We introduce a new and highly tractable structural model for spot and derivative prices in electricity markets. Using a stochastic model of the bid stack, we translate the demand for power and the prices of generating fuels into electricity spot prices. The stack structure allows for a range of generator efficiencies per fuel type and for the possibility of future changes in the merit order of the fuels. The derived spot price process captures important stylized facts of historical electricity prices, including both spikes and the complex dependence upon its underlying supply and demand drivers. Furthermore, under mild and commonly used assumptions on the distributions of the input factors, we obtain closed-form formulae for electricity forward contracts and for spark and dark spread options. As merit order dynamics and fuel forward prices are embedded into the model, we capture a much richer and more realistic dependence structure than can be achieved by classical reduced-form models. We illustrate these advantages by comparing with Margrabe's formula and a simple cointegration model, and highlight important implications for the valuation of power plants.Electricity markets and structural model and forward prices and spread options and power plant valuation JEL Classification Numbers: C60, G12, G13, Q40Partially supported by NSF -DMS-0739195. 1 arXiv:1205.2299v1 [q-fin.PR] 10 May 20121 Alternatively, firms may, in some markets, submit continuous bid curves, which map an amount of electricity to the price at which it is offered. For our purposes this distinction will however not be relevant.2 This description is of course a simplification of the market administrator's complicated unit commitment problem, typically solved by optimization in order to satisfy various operational constraints of generators, as well as transmission constraints. Details vary from market to market and we do not address these issues here, as our goal is to approximate the price setting mechanism and capture the key relationships needed for derivative pricing.ORFE,
Abstract. The purpose of the paper is to present a new pricing method for clean spread options, and to illustrate its main features on a set of numerical examples produced by a dedicated computer code. The novelty of the approach is embedded in the use of structural models as opposed to reduced-form models which fail to capture properly the fundamental dependencies between the economic factors entering the production process.
We present a novel approach to the pricing of financial instruments in emission markets-for example, the European Union Emissions Trading Scheme (EU ETS). The proposed structural model is positioned between existing complex full equilibrium models and pure reduced-form models. Using an exogenously specified demand for a polluting good, it gives a causal explanation for the accumulation of CO 2 emissions and takes into account the feedback effect from the cost of carbon to the rate at which the market emits CO 2 . We derive a forward-backward stochastic differential equation for the price process of the allowance certificate and solve the associated semilinear partial differential equation numerically. We also show that derivatives written on the allowance certificate satisfy a linear partial differential equation. The model is extended to emission markets with multiple compliance periods, and we analyze the impact different intertemporal connecting mechanisms, such as borrowing, banking, and withdrawal, have on the allowance price.
Let S F be a P-martingale representing the price of a primitive asset in an incomplete market framework. We present easily verifiable conditions on the model coefficients which guarantee the completeness of the market in which in addition to the primitive asset, one may also trade a derivative contract S B . Both S F and S B are defined in terms of the solution X to a two-dimensional stochastic differential equation:. From a purely mathematical point of view, we prove that every local martingale under P can be represented as a stochastic integral with respect to the P-martingale S := (S F , S B ). Notably, in contrast to recent results on the endogenous completeness of equilibria markets, our conditions allow the Jacobian matrix of (f, g) to be singular everywhere on R 2 . Hence they cover as a special case the prominent example of a stochastic volatility model being completed with a European call (or put) option.
We present a novel approach to the pricing of financial instruments in emission marketsfor example, the European Union Emissions Trading Scheme (EU ETS). The proposed structural model is positioned between existing complex full equilibrium models and pure reduced-form models. Using an exogenously specified demand for a polluting good, it gives a causal explanation for the accumulation of CO 2 emissions and takes into account the feedback effect from the cost of carbon to the rate at which the market emits CO 2 . We derive a forward-backward stochastic differential equation for the price process of the allowance certificate and solve the associated semilinear partial differential equation numerically. We also show that derivatives written on the allowance certificate satisfy a linear partial differential equation. The model is extended to emission markets with multiple compliance periods, and we analyze the impact different intertemporal connecting mechanisms, such as borrowing, banking, and withdrawal, have on the allowance price.
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