Pricing in Electricity Markets: a Mean Reverting Jump Diffusion Model with Seasonality

In this paper we present a mean-reverting jump diffusion model for the electricity spot price. We obtain a closed-form solution for forward contracts and calibrate it to market data from England and Wales. Finally, based on the calibrated forward curve we present months, quarters, and seasons-ahead forward surfaces.Energy derivatives, mean reversion, jump diffusion, electricity spot and forward.

Spot price modeling and the valuation of electricity forward contracts : the role of demand and capacity.

We propose a model where wholesale electricity prices are explained by two state variables: demand and capacity. We derive analytical expressions to price forward contracts and to calculate the forward premium. We apply our model to the PJM, England and Wales, and Nord Pool markets. Our empirical findings indicate that volatility of demand is seasonal and that the market price of demand risk is also seasonal and positive, both of which exert an upward (seasonal) pressure on the price of forward contracts. We assume that both volatility of capacity and the market price of capacity risk are constant and find that, depending on the market and period under study, it could either exert an upward or downward pressure on forward prices. In all markets we find that the forward premium exhibits a seasonal pattern. During the months of high volatility of demand, forward contracts trade at a premium. During months of low volatility of demand, forwards can either trade at a relatively small premium or, even in some cases, at a discount, i.e. they exhibit a negative forward premiumPower prices; Demand; Capacity; Forward premium; Forward bias; Market price of capacity risk; Market price of demand risk; PJM; England and Wales; Nord Pool;

Modeling asset prices for algorithmic and high frequency trading.

Algorithmic Trading (AT) and High Frequency (HF) trading, which are responsible for over 70% of US stocks trading volume, have greatly changed the microstructure dynamics of tick-by-tick stock data. In this paper we employ a hidden Markov model to examine how the intra-day dynamics of the stock market have changed, and how to use this information to develop trading strategies at ultra-high frequencies. In particular, we show how to employ our model to submit limit-orders to profit from the bid-ask spread and we also provide evidence of how HF traders may profit from liquidity incentives (liquidity rebates). We use data from February 2001 and February 2008 to show that while in 2001 the intra-day states with shortest average durations were also the ones with very few trades, in 2008 the vast majority of trades took place in the states with shortest average durations. Moreover, in 2008 the fastest states have the smallest price impact as measured by the volatility of price innovationsHigh frequency traders; Algorithmic trading; Durations; Hidden Markov model;

Option pricing with Lévy-Stable processes generated by Lévy-Stable integrated variance.

We show how to calculate European-style option prices when the log-stock price process follows a Lévy-Stable process with index parameter 1≤α≤2 and skewness parameter -1≤β≤1. Key to our result is to model integrated variance as an increasing Lévy-Stable process with continuous paths in ΤLévy-Stable processes; Stable Paretian hypothesis; Stochastic volatility; α-stable processes; Option pricing; Time-changed Brownian motion;

UK gas markets : the market price of risk and applications to multiple interruptible supply contracts.

We employ the Schwartz and Smith [Schwartz, E., and J. Smith, 2000, Short-term variations and long-term dynamics in commodity prices, Management Science 46, 893–911.] model to explore the dynamics of the UK gasmarkets. We discuss in detail the short-termand long-termmarket prices of risk borne by the market players and how deviations from expected cyclical storage affect the short-term market price of risk. Finally, we illustrate an application of the model by pricing interruptible supply contracts that are currently traded in the UKInterruptible supply contracts; Gas markets; Commodities; Market price of short-term and long-term risk; Multi-exercise Bermudan options; Convenience yield;

The relationship between the volatility of returns and the number of jumps in financial markets

The contribution of this paper is two-fold. First we show how to estimate the volatility of high frequency log-returns where the estimates are not a affected by microstructure noise and the presence of Lévy-type jumps in prices. The second contribution focuses on the relationship between the number of jumps and the volatility of log-returns of the SPY, which is the fund that tracks the S&P 500. We employ SPY high frequency data (minute-by-minute) to obtain estimates of the volatility of the SPY log-returns to show that: (i) The number of jumps in the SPY is an important variable in explaining the daily volatility of the SPY log-returns; (ii) The number of jumps in the SPY prices has more explanatory power with respect to daily volatility than other variables based on: volume, number of trades, open and close, and other jump activity measures based on Bipower Variation; (iii) The number of jumps in the SPY prices has a similar explanatory power to that of the VIX, and slightly less explanatory power than measures based on high and low prices, when it comes to explaining volatility; (iv) Forecasts of the average number of jumps are important variables when producing monthly volatility forecasts and, furthermore, they contain information that is not impounded in the VIX.Volatility forecasts, High-frequency data, Implied volatility, VIX, Jumps, Microstructure noise

How much should we pay for interconnecting electricity markets? A real options approach

An interconnector is an asset that gives the owner the option to transmit electricity between two locations. In financial terms, the value of an interconnector is the same as a strip of real options written on the spread between power prices in two markets. We model the spread based on a: seasonal trend, mean-reverting Gaussian process, and mean-reverting jump process. We express the value of these real options in closed-form. We apply our valuation tool to five pairs of European neighboring markets to value a hypothetical one-year lease of the interconnector. We show valuations for different assumptions about the seasonal component of the spread, and different liquidity caps which proxy for the depth of the interconnected power markets. We derive no-arbitrage lower bounds for the value of the interconnector in terms of electricity futures contracts. We find that, depending on the depth of the market, the jumps in the spread can account for between 1% and 40% of the total value of the interconnector. The two markets where an interconnector would be most (resp. least) valuable are Germany and the Netherlands (resp. France and Germany).Real options, Bull Call Spread, Interconnector, Electricity prices, Jumps, Jump filter

Derivatives pricing with marked point processes using Tick-by-tick dataR

I propose to model stock price tick-by-tick data via a non-explosive marked point process. The arrival of trades is driven by a counting process in which the waiting-time between trades possesses a Mittag-Leffler survival function and price revisions have an infinitely divisible distribution. I show that the partial-integro-differential equation satisfied by the value of European-style derivatives contains a non-local operator in time-to-maturity known as the Caputo fractional derivative. Numerical examples are provided for a marked point process with conditionally Gaussian and with conditionally CGMY price innovations. Furthermore, the infinitesimal generator of the marked point process derived to price derivatives coincides with that of a Lévy process of either finite or infinite activity.Tick-by-tick data, Waiting-times, Duration, High frequency data, Caputo operator, Marked point process,

Pricing in electricity markets : a mean reverting jump diffusion model with seasonality.

In this paper we present a mean-reverting jump diffusion model for the electricity spot price and derive the corresponding forward in closed-form. Based on historical spot data and forward data from England and Wales we calibrate the model and present months, quarters, and seasons–ahead forward surfacesEnergy derivatives; Electricity; Forward curve; Forward surfaces;

Dynamic hedging of financial instruments when the underlying follows a non-Gaussian process.

Traditional dynamic hedging strategies are based on local information (ie Delta and Gamma) of the financial instruments to be hedged. We propose a new dynamic hedging strategy that employs non-local information and compare the profit and loss (P&L) resulting from hedging vanilla options when the classical approach of Delta- and Gammaneutrality is employed, to the results delivered by what we label Delta- and Fractional- Gamma-hedging. For specific cases, such as the FMLS of Carr and Wu (2003a) and Merton’s Jump-Diffusion model, the volatility of the P&L is considerably lower (in some cases only 25%) than that resulting from Delta- and Gamma-neutrality