Valuation and hedging of a gas storage utility
The problem of valuing gas storage units has been discussed from many angles in the literature, yielding different approaches and numerical methods. Leasing a gas storage unit is equivalent to paying for the right, but not the obligation, to inject or withdraw gas from the unit. Hence the goal of the owner is to optimize the use of the gas storage facility, by injecting or withdrawing gas from the unit and, at the same time, trading gas on the spot and/or futures market. All these decisions have to be made under many operational constraints, such as maximal and minimal volume of the storage, and limited injection and withdrawal rates. This forces the resolution of a constrained stochastic control problem.
The gas spot price is modeled by a process, denoted by S. We suppose that this process is given as a function of a Markov process X in term of which the optimal control problem will be expressed. For example, in the framework (2.4.1), used by Boogert and De Jong , the spot process is a Markov process, so we take obviously X = S.
Dynamic programming equation
From Table 2.1, we recall that Vti+1 (u) only depends on Vti (u) and ui. To emphasize this fact, if Vti = v, we also express Vti+1 (u) by Vbui (v).
At time t, for Xt = x and with current volume level v, the (optimal) value for gas storage will be of course denoted by J?(t; x; v). The dynamic programming principle implies J ? (ti; x; v) = ui2finj; no; withg ui + E hJ ? (ti+1; Xti+1 ; Vbui (v))jXti = x; Vti = vi (2.3.7) max :
The classic way to solve this problem numerically is to use Monte Carlo simula-tions, combined with the Longstaff and Schwartz  algorithm, which approxi-mates the above conditional expectation, using a regression technique. This backward algorithm yields an estimate of the optimal strategy. As noted by Boogert and De Jong , the main difficulty comes from the fact that the value function also depends on volume level, which in turn depends on the optimal strategy.
To circumvent this difficulty, Boogert and De Jong  suggest discretizing the volume into a finite grid, vl = Vmin + l ; l = 0; :::; L = (Vmax Vmin)= where L is the number of volume subintervals. However, the fact that the time grid is discrete and that at each time the storage unit manager has only three possible actions implies that the number of attainable volumes for any strategy is finite. In fact, at each time ti, the set V(i) of possible volumes is given by V(i) = fVi = v0+kainj t+lawith t , such that Vmin Vi Vmax and k; l 2 N ; k+l ig; consequently, it is possible to solve the dynamic programming equation (2.3.7) for all volumes in V(i). The only motivation to use a restricted volume grid would be the reduction of computation time.
Financial hedging strategy
After estimating the optimal strategy, the storage unit manager will follow these op-timal decisions on the sample path revealed by the market. But one should keep in mind that, if one follows this optimal strategy u?, the cumulative wealth is only the realization of a random variable whose expectation equals the initial price J? of the gas storage unit. This motivates the interest in hedging strategies that can reduce the variance of this random variable. This can be done by combining the optimal oper-ating strategy with additional financial trades, so that the expectation of the related cumulative wealth generated by both physical and financial operations is still J?, but its variance (or some other risk criterion) is reduced. Analogously to Bjerksund et al. , who treats the intrinsic value case, this additional financial hedging strategy plays a role analogous to control variates in the variance reduction of Monte Carlo simulations, as it preserves the expected cumulative cash flows and reduces its vari-ance. In order to reduce the variance of a Monte Carlo estimator of a r.v. Y , one adds to it a mean zero control variate, which is highly (negatively) correlated to Y . Since futures contracts are the most liquid assets in the natural gas market, and are strongly correlated to the spot price, they form an ideal hedging instrument. In fact, although a futures contract price F (t; T ) does not converge to the spot price, when the time to maturity T t goes to zero, the correlation between the prompt contract (for example) and the spot price is very high, and often the two contracts move in the same direction. The basic idea of a financial hedging strategy is to add to the physical spot trading,
Table of contents :
1 Change of numeraire in the two-marginals martingale transport problem.
1.2 Basic left-monotone transference plan: existence and uniqueness
1.2.1 Necessary conditions
1.2.2 Sufficient conditions
1.3 Change of numeraire
1.3.1 The symmetry operator S
1.3.2 The symmetric two-marginals martingale problem
1.3.3 Relation to the generalized Spence-Mirrlees condition
1.3.4 Symmetry and model risk
1.4 Construction of the basic right-monotone transference map via change of numeraire
1.5 Symmetry: Hobson and Klimmek  revisited
1.5.1 Sufficient conditions
1.6 Two new transference plans
1.6.1 What are the payoffs for which this transference plan is optimal ?
1.7.1 Symmetrized payoffs have a lower model risk
1.7.2 Example: the symmetric log normal case
Appendix 1.A Proof of Lemma 1.2.3
Appendix 1.B Proof of Proposition 1.2.4
Appendix 1.C Proof of Lemma 1.2.6
Appendix 1.D Proof of Lemma 1.5.4
2 Gas storage valuation and hedging. A quantification of model risk.
2.2 Natural gas stylized facts
2.3 Valuation and hedging of a gas storage utility
2.3.1 Gas storage specification
2.3.2 Dynamic programming equation
2.3.3 Financial hedging strategy
2.4 Literature on price processes
2.4.1 Spot price processes
2.5 Our modeling framework
2.5.1 Modeling the futures curve
2.5.2 Modeling spot price
2.6 Numerical results
2.7 Model risk
2.7.1 Spot modeling
2.7.2 Model risk measure
Appendix 2.A Different types of gas storage facilities
Appendix 2.B Futures-based valuation methods
3 BSDEs, c`adl`ag martingale problems and mean-variance hedging under basis risk.
3.2 Strong inhomogeneous martingale problem
3.2.1 General considerations
3.2.2 The case of Markov semigroup
3.2.3 Diffusion processes
3.2.4 Variant of diffusion processes
3.2.5 Exponential of additive processes
3.3 The basic BSDE and the deterministic problem
3.3.1 General framework
3.3.2 The forward-backward case and the deterministic problem .
3.3.3 Illustration 1: the Markov semigroup case
3.3.4 Illustration 2: the diffusion case
3.4 Explicit solution for F¨ollmer-Schweizer decomposition in the basis risk
3.4.1 General considerations
3.4.2 Application: exponential of additive processes
3.4.3 Diffusion processes
Appendix 3.A Proof of Proposition 3.2.8
Appendix 3.B Proof of Theorem 3.2.18