Estimating Early Exercise Premiums on Gold and Copper Options Using a Multifactor Model and Density Matched Lattices

We use the standard geometric Brownian motion augmented by jumps to describe the spot underlying and mean regressive models of interest rates and convenience yields as state variables for gold and copper prices. Estimates of parameters of the diffusion processes are obtained by the Kalman filter. Using these estimates, jump parameters are estimated in the second stage by least squares. Early exercise premia on puts and calls are computed using a lattice with probabilities assigned by the density matching technique. We find that while deep in the money options have greater absolute early exercise premiums, the early exercise premium is roughly constant as a percent of option price. Our findings also confirm that gold behaves like an investment asset and copper behaves like a commodity.     

The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging

In this article we compare three models of the stochastic behavior of commodity prices that take into account mean reversion, in terms of their ability to price existing futures contracts, and their implication with respect to the valuation of other financial and real assets. The first model is a simple one-factor model in which the logarithm of the spot price of the commodity is assumed to follow a mean reverting process. The second model takes into account a second stochastic factor, the convenience yield of the commodity, which is assumed to follow a mean reverting process. Finally, the third model also includes stochastic interest rates. The Kalman filter methodology is used to estimate the parameters of the three models for two commercial commodities, copper and oil, and one precious metal, gold. The analysis reveals strong mean reversion in the commercial commodity prices. Using the estimated parameters, we analyze the implications of the models for the term structure of futures prices and volatilities beyond the observed contracts, and for hedging contracts for future delivery. Finally, we analyze the implications of the models for capital budgeting decisions.     

Arbitrage, Risk Premium, and Cointegration Tests of the Efficiency of Futures Markets

This article provides a new perspective on the efficiency of futures markets in a cointegration framework. Under the conventional risk premium hypothesis, if futures and spot prices are non-stationary, they must be cointegrated if futures markets are efficient. Alternatively, the cost-of-carry model implies that there should be a cointegration relationship among spot prices, futures prices and interest rates assuming all the series contain a unit root. Market efficiency further implies specific parameter restrictions under these two models. Using data on the futures markets for gold, silver, palladium and platinum, this article first establishes that interest rates, spot and futures prices are unit root non-stationary. The evidence on cointegration is somewhat mixed: the gold futures market is consistent with the cost-of-carry model, and the silver futures market satisfies the risk premium hypothesis, but the evidence for the other two markets is inconclusive.     

Risk premia in option markets

Risk premia are related to price probability ratios or for continuous time pure jump processes the ratios of jump arrival rates under the pricing and physical measures. The variance gamma model is employed to synthesize densities with risk premia seen as the ratio of the three parameters. The premia are shown to be mean reverting, predictable, focused on crashes at shorter horizons and rallies at the longer horizon. Predicted premia may be used to adjust physical parameters to develop option prices based on time series data.     

Electricity Forward Prices: A High-Frequency Empirical Analysis

We conduct an empirical analysis of forward prices in the PJM electricity market using a high‐frequency data set of hourly spot and day‐ahead forward prices. We find that there are significant risk premia in electricity forward prices. These premia vary systematically throughout the day and are directly related to economic risk factors, such as the volatility of unexpected changes in demand, spot prices, and total revenues. These results support the hypothesis that electricity forward prices in the Pennsylvania, New Jersey, and Maryland market are determined rationally by risk‐averse economic agents.     

Rational Expectations and Risk Premia in Forward Markets: Primary Metals at the London Metals Exchange

This paper tests whether forward prices equal the traders' expectations of the future spot prices at maturity, under two different models of expectations formation: full information rational expectations and incomplete information mechanical forecasting rule. The tests are performed, over the period January 1970 through September 1980, on the forward markets for the primary metals—copper, tin, lead, and zinc-traded in the London Metals Exchange. We find evidence consistent with the existence of time varying risk premia.     

Monetary policy,bond risk premia,and the economy

Within an affine model of the term structure of interest rates, where bond yields get driven by observable and unobservable macroeconomic factors, parameter restrictions help identify the effects of monetary policy and other structural disturbances on output, inflation, and interest rates and decompose movements in long-term rates into terms attributable to changing expected future short rates versus risk premia. When estimated, the model highlights a broad range of channels through which monetary policy affects risk premia and the economy, risk premia affect monetary policy and the economy, and the economy affects monetary policy and risk premia.     

Cross‐sectional Restrictions on the Spot and Forward Term Structures of Interest Rates and Panel Unit Root Tests

In this paper we examine the stationarity of all the rates comprising the USD, GBP, DM and JPY spot and forward term structures. Instead of focussing on short maturity interest rates, as most other papers do, we perform a detailed analysis of the whole range of spot and forward interest rates of the 4 main currencies. We investigate the issue of stationarity within the framework of an equilibrium interest rate model such as Vasicek (1977), that defines the cross-sectional and time series properties that interest rates of various maturities must satisfy. We show that within a one-factor interest rate model, such as Vasicek, all interest rates are restricted to exhibit the same mean reverting behaviour. This restriction allows us to apply more powerful panel unit root tests. This methodology increases considerably the number of observations available and as a result the power of the unit root tests. The higher power of these tests allows us to demonstrate that there does exist mean reversion on the spot and forward US interest rates and the forward DM and GBP interest rates.     

Interpreting Implied Risk-Neutral Densities: The Role of Risk Premia

This paper examines differences between risk-neutral and objective probability densities of future interest rates. The identification and quantification of these differences are important when risk-neutral densities (RNDs), such as option-implied RNDs, are used as indicators of actual beliefs of investors. We employ a multi-factor essentially affine modeling framework applied to German time-series and cross-section term structure data in order to identify both the risk-neutral and the objective term structure dynamics. We find important differences between risk-neutral and objective distributions due to risk premia in bond prices. Moreover, the estimated premia vary over time in a quantitatively significant way, which implies that the differences between the objective and the risk-neutral distributions also vary over time. We therefore conclude that one should be cautious in interpreting RNDs in terms of expectations. The method used in this paper provides an alternative approach to identifying objective probabilities of future interest rates.     

Momentum and reversion in risk neutral martingale probabilities

A time homogeneous, purely discontinuous, parsimonous Markov martingale model is proposed for the risk neutral dynamics of equity forward prices. Transition probabilities are in the variance gamma class with spot dependent parameters. Markov chain approximations give access to option prices. The model is estimated on option prices across strike and maturity for five days at a time. Properties of the estimated processes are described via an analysis of return quantiles, momentum functions that measure the response of tail probabilities to such moves. Momentum and reversion are also addressed via the construction of reverse conditional expectations. Term structures for the moments of marginal distributions support a decay in skewness and excess kurtosis with maturity at rates slower than those implied by Lévy processes. Out of sample performance is additionally reported. It is observed that risk neutral dynamics by and large reflect the presence of momentum in numerous probabilities. However, there is some reversion in the upper quantiles of risk neutral return distributions.     

Implied Foreign Exchange Risk Premia

This paper uses implied volatilities from foreign exchange option prices and the results of no‐arbitrage theory to estimate foreign exchange risk premia. In particular, under the assumption of no‐arbitrage, the foreign exchange risk premium is driven by the difference between investors’ market prices of risk in the two currencies. In an international economy with three currencies, sterling, US dollar and Deutschemark, we can use the information on implied volatilities of the three cross rates to derive estimates of implied or ex ante market prices of risk and of foreign exchange risk premia. The foreign exchange risk premia estimates are then compared to survey‐based risk premia.     

Macroeconomic Conditions and the Puzzles of Credit Spreads and Capital Structure

I build a dynamic capital structure model that demonstrates how business cycle variation in expected growth rates, economic uncertainty, and risk premia influences firms' financing policies. Countercyclical fluctuations in risk prices, default probabilities, and default losses arise endogenously through firms' responses to macroeconomic conditions. These comovements generate large credit risk premia for investment grade firms, which helps address the credit spread puzzle and the under‐leverage puzzle in a unified framework. The model generates interesting dynamics for financing and defaults, including market timing in debt issuance and credit contagion. It also provides a novel procedure to estimate state‐dependent default losses.     

Oil convenience yields estimated under demand/supply shock

This paper extends the call option model of Milonas and Thomadakis (1997) to estimate oil convenience yields with futures prices. We define the business cycle of a seasonal commodity with demand/supply shocks and find that the convenience yield for crude oil exhibits seasonal behavior. The convenience yield for West Texas Intermediate (WTI) crude oil is the highest in the summer, while that for Brent crude oil is the highest in the winter. This implies that WTI crude oil is more sensitive to high summer demand and that Brent crude oil is more sensitive to shortages in winter supply. Convenience yields are negatively related to the inventory level of the underlying crude oil and positively related to interest rates due to the business cycle. We also show that convenience yields may explain price spread between WTI crude oil and Brent crude oil. Our computed convenience yields are consistent with Fama and French (1988) in that oil prices are more volatile than futures prices at low inventory level, verifying the Samuelson (1965) hypothesis that future prices are less variables than spot prices at lower inventory levels.

Business Cycles and the Behavior of Metals Prices

The theory of storage says that the marginal convenience yield on inventory falls at a decreasing rate as inventory increases. The authors test this hypothesis by examining the relative variation of spot and futures prices for metals. As the hypothesis implies, futures prices are less variable than spot prices when inventory is low, but spot and futures prices have similar variability when inventory is high. The theory of storage also explains inversions of “normal” futures-spot price relations around business-cycle peaks. Positive demand shocks around peaks reduce metal inventories and, as the theory predicts, generate large convenience yields and price inversions.     

Equity yields

We study a new data set of dividend futures with maturities up to ten years across three world regions: the US, Europe, and Japan. We use these asset prices to construct equity yields, analogous to bond yields. We decompose the equity yields to obtain a term structure of expected dividend growth rates and a term structure of risk premia, which decomposes the equity risk premium by maturity. We find that the slope of the term structure of risk premia is pro-cyclical, whereas the slope of the term structure of expected dividend growth rates is counter-cyclical. The comovement of yields across regions is, on average, higher for long-maturity yields than for short-maturity yields, whereas the variation in this comovement is much higher for short-maturity yields.     

Pricing the term structure of inflation risk premia: Theory and evidence from TIPS

In this paper, we study inflation risk and the term structure of inflation risk premia in the United States' nominal interest rates through the Treasury Inflation Protection Securities (TIPS) with a multi-factor, modified quadratic term structure model with correlated real and inflation rates. We derive closed form solutions to the real and nominal term structures of interest rates that drastically facilitate the estimation of model parameters and improve the accuracy of the valuation of nominal rates and TIPS prices. In addition, we contribute to the literature by estimating the term structure of inflation risk premia implied from the TIPS market. The empirical evidence using data from the period of January 1998 through October 2007 indicates that the expected inflation rate, contrary to data derived from the consumer price indices, is very stable and the inflation risk premia exhibit a positive term structure.     

The forward premium in electricity futures

Understanding the nature of the forward premium is particularly crucial, but rather elusive, for a non-storable commodity such as wholesale electricity. Whilst forward prices emerge as the expectation of spot plus, or minus, an ex ante premium for risk, the manifestation and empirical analysis must focus upon realised ex post premiums. This presents modelling requirements to control for shocks to the spot expectation as well as the endogeneity of ex post premia with spot price outcomes. In addition, because electricity is a derived commodity in the sense that market prices are often set by technologies that convert gas or coal into power, it is an open question whether much of the premia in power may actually be a pass-through of the premia in gas (or coal). Using a four dimensional VAR model we are able to distinguish fundamental and behavioural aspects of price formation in both the daily and monthly forward premia from the British market. We present new evidence on daily and seasonal sign reversals, associated with demand cycles, the greater importance of behavioural adaptations in the risk premia than fundamental or spot market risk measures, and the substantial fuel risk pass-through. We also show the value of a nonlinear specification in this context.     

Extracting inflation expectations and inflation risk premia from the term structure: A joint model of the UK nominal and real yield curves

This paper analyses the UK interest rate term structure over the period since October 1992, when the United Kingdom adopted an explicit inflation target, using an affine term structure model estimated using both government bond yields and survey data. The model imposes no-arbitrage restrictions across nominal and real yields, which enables interest rates to be decomposed into expected real policy rates, expected inflation, real term premia and inflation risk premia. The model is used to shed light on major developments over the period, including the impact of Bank of England independence and the low real bond yield ‘conundrum’.     

Modelling electricity prices: a time change approach

To capture mean reversion and sharp seasonal spikes observed in electricity prices, this paper develops a new stochastic model for electricity spot prices by time changing the Jump Cox-Ingersoll-Ross (JCIR) process with a random clock that is a composite of a Gamma subordinator and a deterministic clock with seasonal activity rate. The time-changed JCIR process is a time-inhomogeneous Markov semimartingale which can be either a jump-diffusion or a pure-jump process, and it has a mean-reverting jump component that leads to mean reversion in the prices in addition to the smooth mean-reversion force. Furthermore, the characteristics of the time-changed JCIR process are seasonal, allowing spikes to occur in a seasonal pattern. The Laplace transform of the time-changed JCIR process can be efficiently computed by Gauss–Laguerre quadrature. This allows us to recover its transition density through efficient Laplace inversion and to calibrate our model using maximum likelihood estimation. To price electricity derivatives, we introduce a class of measure changes that transforms one time-changed JCIR process into another time-changed JCIR process. We derive a closed-form formula for the futures price and obtain the Laplace transform of futures option price in terms of the Laplace transform of the time-changed JCIR process, which can then be efficiently inverted to yield the option price. By fitting our model to two major electricity markets in the US, we show that it is able to capture both the trajectorial and the statistical properties of electricity prices. Comparison with a popular jump-diffusion model is also provided.     

Pricing Commodity Spread Options with Stochastic Term Structure of Convenience Yields and Interest Rates

The purpose of this research is to provide a valuation formula for commodity spread options. Commodity spread options are options written on the difference of the prices (spread) of two commodities. From the aspect of commodity contingent claims, it is considered that commodity spread options are difficult to evaluate with accuracy because of the existence of the convenience yield. Hence, the model of the convenience yield is the key factor to price commodity spread options. We use the concept of future convenience yields to develop the model that enriches the stochastic behavior of convenience yield. We also introduce Heath-Jarrow-Morton interest rate model to the valuation framework. This general model not only captures the mean reverting feature of the convenience yield, but also allows us to handle a very wide range of shape that the term structure of convenience yield can take. Therefore our model provides various types of models. The numerical analysis presented in this paper provides some unique features of commodity spread options in contrast to normal options. These characteristics have never been addressed in previous studies. Moreover, it suggests that the existing model overprice commodity spread options through neglecting the effect of interest rates.