Electricity Prices and Power Derivatives: Evidence from the Nordic Power Exchange

This paper examines the importance of the regular pattern in the behavior of electricity prices, and its implications for the purposes of derivative pricing. We analyze the Nordic Power Exchange's spot, futures, and forward prices. We conclude that the seasonal systematic pattern throughout the year, in particular, is of crucial importance in explaining the shape of the futures/forward curve. Moreover, in the context of the oneand two factor models analyzed in this paper, a simple sinusoidal functionis adequate in order capture the seasonal pattern of the features and forwardcurve directly implied by the seasonal behavior of spot electricity prices.     

On the seasonality in the implied volatility of electricity options

Seasonality is an important topic in electricity markets, as both supply and demand are dependent on the time of the year. Clearly, the level of prices shows a seasonal behaviour, but not only this. Also, the price fluctuations are typically seasonal. In this paper, we study empirically the implied volatility of options on electricity futures, investigate whether seasonality is present and we aim at quantifying its structure. Although typically futures prices can be well described through multi-factor models including exponentially decreasing components, we do not find evidence of exponential behaviour in our data set. Generally, a simple linear shape reflects the squared volatilities very well as a curve depending on the time to maturity. Moreover, we find that the level of volatility exhibits clear seasonal patterns that depend on the delivery month of the futures. Furthermore, in an out-of-sample analysis we compare the performance of several implementations of seasonality in the one-factor framework.     

A spot market model for pricing derivatives in electricity markets

Abstract

In this paper, we analyse the evolution of prices in deregulated electricity markets. We present a general model that simultaneously takes into account the following features: seasonal patterns, price spikes, mean reversion, price dependent volatilities and long term non-stationarity. We estimate the parameters of the model using historical data from the European Energy Exchange. Finally, we demonstrate how it can be used for pricing derivatives via Monte Carlo simulation.     

Futures volatility forecasting based on big data analytics with incorporating an order imbalance effect

Future markets play vital roles in supporting economic activities in modern society. For example, crude oil and electricity futures markets have heavy effects on a nation’s energy operation management. Thus, volatility forecasting of the futures market is an emerging but increasingly influential field of financial research. In this paper, we adopt big data analytics, called Extreme Gradient Boosting (XGBoost) from computer science, in an attempt to improve the forecasting accuracy of futures volatility and to demonstrate the application of big data analytics in the financial spectrum in terms of volatility forecasting. We further unveil that order imbalance estimation might incorporate abundant information to reflect price jumps and other trading information in the futures market. Including order imbalance information helps our model capture underpinned market rules such as supply and demand, which lightens the information loss during the model formation. Our empirical results suggest that the volatility forecasting accuracy of the XGBoost method considerably beats the GARCH-jump and HAR-jump models in both crude oil futures market and electricity futures market. Our results could also produce plentiful research implications for both policy makers and energy futures market participants.     

An empirical comparison of continuous-time models of implied volatility indices

We explore the ability of alternative popular continuous-time diffusion and jump-diffusion processes to capture the dynamics of implied volatility indices over time. The performance of the various models is assessed under both econometric and financial metrics. To this end, data are employed from major European and American implied volatility indices and the rapidly growing CBOE volatility futures market. We find that the addition of jumps is necessary to capture the evolution of implied volatility indices under both metrics. Mean reversion is of second-order importance though. The results are consistent across the various metrics, markets, and construction methodologies.     

An Equilibrium Analysis of Hedging with Liquidity Constraints, Speculation, and Government Price Subsidy in a Commodity Market

We develop a simple commodity model to analyze (i) the effects of hedging with liquidity constraints, due to producers' inability to bear unlimited trading losses, (ii) the role of speculation in the process of risk allocation between consumers and producers, and (iii) the equilibrium implications of government price subsidies to the producers. We find that (1) liquidity constraints can cause futures prices to exhibit mean reversion, which then makes speculation profitable; (2) speculation tends to make futures price volatility an increasing function of futures price; and (3) government price subsidy, if actively hedged by the producers, serves to lower the futures risk premium and reduce futures volatility.     

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.     

The relative pricing of European dividend futures and their predictive abilities for index returns

We use dividend futures prices to derive a dividend future discount model. Arbitrage arguments postulate that the sum of discounted dividend futures prices should equal the index price, i.e. the sum of discounted dividends. We analyze whether this relation holds and find that the two valuation approaches lead to a different valuation of expected dividends. These observations indicate that dividend futures and index prices seem to provide the investor with different information on future dividends. We further show that the difference in valuation can be used to forecast index returns and show how an investment strategy can exploit this predictability.     

The affine styled-facts price dynamics for the natural gas: evidence from daily returns and option prices

This study analyzes affine styled-facts price dynamics of Henry Hub natural gas price by incorporating the price features of jump risk, and seasonality within stochastic volatility framework. Affine styled-facts dynamics has the advantage of being able to incorporate mean reversion (MR), stochastic volatility (SV), seasonality trends (S), and jump diffusion (J) in a standardized inclusive framework. Our main finding is that models that incorporate jumps significantly improve overall out-of-sample option pricing performance. The combined MRSVJS model provides the best fit of both daily gas price returns and the related cross section of option prices. Incorporating seasonal effects tend to provide more stable pricing ability, especially for the long-term option contracts.     

Information transfer,microstmctures and intraday price return spikes

Previous research has identified overnight public information as the cause of higher opening returns and mean reversion in security markets. This paper tests this hypothesis by using an intervention and transfer function time series model to filter out the dynamic effects of an overnight information set on the opening, and subsequent, intraday AOI stock and SPI futures intraday price returns. A further research objective was to analyse the process by which information is transferred into prices and whether there is a differential impact across stock and futures markets. It was determined that the information contained in the overnight US stock market had: (i) a differential impact on the Australian stock and futures market, and (ii) after filtering out the impact of overnight information, a significant reversal tendency remained in both markets after opening. Further analysis supported the conclusion that price spikes at opening were not wholly related to overnight information. Other possible explanations, such as different trading mechanisms, did not provide a satisfactory explanation. Overall, it appears that the uncertainty participants face at the beginning of a trading session may induce a number of subtle market reactions (both rational and irrational), in markets with different microstmctures and trading clientele.     

How Crashes Develop: Intradaily Volatility and Crash Evolution

This paper explores whether affine models with volatility jumps estimated on intradaily S&P 500 futures data over 1983 to 2008 can capture major daily outliers such as the 1987 stock market crash. Intradaily jumps in futures prices are typically small; self‐exciting but short‐lived volatility spikes capture intradaily and daily returns better. Multifactor models of the evolution of diffusive variance and jump intensities improve fits substantially, including out‐of‐sample over 2009 to 2016. The models capture reasonably well the conditional distributions of daily returns and realized variance outliers, but underpredict realized variance inliers. I also examine option pricing implications.     

COS method for option pricing under a regime-switching model with time-changed Lévy processes

We extend the regime-switching model to the rich class of time-changed Lévy processes and use the Fourier cosine expansion (COS) method to price several options under the resulting models. The extension of the COS method to price under the regime-switching model is not straightforward because it requires the evaluation of the characteristic function which is based on a matrix exponentiation which is not an easy task. For a two-state economy, we give an analytical expression for computing this matrix exponential, and for more than two states, we use the Carathéodory–Fejér approximation to find the option prices efficiently. In the new framework developed here, it is possible to allow switches not only in the model parameters as is commonly done in literature, but we can also completely switch among various popular financial models under different regimes without any additional computational cost. Calibration of the different regime-switching models with real market data shows that the best models are the regime-switching time-changed Lévy models. As expected by the error analysis, the COS method converges exponentially and thus outperforms all other numerical methods that have been proposed so far.     

Pricing swing options in the electricity markets under regime-switching uncertainty

The spot price market for electricity is highly volatile. The time series of the daily average electricity price is characterised by seasonality, mean reversion, jumps, and regime-switching processes. In electricity markets, ‘swing’ contracts, which can provide some protection against the day-to-day price fluctuations, are used to incorporate flexibility in acquiring given quantities of electricity. We develop a lattice approach for the valuation of swing options by modelling the daily average price of electricity by a regime-switching process that utilises three regimes, consisting of Brownian motions and a mean-reverting process. Various numerical examples are presented to illustrate the methodology.     

Asian options and meromorphic Lévy processes

One method to compute the price of an arithmetic Asian option in a Lévy driven model is based on an exponential functional of the underlying Lévy process: If we know the distribution of the exponential functional, we can calculate the price of the Asian option via the inverse Laplace transform. In this paper, we consider pricing Asian options in a model driven by a general meromorphic Lévy process. We prove that the exponential functional is equal in distribution to an infinite product of independent beta random variables, and its Mellin transform can be expressed as an infinite product of gamma functions. We show that these results lead to an efficient algorithm for computing the price of the Asian option via the inverse Mellin–Laplace transform, and we compare this method with some other techniques.     

Pricing Efficiency and Arbitrage in the Bitcoin Spot and Futures Markets

This study examines the pricing efficiency for the leading cryptocurrency, Bitcoin using spot prices and all CBOE and CME futures contracts traded from January 2018 to March 2019. We find that the futures basis provide some predictive power for future changes in the spot price and in the risk premium. However, the basis of Bitcoin is a biased predictor of the future spot price changes. Cointegration tests also demonstrate that futures prices are biased predictors of spot prices. Deviations from no-arbitrage between spot and futures markets are persistent and widen significantly with Bitcoin thefts (hacks, frauds) as well as alternative cryptocurrency issuances.     

The role of gold futures in mitigating the impact of economic uncertainty on spot prices: Evidence from China

This paper studies the co-integration relationship and volatility spillover effect between China's gold futures and spot prices through the VECM-BEKK-GARCH model. Then, MSGARCH and DCCE-GARCH are applied to study the relationship among China's gold futures market, spot market price volatility and the stabilization effect in uncertain economic environments. This paper enriches the current research, providing gold market participants with hints to address economic uncertainty. The empirical results show that China's gold futures market has a weak stabilization effect on spot price volatility. In scenarios with uncertain economic information and uncertain macroeconomic changes, the correlation between gold futures and spot price volatility is reduced in China, and the role of gold futures in stabilizing the spot price weakens. Furthermore, with economic uncertainty, the fluctuation range of the gold futures price is greater than that of the spot price, with a tendency of more frequent fluctuations. This also means that the effectiveness of the futures market in regulating the spot price will be reduced, and gold market regulators need to stabilize the market through alternative methods to futures.     

A continuous time equilibrium model of forward prices and futures prices in a multigood economy

This paper is a theoretical investigation of equilibrium forward and futures prices. We construct a rational expectations model in continuous time of a multigood, identical consumer economy with constant stochastic returns to scale production. Using this model we find three main results. First, we find formulas for equilibrium forward, futures, discount bond, commodity bond and commodity option prices. Second, we show that a futures price is actually a forward price for the delivery of a random number of units of a good; the random number is the return earned from continuous reinvestment in instantaneously riskless bonds until maturity of the futures contract. Third, we find and interpret conditions under which normal backwardation or contango is found in forward or futures prices; these conditions reflect the usefulness of forward and futures contracts as consumption hedges.     

Pricing double barrier options under a volatility regime-switching model with psychological barriers

The prices of lots of assets have been proved in literature to exhibit special behaviors around psychological barriers, which is an important fact needed to be considered when pricing derivatives. In this paper, we discuss the valuation problem of double barrier options under a volatility regime-switching model where there exist psychological barriers in the prices of underlying assets. The volatility can shift between two regimes, that is to say, when the asset price rises up or falls down through the psychological barrier, the volatility takes two different values. Using the Laplace transform approach, we obtain the price of the double barrier knock-out call option as well as its delta. We also provide the eigenfunction expansion pricing formula and examine the effect of the psychological barrier on the option price and delta, finding that the gamma of the option is discontinuous at such barriers.     

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.     

The smirk in the S&P500 futures options prices: a linearized factor analysis

In the S&P500 futures options, we identify three factors, corresponding to movements in the underlying, parallel movements, and tilting of the cross section of implied volatilities (the “smirk factor”). We relate these factors non-linearly to movements in the option prices. They seem to be diffusive in nature, have significant associated risk premia, and can account for an overwhelming part of the option price movements. We interpret the options smirk, which is the notion that out-of-the-money (OTM) puts seem expensive relative to OTM calls, in terms of the prices of these risk factors. Going short OTM puts and long OTM calls, corresponding to the third factor, makes a profit on average, but this corresponds to its risk premium, and does not represent a market inefficiency. Our smirk factor is useful for hedging option portfolios, but seems unrelated to movements in the underlying, and does not fit into the framework of the jump-diffusion models.