Modeling and Pricing in Financial Markets for Weather Derivatives

Krühner

2015

SIAM J. Finan. Math.

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ABSTRACT. Based on forward curves modelled as Hilbert-space valued processes, we analyse the pricing of various options relevant in energy markets. In particular, we connect empirical evidence about energy forward prices known from the literature to propose stochastic models. Forward prices can be represented as linear functions on a Hilbert space, and options can thus be viewed as derivatives on the whole curve. The value of these options are computed under various specifications, in addition to their deltas. In a second part, crosscommodity models are investigated, leading to a study of square integrable random variables with values in a "two-dimensional" Hilbert space. We analyse the covariance operator and representations of such variables, as well as presenting applications to pricing of spread and energy quanto options.

Section: Hilbert-space Realization Of Energy Forwards and Futuresmentioning

confidence: 99%

Derivatives Pricing in Energy Markets: An Infinite-Dimensional Approach

Krühner

2015

SIAM J. Finan. Math.

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“…Such processes have been applied to model not only the stochastic dynamics of energy prices (see Barndorff-Nielsen et al [4]), but also other random phenomena like wind speed and temperature (see Benth and Šaltytė Benth [9]) and turbulence (see BarndorffNielsen and Schmiegel [5]). We may discuss similar specifications of the drift β in light of these different special classes of spot models.…”

Section: Theorem 38 If L Is a Subordinated Brownian Motion Ie L(mentioning

confidence: 99%

“…Note that we are free to choose the covariance kernel q and the θ , which means that we can have many different forward curve models resulting in the same CARMA spot model. CARMA-processes have been applied to temperature models (see Benth et al [9] and Härdle and Lopez-Cabrera [30]), spot power prices (see Garcia et al [28]) and the prices of oil (see Paschke and Prokopczuk [39]). For a detailed analysis of CARMA processes, we refer the reader to Brockwell [16].…”

Section: Remark 47mentioning

confidence: 99%

Representation of Infinite-Dimensional Forward Price Models in Commodity Markets

Krühner

2014

Commun. Math. Stat.

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We study the forward price dynamics in commodity markets realised as a process with values in a Hilbert space of absolutely continuous functions defined by Filipović (Consistency problems for Heath-Jarrow-Morton interest rate models, 2001). The forward dynamics are defined as the mild solution of a certain stochastic partial differential equation driven by an infinite-dimensional Lévy process. It is shown that the associated spot price dynamics can be expressed as a sum of OrnsteinUhlenbeck processes, or more generally, as a sum of certain stationary processes. These results link the possibly infinite-dimensional forward dynamics to classical commodity spot models. We continue with a detailed analysis of multiplication and integral operators on the Hilbert spaces and show that Hilbert-Schmidt operators are essentially integral operators. The covariance operator of the Lévy process driving the forward dynamics and the diffusion term can both be specified in terms of such operators, and we analyse in several examples the consequences on model dynamics and their probabilistic properties. Also, we represent the forward price for contracts delivering over a period in terms of an integral operator, a case being relevant for power and gas markets. In several examples, we reduce our general model to existing commodity spot and forward dynamics.

“…Following Benth andŠaltytė Benth [4], we let Y i (t) be a CARMA(p i , q i ) process, for p i > q i , i = 1, 2 defined as…”

Section: A Pricing Measure Preserving Cointegrationmentioning

confidence: 99%

Pricing of Forwards and Other Derivatives in Cointegrated Commodity Markets

Koekebakker²

2014

SSRN Journal

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ABSTRACT. We analyse cointegration in commodity markets, and propose a parametric class of pricing measures which preserves cointegration for forward prices with fixed time to maturity. We present explicit expressions for the term structure of volatility and correlation in the context of our spot price models based on continuous-time autoregressive moving average dynamics for the stationary components. The term structures have many interesting shapes, and we provide some empirical evidence from refined oil futures prices at NYMEX defending our modelling idea. Motivated from these results, we present a cointegrated forward price dynamics using the Heath-Jarrow-Morton modelling idea. In this setting, the concept of cointegration is extended to what we call cointegration in the limit, which is an asymptotic form of the notion. The Margrabe formula for spread option prices is shown to hold, with an explicit plug-in volatility. We present several numerical examples showing that cointegration leads to significantly cheaper spread options compared to the complete market case, where cointegration disappears for the pricing measure.