We assume that the log-pricesXj,i := log Pj,i andYj,i := log Ij,ifollow the discretized version of the mean-reverting dynamics
dXt= θtP − aPXt
dt + σPdWtP dYt= θtI− aIYt
dt + σIdWtI
where WtP and WtI are two Brownian motions with mutual correlation ρ: these pro-cesses are particular cases of the model in [24] and are rather standard models for en-ergy prices (see for example [14, Chapter 23.3].
In the discretized version, bothXj,i andYj,ichange at the beginning of every sub-period (i.e. at the beginning of every month). This is exactly what happens for the indexI, and it is an acceptable simplification for the gas price P . In particular, we dis-cretize the prices(Pj,i)j,iand(Ij,i)j,iby building two trinomial trees with the procedure explained in [9, 14] and here summarized.
The first step is to build trinomial trees forX and Y by discretizing the dynamics of processes
dXt∗ = −aXt∗dt + σdWt, X0∗= 0 (33) with(a, σ) = (aP, σP), or (a, σ) = (aI, σI) in the analogous specification for Y∗. The trees for these processes are symmetric around0 and their nodes are evenly spaced in time and value at intervals of predetermined length∆t and ∆X∗ = σ√
3∆t.
As usual, we denote by (i, j) the node xi,j in the tree for which xi,j = Xt∗ with t = i∆t and Xi∆t∗ = j∆X∗6. Hull and White proved [15, 17] that the probabilities to switch from node(i, j) to node (i + 1, k) are always nonnegative if −j 6 j 6 j, where j is the smallest integer greater than 0.184/(a∆t). This means that at every time step i = 0, . . . , N we have a finite number of nodes (i, j) placed at points j∆X∗ for every integerj ∈ {−j∗, . . . , 0, . . . , j∗}, with j∗ := min
j, 2i − 1
. Thus, the total width of the tree depends ona, σ and ∆t.
The second step is to put together the two trinomial trees in a 2-dimensional tree for (X∗, Y∗): this is done at each node in such a way to preserve the marginal distributions ofX∗andY∗and the covariance structure induced by the correlated Brownian motions WP andWI, as in [16] (see also [9, Appendix F]).
The third step is aimed to calibrate the previous symmetric tree to the term structure Fi one has, Fi standing for the value of the forward with maturity i∆t: this step is used to incorporate into the tree mean reversion to levels different from zero, and in particular can be used here to introduce seasonality effects. This is obtained by adding a quantityαi to the valuexi,j of all nodes(i, j). For every step i we have a value for αi
6Notice that in this Appendix the notation (i, j) is not referred to the notation “year j, month i” used until now in the paper. Here we not distinguish between year and months, having a unique time index i that varies between 0 and N · D. However, for sake of notation, in this appendix we suppose that i = 0, . . . , N, being N the appropriate number.
such that: X
j
Qi,jeαi+xi,j = Fi
that leads to
αi = log(Fi) − log
X
j
Qi,jexi,j
having denoting withQi,j the probability to reach the node(i, j) starting from the node (0, 0). Once we have the values for αiwe obtain the final tree which has, at stepi, the nodes with valueeαi+xi,j.
An example of two possible final results for the two trees, obtained for some values ofa and σ, is plotted in Figure 13. Notice that the higher aI(oraP) is, the less nodes the respective tree have.
In order to calibrate for the parameters of Equation (33), we use a procedure inspired by [8]. The main idea is to use the discrete time version of the solution of Equation (33):
X∗(t) = X∗(s)e−a(t−s)+ σe−at Z t
s eaudWu, 0 6 s < t, (34) which gives
x(ti) = bx(ti−1) + δε (ti) (35) with
b = e−a∆t, δ = σ
r1 − e−2a∆t
2a (36)
andε is a Gaussian white noise (ε(ti) ∼ N(0, 1) for all i). Then, in order to provide the maximum likelihood estimator for the parametersb and δ, perform a least squares regression of the time seriesx(ti) on its lagged value x (ti−1), as in Equation (35). Once we haveb and δ, we can invert Equation (36) and derive the original parameter a and σ.
References
[1] F. Aasche, P. Osmundsen, R. Tveter˚as, European market integration for gas? Volume flexibility and political risk. Energy Economics 24 (3), 249–265 (2002)
[2] Alba Soluzioni http://www.albasoluzioni.com, last accessed: 14/1/2011 [3] O. Bardou, S. Bouthemy, G. Pag`es, Optimal quantization for the pricing of swing options.
Applied Mathematical Finance 16 (2), 183–217 (2009)
[4] O. Bardou, S. Bouthemy, G. Pag`es, When are swing options bangbang and how to use it?
Pre-print LPMA-1141 (2007)
0 5 10 15 20 25 30 35 40
(a) Strong mean reversion, low volatility:
aP = 3, aI= 10, σP = 0.3, σI= 0.1
(b) Small mean reversion, high volatility:
aP = 0.1, aI= 0.1, σP = 0.7, σI = 0.2
Figure 13: Trees for prices for different values of parameters. Notice that the higher aI andaI are, the less nodes the respective trees have: in subfigure (a) we have trees obtained with high aI, aP and few nodes in both the trees, while in subfigure (b) we have the converse situation.
[5] C. Barrera-Esteve, F. Bergeret, C. Dossal, E. Gobet, A. Meziou, R. Munos, D. Reboul-Salze, Numerical methods for the pricing of swing options: a stochastic control approach.
Methodology and Computing in Applied Probability 8(4), 517–540 (2006)
[6] F. E. Benth, J. Lempa, T. K. Nilssen On optimal exercise of swing options in electricity markets. Working paper
[7] D. P. Bertsekas, S. E. Shreve, Stochastic Optimal Control: The Discrete Time Case. Math-ematics in Science and Engineering, vol. 139. Academic: New York, [Harcourt Brace Jovanovich Publishers], 1978.
[8] D. Brigo, A. Dalessandro, M. Neugebauer, F. Triki A stochastic processes toolkit for risk management: Geometric Brownian motion, jumps, GARCH and variance gamma models.
Journal of Risk Management in Financial Institutions 2, 365-393 (2009)
[9] D. Brigo, F. Mercurio Interest rate models - theory and practice, 2nd edition. Springer, 2006
[10] L. Clewlow, C. Strickland Energy derivatives: pricing and risk management. Lacima Publications, 2000
[11] European Commission’s Eurostat. http://epp.eurostat.ec.europa.eu/portal/page/portal/statistics/search_database, last accessed: 6 / 9 / 2011
[12] S. Fiorenzani, Quantitative methods for electricity trading and risk management. Pal-grave Macmillan, 2006
[13] L. Holden, A. Løland, O. Lindqvist, Valuation of Long-Term Flexible Gas Contracts.
The Journal of Derivatives, 18, 75–85 (2011)
[14] J. C. Hull, Options, futures and other derivativs, 6th edition. Prentice Hall, 2006 [15] J.C. Hull, A. White, Numerical procedures for implementing term structures I: single
factor models. Journal of Derivatives 2, 7–16 (1994)
[16] J.C. Hull, A. White, Numerical procedures for implementing term structures II: two fac-tor models. Journal of Derivatives 2, 37–47 (1994)
[17] J.C. Hull, A. White, Using Hull-White interest rate trees. Journal of Derivatives 3, 26-36 (1996)
[18] IEA Natural Gas Information Statistics. OECD Publishing, 2010
http://www.oecdbookshop.org/oecd/display.asp?sf1=identifiers&st1=SUB-64041S2, last accessed: 6 / 9 / 2011
[19] P. Jaillet, E. I. Ronn, S. Tompaidis, Valuation of commodity-based swing options. Man-agement Science 50, 909–921 (2004)
[20] M. Kanai, Decoupling the Oil and the Gas Prices. IFRI papers (2011).
www.ifri.org/downloads/noteenergiemiharukanai.pdf
[21] A. Løland, O. Lindqvist, Valuation of commodity-based swing options:
a survey. Note SAMBA/38/80, Norwegian Computing Center (2008) http://publications.nr.no/swingoptionsurvey.pdf
[22] M. Moreno, J. F. Navas, On the Robustness of Least - Squares Monte Carlo (LSM) for Pricing American Derivatives. Review of Derivatives Research, 6, 2 (2003)
[23] A. Pokorn´a, Pricing of Gas Swing Options. Diploma Thesis, Charles University, Prague (2009) http://ies.fsv.cuni.cz/default/file/download/id/11116 [24] E. Schwartz, J. E. Smith, Short-Term Variations and Long-Term Dynamics in
Commod-ity Prices. Management Science 46 (7), 893–911 (2000)
[25] http://www.theodora.com/pipelines, last accessed: 14/1/2011 [26] World Energy Outlook. IEA 2009