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Register a new userValuation of electricity storage contracts using the COS method
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Storage of electricity has become increasingly important, due to the gradual replacement of fossil fuels by more variable and uncertain renewable energy sources. In this paper, we provide details on how to mathematically formalize a corresponding electricity storage contract, taking into account the physical limitations of a storage facility and the operational constraints of the electricity grid. We give details of a valuation technique to price these contracts, where the electricity prices follow a structural model based on a stochastic polynomial process. In particular, we show that the Fourier-based COS method can be used to price the contracts accurately and efficiently.
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Applying the Cherny-Shiryaev-Yor invariance principle, we introduce a generalized Jarrow-Rudd (GJR) option pricing model with uncertainty driven by a skew random walk. The GJR pricing tree exhibits skewness and kurtosis in both the natural and risk-neutral world. We construct implied surfaces for the parameters determining the GJR tree. Motivated by Mertons pricing tree incorporating transaction costs, we extend the GJR pricing model to include a hedging cost. We demonstrate ways to fit the GJR pricing model to a market driver that influences the price dynamics of the underlying asset. We supplement our findings with numerical examples.
Various valuation adjustments, or XVAs, can be written in terms of non-linear PIDEs equivalent to FBSDEs. In this paper we develop a Fourier-based method for solving FBSDEs in order to efficiently and accurately price Bermudan derivatives, including options and swaptions, with XVA under the flexible dynamics of a local Levy model: this framework includes a local volatility function and a local jump measure. Due to the unavailability of the characteristic function for such processes, we use an asymptotic approximation based on the adjoint formulation of the problem.
In this article, we show how the scaling symmetry of the SABR model can be utilized to efficiently price European options. For special kinds of payoffs, the complexity of the problem is reduced by one dimension. For more generic payoffs, instead of solving the 1+2 dimensional SABR PDE, it is sufficient to solve $N_V$ uncoupled 1+1 dimensional PDEs, where $N_V$ is the number of points used to discretize one dimension. Furthermore, the symmetry argument enables us to obtain prices of multiple options, whose payoffs are related to each other by convolutions, by valuing one of them. The results of the method are compared with the Monte Carlo simulation.
We present the closed-form solution to the problem of hedging price and quantity risks for energy retailers (ER), using financial instruments based on electricity price and weather indexes. Our model considers an ER who is intermediary in a regulated electricity market. ERs buy a fixed quantity of electricity at a variable cost and must serve a variable demand at a fixed cost. Thus ERs are subject to both price and quantity risks. To hedge such risks, an ER could construct a portfolio of financial instruments based on price and weather indexes. We construct the closed form solution for the optimal portfolio for the mean-Var model in the discrete setting. Our model does not make any distributional assumption.
This paper proposes a paradigm shift in the valuation of long term annuities, away from classical no-arbitrage valuation towards valuation under the real world probability measure. Furthermore, we apply this valuation method to two examples of annuity products, one having annual payments linked to a mortality index and the savings account and the other having annual payments linked to a mortality index and an equity index with a guarantee that is linked to the same mortality index and the savings account. Out-of-sample hedge simulations demonstrate the effectiveness of real world valuation. In contrast to risk neutral valuation, which is a form of relative valuation, the long term average excess return of the equity market comes into play. Instead of the savings account, the numeraire portfolio is employed as the fundamental unit of value in the analysis. The numeraire portfolio is the strictly positive, tradable portfolio that when used as benchmark makes all benchmarked nonnegative portfolios supermartingales. The benchmarked real world value of a benchmarked contingent claim equals its real world conditional expectation. This yields the minimal possible value for its hedgeable part and minimizes the fluctuations for its benchmarked hedge error. Under classical assumptions, actuarial and risk neutral valuation emerge as special cases of the proposed real world valuation. In long term liability and asset valuation, the proposed real world valuation can lead to significantly lower values than suggested by classical approaches when an equivalent risk neutral probability measure does not exist.
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