A closed-form solution for the price of cross-commodity electricity derivatives

Valuing Plug-In Hybrid Electric Vehicles’ Battery Capacity Using a Real Options Framework

“…Also, let the price processes follow geometric mean-reverting processes: (33) where P i (t) is the spot price of commodity i at time t, κ i (t) and s i (t) are the speed of mean reversion and volatility parameters as in (7), and B 1 (t) and B 2 (t) are independent Brownian motions. As in (7), a i (t) may be interpreted as providing at time t the long-term mean of the log price process for commodity i. a i (t) was implicitly equal to zero in Tsitakis et al (2006). We want to price the European call option with the following payoff at maturity T:…”

“…The simplest pricing technique is to model the evolution of the spread directly, but as Carmona and Durrleman (2003) and Eydeland and Wolyniec (2003) the assumptions required by this approach are probably not valid in energy markets. Deng et al (2001) price spark spread options in the case where the futures prices are meanreverting, but we use the analytical solution of Tsitakis et al (2006) because they assume only that the spot prices of the underlying assets follow mean-reverting exponential Ornstein-Uhlenbeck processes, that each commodity has a price of risk, and that there is no arbitrage. However, they only consider the case in which the long-term mean of each logarithmic price process is zero.…”

Specifying An Efficient Renewable Energy Feed-in Tariff

“…Augmentation of GBM when applying FiT pricing model in an incomplete market Application of the Black-Scholes model assumes that the underlying market is complete, in that, there exist contracts to insure against all possible eventualities. However, electricity is a unique commodity as it cannot be stored and thus demand must equal supply at all moments in time (Lyle and Elliott, 2009;Burger et al, 2004;Tsitakis and Yannacopoulos, 2006). This characteristic affects the ability to hedge and thus the market for electricity is incomplete (Burger et al, 2004;Tsitakis and Yannacopoulos, 2006).…”

“…However, electricity is a unique commodity as it cannot be stored and thus demand must equal supply at all moments in time (Lyle and Elliott, 2009;Burger et al, 2004;Tsitakis and Yannacopoulos, 2006). This characteristic affects the ability to hedge and thus the market for electricity is incomplete (Burger et al, 2004;Tsitakis and Yannacopoulos, 2006). The market price of risk may thus need to be incorporated into the Black-Scholes formula (Lemoine, 2009;Lyle and Elliott, 2009) and the SDE must be adjusted to account for this in the following way:…”

“…Electricity is a unique commodity as it cannot be stored and thus demand must equal supply at all moments in time. This characteristic affects the ability to hedge and to profit from arbitrage and so the market for electricity is incomplete (Lyle and Elliott, 2009;Burger et al, 2004;Tsitakis et al, 2006). We follow the approach of Lyle and Elliott (2009), where all options in this paper are priced under the physical or real world measure.…”