A new approach to wind power futures pricing

Abstract

We propose a new model for the pricing of wind power futures written on the wind power production index. Our approach is based on an arithmetic multi-factor pure-jump Ornstein–Uhlenbeck setup with time-dependent coefficients. We express the wind power production index and the corresponding futures price in terms of Fourier integrals and derive the related time dynamics. We conclude the paper by an investigation of the risk premium associated with our wind power model.

Appendix

In what follows, we present an alternative modeling approach for the wind power production index. First of all, note that for $x \ge 0$ the function $\eta \left( x \right): = e^{ - x}$ takes values in the interval $\left] {0,1} \right]$. Hence, instead of (2.8), we could alternatively define the wind power production index $P_{t}$ via

$$ P_{t} : = \eta \left( {Y_{t} } \right) = e^{{ - Y_{t} }} $$

(A.1)

where $t \ge 0$. If we do so, we find $0 < P_{t} \le 1$ ${\mathbb{P}}$-a.s. $\forall t \ge 0$, because $Y_{t}$ in (2.7) is non-negative. Substituting (A.1) and (2.7) into (3.1), we get

$$ f_{t} \left( T \right) = {\mathbb{E}}_{{\mathbb{Q}}} \left( {\exp \left\{ { - \mathop \sum \limits_{k = 1}^{n} X_{T}^{k} } \right\}{|}{\mathcal{F}}_{t} } \right) $$

(A.2)

for all $0 \le t \le T$. We next put (2.6) [with $t$ therein replaced by $T$] into (A.2) and obtain

$$ f_{t} \left( T \right) = \exp \left\{ { - \mathop \sum \limits_{k = 1}^{n} \left( {x_{k} e^{{ - \lambda_{k} T}} + \mathop \int \limits_{0}^{t} \mathop \int \limits_{{D_{k} }}^{ } \theta_{k} \left( {s,z,T} \right) N_{k} \left( {{\text{d}}s,{\text{d}}z} \right)} \right)} \right\} \times {\mathbb{E}}_{{\mathbb{Q}}} \left[ {\exp \left\{ { - \mathop \sum \limits_{k = 1}^{n} \mathop \int \limits_{t}^{T} \mathop \int \limits_{{D_{k} }}^{ } \theta_{k} \left( {s,z,T} \right) N_{k} \left( {{\text{d}}s,{\text{d}}z} \right)} \right\}} \right] $$

(A.3)

where $\theta_{k} \left( {s,z,T} \right)$ is such as defined in (3.3). In the last step, we used the independent increment property of the processes $L^{1} , \ldots ,L^{n}$. An application of the Lévy–Khinchin formula yields

$$ {\mathbb{E}}_{{\mathbb{Q}}} \left[ {\exp \left\{ { - \mathop \sum \limits_{k = 1}^{n} \mathop \int \limits_{t}^{T} \mathop \int \limits_{{D_{k} }}^{ } \theta_{k} \left( {s,z,T} \right) N_{k} \left( {{\text{d}}s,{\text{d}}z} \right)} \right\}} \right] = \exp \left\{ {\mathop \sum \limits_{k = 1}^{n} \mathop \int \limits_{t}^{T} \mathop \int \limits_{{D_{k} }}^{ } \left[ {e^{{ - \theta_{k} \left( {s,z,T} \right)}} - 1} \right] e^{{h_{k} \left( {s,z} \right)}} \rho_{k} \left( s \right) \nu_{k} \left( {{\text{d}}z} \right){\text{d}}s} \right\} . $$

Merging the latter equation into (A.3), we deduce the following representation for the wind power futures price

$$ f_{t} \left( T \right) = \exp \left\{ { - \mathop \sum \limits_{k = 1}^{n} \left( {x_{k} e^{{ - \lambda_{k} T}} + \mathop \int \limits_{t}^{T} \mathop \int \limits_{{D_{k} }}^{ } \left[ {1 - e^{{ - \theta_{k} \left( {s,z,T} \right)}} } \right] e^{{h_{k} \left( {s,z} \right)}} \rho_{k} \left( s \right) \nu_{k} \left( {{\text{d}}z} \right){\text{d}}s + \mathop \int \limits_{0}^{t} \mathop \int \limits_{{D_{k} }}^{ } \theta_{k} \left( {s,z,T} \right) N_{k} \left( {{\text{d}}s,{\text{d}}z} \right)} \right)} \right\} $$

(A.4)

which corresponds to Eq. (3.4). Applying Itô’s formula on (A.4), for all $0 \le t \le T$ we infer the geometric $\left( {{\mathcal{F}},{\mathbb{Q}}} \right)$-martingale dynamics

$$ {\text{d}}f_{t} \left( T \right) = f_{t - } \left( T \right)\mathop \sum \limits_{k = 1}^{n} \mathop \int \limits_{{D_{k} }}^{ } \left[ {e^{{ - \theta_{k} \left( {t,z,T} \right)}} - 1} \right] \tilde{N}_{k}^{\mathbb{Q}} \left( {{\text{d}}t,{\text{d}}z} \right) $$

(A.5)

where the initial value $f_{0} \left( T \right)$ can immediately be derived from (A.4). We recall that (A.5) corresponds to Eq. (3.13). We stress that the wind power production index $P$ defined in (A.1) is always strictly positive which ought to be regarded as a disadvantage from a practitioner’s perspective, as in reality the wind power production index can be equal to zero during certain time periods with no or little wind. We recall that, on the contrary, the wind power production index defined in (2.8) also takes zero values yielding a more realistic modeling setup.

In the next step, we derive the risk premium formula associated with the geometric model (A.1)–(A.5). Parallel to Sect. 3, we define

$$ f_{t}^{{\mathbb{P}}} \left( T \right) = {\mathbb{E}}_{\mathbb{P}} \left( P_T| {\mathcal{F_t}} \right). $$

Using (A.1), (2.7) and (2.6), we then get

$$ f_{t}^{{\mathbb{P}}} \left( T \right) = \exp \left\{ { - \mathop \sum \limits_{k = 1}^{n} \left( {x_{k} e^{{ - \lambda_{k} T}} + \mathop \int \limits_{0}^{t} \mathop \int \limits_{{D_{k} }}^{ } \theta_{k} \left( {s,z,T} \right) N_{k} \left( {{\text{d}}s,{\text{d}}z} \right)} \right)} \right\} \times {\mathbb{E}}_{{\mathbb{P}}} \left[ {\exp \left\{ { - \mathop \sum \limits_{k = 1}^{n} \mathop \int \limits_{t}^{T} \mathop \int \limits_{{D_{k} }}^{ } \theta_{k} \left( {s,z,T} \right) N_{k} \left( {{\text{d}}s,{\text{d}}z} \right)} \right\}} \right] $$

which closely resembles (A.3), but with ${\mathbb{Q}}$ therein replaced by ${\mathbb{P}}$. The appearing ${\mathbb{P}}$-expectation can be computed by the Lévy–Khinchin formula yielding

$$ f_{t}^{{\mathbb{P}}} \left( T \right) = \exp \left\{ { - \mathop \sum \limits_{k = 1}^{n} \left( {x_{k} e^{{ - \lambda_{k} T}} + \mathop \int \limits_{t}^{T} \mathop \int \limits_{{D_{k} }}^{ } \left[ {1 - e^{{ - \theta_{k} \left( {s,z,T} \right)}} } \right] \rho_{k} \left( s \right) \nu_{k} \left( {{\text{d}}z} \right){\text{d}}s + \mathop \int \limits_{0}^{t} \mathop \int \limits_{{D_{k} }}^{ } \theta_{k} \left( {s,z,T} \right) N_{k} \left( {{\text{d}}s,{\text{d}}z} \right)} \right)} \right\} $$

which equals (A.4), but with $h_{k} \left( {s,z} \right) \equiv 0$ therein. We now substitute the latter equation as well as (A.4) into (3.14) which leads us to the risk premium formula

$$ R_{t}^{{{\mathbb{P}},{\mathbb{Q}}}} \left( T \right) = \exp \left\{ { - \mathop \sum \limits_{k = 1}^{n} \left( {x_{k} e^{{ - \lambda_{k} T}} + \mathop \int \limits_{0}^{t} \mathop \int \limits_{{D_{k} }}^{ } \theta_{k} \left( {s,z,T} \right) N_{k} \left( {ds,dz} \right)} \right)} \right\} \times \left[ {\exp \left\{ {\mathop \sum \limits_{k = 1}^{n} \mathop \int \limits_{t}^{T} \mathop \int \limits_{{D_{k} }}^{ } \left[ {e^{{ - \theta_{k} \left( {s,z,T} \right)}} - 1} \right] e^{{h_{k} \left( {s,z} \right)}} \rho_{k} \left( s \right) \nu_{k} \left( {{\text{d}}z} \right){\text{d}}s} \right\} - \exp \left\{ {\mathop \sum \limits_{k = 1}^{n} \mathop \int \limits_{t}^{T} \mathop \int \limits_{{D_{k} }}^{ } \left[ {e^{{ - \theta_{k} \left( {s,z,T} \right)}} - 1} \right] \rho_{k} \left( s \right) \nu_{k} \left( {{\text{d}}z} \right){\text{d}}s} \right\}} \right] $$

where $0 \le t \le T$. Note that the latter equation corresponds to (3.15). By the Itô formula we eventually deduce the geometric $\left( {{\mathcal{F}},{\mathbb{P}}} \right)$-martingale dynamics

$$ {\text{d}}f_{t}^{{\mathbb{P}}} \left( T \right) = f_{t - }^{{\mathbb{P}}} \left( T \right)\mathop \sum \limits_{k = 1}^{n} \mathop \int \limits_{{D_{k} }}^{ } \left[ {e^{{ - \theta_{k} \left( {t,z,T} \right)}} - 1} \right] \tilde{N}_{k}^{{\mathbb{P}}} \left( {{\text{d}}t,{\text{d}}z} \right) $$

where we used (2.4). The latter SDE corresponds to (A.5).

In the sequel, we show that the model proposed in Benth and Pircalabu (2018) is contained as a sub-case in our current geometric model associated with (A.1). If we take ${n = 1}$, $w_{1} = 1$ and $\mu \left( t \right) = - \ln \Lambda \left( t \right)$ in (2.1), we get

$$ Y_{t} = X_{t}^{1} - \ln \Lambda \left( t \right) $$

and hence,

$$ P_{t} = \Lambda \left( t \right) e^{{ - X_{t}^{1} }} $$

due to (A.1). This representation for $P_{t}$ possesses the same structure as Eq. (1) in Benth and Pircalabu (2018). If we further take $\lambda_{1} = \alpha$ and $\sigma_{1} \left( t \right) \equiv 1$, then (2.2) translates into

$$ {\text{d}}X_{t}^{1} = - \alpha X_{t}^{1} {\text{d}}t + {\text{d}}L_{t}^{1} $$

which implies

$$ {\text{d}}Y_{t} = \left( { - \frac{{\Lambda^{{\prime }} \left( t \right)}}{\Lambda \left( t \right)} - \alpha X_{t}^{1} } \right){\text{d}}t + {\text{d}}L_{t}^{1} . $$

(A.6)

Comparing (A.6) with Eq. (2) in Benth and Pircalabu (2018), we see that we get correspondence between these SDEs, if we take

$$ \Lambda \left( t \right) = e^{ - \alpha \mu t},\quad X_{t}^{1} = X_{t},\quad L_{t}^{1} = L_{t}. $$

We conclude that the model proposed in Benth and Pircalabu (2018) is a sub-case of the geometric model presented in (A.1)–(A.5) above.