Abstract
The purpose of this paper is twofold: Firstly, we analyse the option value approximation of traded options in the presence of a volatility term structure. The options are identified as: (a) “European” (written on the forward price of a future flow delivery); and (b) “Asian”. Both types are in fact written on (arithmetic) price averages. Secondly, adopting a 3-factor model for market risk which is compatible with the valuation results, we discuss risk management in the electricity market within the Value at Risk concept. The analysis is illustrated by numerical cases from the Norwegian electricity derivatives market.
This chapter is a corrected version of Bjerksund, Rasmussen, and Stensland (2000).
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Notes
- 1.
To establish the first equality, apply Ito’s lemma
$${ \mbox{ Var}}_{t}^{{_\ast}}\left [\ln \left (\frac{f(\tau,T)} {f(t,T)} \right )\right ] ={ \mbox{ Var}}_{t}^{{_\ast}}\left [{\int \nolimits \nolimits }_{s=t}^{s=\tau }\left (\frac{df(s,T)} {f(s,T)} \right ) -{\int \nolimits \nolimits }_{s=t}^{s=\tau }\mbox{ $\frac{1} {2}$}{\left (\frac{df(s,T)} {f(s,T)} \right )}^{2}\right ],$$insert the assumed forward price dynamics, and observe that the second integral is deterministic as of date t. The second equality follows from the fact that Brownian motions have independent increments across time.
- 2.
Observe that w(s; r) > 0 ∀s ∈ [T 1, T 2] and \({\int \nolimits \nolimits }_{{T}_{1}}^{{T}_{2}}w(s; r)ds = 1\).
- 3.
The approximation proceeds in the following two steps
$$\frac{\mathit{dF}(t,{T}_{1},{T}_{2})} {F(t,{T}_{1},{T}_{2})} \approx {\int \nolimits \nolimits }_{s={T}_{1}}^{s={T}_{2}}w(s; r)\frac{\mathit{df }(t,s)} {f(t,s)} \mathit{ds} \approx {\int \nolimits \nolimits }_{s={T}_{1}}^{s={T}_{2}}w(s; 0)\frac{\mathit{df }(t,s)} {f(t,s)} \mathit{ds}.$$ - 4.
We have corrected the typos in a previous version of this paper pointed out in Lindell and Raab (2008).
- 5.
The function is approximated numerically by
$$\mbox{ dilog }(x) = \left \{\begin{array}{l@{\quad }l} { \sum \nolimits }_{k=1}^{n}\frac{{(x-1)}^{k}} {{k}^{2}} \quad \mbox{ for $0 \leq x \leq 1$ } \\ -\mbox{ $\frac{1} {2}$}{\left (\ln (x)\right )}^{2} -{\sum \nolimits }_{k=1}^{n}\frac{{\left ((1/x)-1\right )}^{k}} {{k}^{2}} \quad \mbox{ for $x > 1$ } \end{array} \right.$$where n is a sufficiently large positive integer.
- 6.
Assume for the moment a discrete time model where the delivery period [T 1, T 2] is divided into n time intervals of time length Δt. Consider the following strategy: At the evaluation date t, buy \({e}^{-r({T}_{2}-({T}_{1}+i\cdot \Delta t))}(1/n)\) units forward for each delivery T 1 + i ⋅Δt, i = 1, …, n. As time passes and the contracts are settled, invest (or finance) the proceeds at the riskless interest rate r. At the delivery date τ ≥ T 2, the pay-off from the strategy is ∑ i = 1 n(1 ∕ n)f(T 1 + i ⋅Δt, T 1 + i ⋅Δt) − ∑ i = 1 n(1 ∕ n)f(t, T 1 + i ⋅Δt), where the first term represents the desired spot price, and the second (riskless) term may be interpreted as the forward price as of date t.
- 7.
It is straightforward to verify these results using the fact that
$$\frac{\partial } {\partial x}\mbox{ dilog }(x) = \frac{\ln (x)} {1 - x}.$$
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Acknowledgements
We thank the editors, anonymous referees, and P.E. Manne for useful comments.
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Bjerksund, P., Rasmussen, H., Stensland, G. (2010). Valuation and Risk Management in the Norwegian Electricity Market. In: Bjørndal, E., Bjørndal, M., Pardalos, P., Rönnqvist, M. (eds) Energy, Natural Resources and Environmental Economics. Energy Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12067-1_11
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