameters to exhibit usv we use a joint eurodollar futures and options data set to estimate affine three factor models with and without usv restrictions the unrestricted model captures prices of futures and options well option pricing errors are much larger in the usv model the usv model is rejected in favor of the unrestricted model based on the likelihood ratio and wald tests we use the implications of the unrestricted model as a benchmark for understanding the extant evidence that favors usv specifically we replicate extant tests in samples simulated from the unrestricted model we show that none of the existing findings contradict the model without usv restrictions

stochastic risk premiums stochastic skewness in currency options and stochastic discount factors in international economies abstract we develop models of stochastic discount factors in international economies that produce stochastic risk premiums and stochastic skewness in currency options we estimate the models using time series returns and option prices on three currency pairs that form a triangular relation estimation shows that the average risk premium in japan is larger than that in the us or the uk the global risk premium is more persistent and volatile than the country specific risk premiums and investors respond differently to different shocks we also identify high frequency jumps in each economy but find that only downside jumps are priced finally our analysis shows that the risk premiums are economically compatible with movements in stock and bond market fundamentals

satiation and habit persistence or the dieter s dilemma abstract frequently individual consumption varies cyclically over time in order to model such behavior we have to generalize the intertemporal utility function so that it depends not only on current consumption but on total previous consumption and on the change in current consumption solving the resulting model requires a generalization of the euler lagrange equations