I am interested in pricing a gas storage. Most approaches in the literature fit some process to gas prices (either futures or spot) and then try to find a (approximate) optimal policy (in expectation) of injections and withdrawals given the fitted process. In all papers that I found till now, a risk neutral version of the price process is used to set up the problem. This is argued by loose references to risk neutral pricing, where the value is computed as the expectation under the risk neutral probability measure.My understanding of no arbitrage pricing is the following. If the pay-off of a derivative security can be replicated and there is an equivalent risk neutral measure, then the value of the security today is just the expectation with respect to the risk neutral measure (which is unique according to the second part of the fundamental theorem of asset pricing).Hence, completeness of the market, i.e., the ability to replicate every derivative security, is essential for risk neutral pricing. At the heart of every replication strategy is the ability to store the underlying. Hence, in storage valuation the situation is, in my opinion, not so clear: gas storage cannot be replicated with spot or futures trading without owning storage in the first place. Hence, the market seems to be incomplete.My question therefore is: Why not work with the physical measure in the pricing problem? In this case, one can not claim to have found the 'one true' arbitrage free price and the risk preference of the person solving the pricing problem would play a role in determining the price. However, this price anyway does not exist, if the storage can not be replicated.

Replication is a red herring, an accident of history (BS theory), and not necessary for risk-neutral pricing.Let X be anything to bet on, let's say the temperature at noon in Des Moines in a year.Introduce a market: some derivatives on X, say futures or options. It's pretty clear these securities are not something that can be 'replicated'. Hire a team of the world's best meteorologists to develop the real-world (actuarial) probability distribution for X, and some associated expected value for the derivatives.Need the market agree? No, the market can use its own measure (the risk neutral measure). It will generally use a different one if there are systematic risks associated tomoves in X, such that large moves in X are correlated with moves in broader investment categories, like equities. Now let X be gas prices.

I don't disagree with the main thrust of Alan's argument, but it is fair to say that the ability to replicate the cash flow pattern of a contract by a self-financing trading strategy employing traded instruments is a sufficient condition for risk neutral pricing to apply. While formally equivalent, in cases like yours I am inclined to try and do the modeling under the physical measure and explicitly introduce a risk premium (vector if necessary) into the discounting. It simplifies the statistical analysis, or at least its interpretation, and cuts down on confusion. Of course, I know nothing about gas storage....

Thank you for your answers.However, I cannot see the point with the red herring (maybe also due to my rather basic finance education).In your example, an option on the temperatures in Des Moines can be traded on the market. I agree that the price does not have to agree to the expected price with respect to the physical measure, however carefully calibrated to the reality.To my understanding this does not mean that there is a risk free measure (martingale) which can be used to price the option. If there is a martingale that is equivalent to the process governing the underlying (in your example the temperature), this martingale need not be unique. As soon as uniqueness is lost, the pricing argument brakes down, since the expectation with respect to every equivalent martingale is a valid arbitrage free price.The only condition that I know that ensures uniqueness of the martingale measure is the completeness of the market, i.e., the condition that every derivative can be replicated.Maybe this is a little naive way to look at it and I would be thankful if you could point out to me where I took a wrong turn and what other sufficient conditions there are for risk neutral pricing.

It's confusing to me exactly what you are objecting to. You yourself mention the "second part of the fundamental theorem of asset pricing".The first part would seem to answer your question. Forget my "temperature in Des Moines" for the moment and just take a traded securitywith derivatives in an incomplete market. I am not an expert on that theorem but it asserts, roughly, that if that securities market is described by a stochastic process with a measure and there are no arbitrage opportunities, then equivalent (risk neutral) measures exist.So, uniqueness is not necessary in the sense of "uniquely determined by the physical measure".Given a physical process/measure and no arbitrage, there are generally many (mathematically) possible equivalent measures.Then, the idea is that the market picks out one of those. So, when you say "this does not mean there is a risk-free measure" ,at least in this context, I don't get the objection. The theorem is asserting there *is* at least one risk neutral measure.Perhaps the confusion is in two possible meanings of uniqueness? ("uniquely determined by the physical measure" vs "the market picks out one")??This is an aside, and will likely further confuse the issue, but -- strictly speaking -- there really is little justification for a 'physical process' beyond modeling convenience; after all, finance is generally not physics.Even if no physical processes exist (and so the above theorems are empty of content), the market's probability distributions *do* exist and can be read off any existentoption chains: just apply the Breeden-Litzenberger relation. Calling these "risk neutral" distributions is perhaps unfortunate language, but common.

QuoteOriginally posted by: dawoI am interested in pricing a gas storage. Most approaches in the literature fit some process to gas prices (either futures or spot) and then try to find a (approximate) optimal policy (in expectation) of injections and withdrawals given the fitted process. In all papers that I found till now, a risk neutral version of the price process is used to set up the problem. This is argued by loose references to risk neutral pricing, where the value is computed as the expectation under the risk neutral probability measure.My understanding of no arbitrage pricing is the following. If the pay-off of a derivative security can be replicated and there is an equivalent risk neutral measure, then the value of the security today is just the expectation with respect to the risk neutral measure (which is unique according to the second part of the fundamental theorem of asset pricing).Hence, completeness of the market, i.e., the ability to replicate every derivative security, is essential for risk neutral pricing. At the heart of every replication strategy is the ability to store the underlying. Hence, in storage valuation the situation is, in my opinion, not so clear: gas storage cannot be replicated with spot or futures trading without owning storage in the first place. Hence, the market seems to be incomplete.My question therefore is: Why not work with the physical measure in the pricing problem? In this case, one can not claim to have found the 'one true' arbitrage free price and the risk preference of the person solving the pricing problem would play a role in determining the price. However, this price anyway does not exist, if the storage can not be replicated.a (approximate) optimal policy (in expectation) of injections and withdrawals given the fitted process/// 1. in a single deal we have a buy price at t and a sell price at T , T > t what does it means optimal. If T is random moment the solution of the problem when T is fixed deterministic will be generalized.2. what class of the derivatives is implied by storage pricing ///no arbitrage pricing is the following. If the pay-off of a derivative security ///3. risk neutral pricing/// is implied by option pricing. whether does the storage is defined by the option price

Thanks for your clarification, Alan.I see your point: Since there is no arbitrage, the market price is the expectation w.r.t. some martingale (out of a set of equivalent martingale measures). The martingale can be found by calibrating a stochastic model to observed option prices in some way or the other (the Breeden-Litzenberger relation was new to me, but it seems one needs a market with many liquidly traded options to actually use it).I am not an expert in calibration, but it seems to me there are many pitfalls and it is hard to actually find a model/measure that prices all the observed options correctly. Hence, while calibration in incomplete markets is certainly an option, it is much less convincing to me than the complete market case.One of the reasons for this is that prices calculated like this are no longer 'risk neutral', i.e., independent of the preferences of agents. To put it differently: agent A might value a gas storage at a higher price than the market price, while agent B puts the value of the same storage at a price lower than the market price. That is perfectly possible in incomplete markets if A and B have different risk preferences and initial endowments. Therefore the proposed method is theoretically valid to calculate market prices of securities but it can not be used to determine whether any given agent would want to buy/sell it for that price.

@List1To answer your questions:(1) Unlike in the case of a (european) option, holding a storage involves a series non-trivial decisions: One has to decide when to buy and inject and when to sell and withdraw from the storage. This decision is usually taken solving an (stochastic) optimization problem. When no decision is taken (no withdrawals and injections) the storage is worthless.(2) There is no one-to-one correspondence between any standardly traded derivate and a storage. One can view a storage as a basket of (time) spread options, but even in this case there are many different ways to choose options, i.e., one again has to solve an optimization problem.(3) I am afraid, I don't understand this comment.

Buying and selling is always optimization problem. The problem is to get maximum return. In storage problem it would look like to get maximum return minus storage expenses. If V ( t ) is a volume of the gas at t which does not exceed Vo we should define a portion r _ 1 of V ( t ) to be sold keeping in account an estimate of a distribution of the random time of next selling moment. At the selling moment we should define the portion of the r_1V ( t ) to be sold. Similar calculation should be done with buying gas bearing in mind that V ( t ) < Vo . IT is possible that some other factors should be taking into account. This process takes place in continuous time. Of course this is of course complex process but if you do not use GBM as underlying of the hedged position there is no evidence why one should know about risk neutral measure.

QuoteOriginally posted by: dawoThanks for your clarification, Alan.I see your point: Since there is no arbitrage, the market price is the expectation w.r.t. some martingale (out of a set of equivalent martingale measures). The martingale can be found by calibrating a stochastic model to observed option prices in some way or the other (the Breeden-Litzenberger relation was new to me, but it seems one needs a market with many liquidly traded options to actually use it).I am not an expert in calibration, but it seems to me there are many pitfalls and it is hard to actually find a model/measure that prices all the observed options correctly. Hence, while calibration in incomplete markets is certainly an option, it is much less convincing to me than the complete market case.One of the reasons for this is that prices calculated like this are no longer 'risk neutral', i.e., independent of the preferences of agents. To put it differently: agent A might value a gas storage at a higher price than the market price, while agent B puts the value of the same storage at a price lower than the market price. That is perfectly possible in incomplete markets if A and B have different risk preferences and initial endowments. Therefore the proposed method is theoretically valid to calculate market prices of securities but it can not be used to determine whether any given agent would want to buy/sell it for that price.Yes, we are in agreement on all these points. All markets are incomplete and models are always crude descriptions of reality; the best you can do is make them more realistic incrementally.If you are primarily a researcher (and not a trader), to improve your models, sit down with the traders and learn what they know.Joint P/Q calibrations, in principle, let you discover aggregate risk premiums (as mentioned by bearish), which are in some sense the wealth-weighted effect of preferences of market participants. These are generally stochastic.In futures markets, some try to use the CFTC "Commitments of Traders" (COT) reports, which categorize agents into afew small categories (hedgers/commercials, speculators/retail), to deduce something useful. I know nothing about your market, but if there is data on participants, perhaps some type of COT analysis can be developed. Just a thought...p.s. Re Breeden-Litzenberger (BL), you can get surprising far with just a few options: fit their smile to Gatheral's SVI, apply BL and you're good to go.

I suppose there are 2 big reasons to model gas storage optionality:1) You own storage, and want to know how to realize the value of your storage optionality by hedging with options on futures2) You want to understand how storage owners are likely to behave, assuming that they act "rationally"Which of these you are interested in will have quite a strong influence on how you want to model things. If you are interested in 1) then, to the extent that options on futures and futures can replicate your payoffs, you may well want to use risk-neutral measures, since you may actually be able to trade out of your optionality quite effectively. However, NG basis is horrifically volatile and the only liquid options in the US are on HH gas, so you have to be careful with this assumption.If you are interested in 2), you need to think more carefully about how (and in what sense) storage operators are rational. You are likely to find that worrying about the measure is going to be less important than understanding their operational constraints and mechanical/pipeline constraints. As a starting point for either, though, I think bearish's advice is good: you want to understand the physical price process (being careful about the basis between where your storage is located and the traded markets you are hedging with) and then explicitly thinking about what the risk-premium should be.That being said, your answer will also be MASSIVELY model-dependent. The kind of seasonality and jumps in gas prices that you think there are, and the nature of your assumptions about summer/winter price spreads is going to have a huge effect on your optimal injection/withdrawal strategy and, therefore, your option-equivalent price. Modeling gas storage optionality is what we call a "hard" problem: choosing which measure to use is not the biggest problem you face

@Alan: Thanks, for your answer and the references to calibration techniques. I will have a careful look at the methods you suggest.@crmorcom: I assume that when you talk about the value of optionality, you are referring to what is usually called the 'extrinsic value' in the literature, i.e., the value that you can get on top of the 'intrinsic value' (which is the value you can replicate with today's futures not taking account possible future developments).However, according to the above discussion, in an incomplete market, you can compute the market value of a storage, but this value is not (exactly) replicable. What is more: the market value may be an understatement or overstatement of the value for any given agent.This was my initial problem with risk free pricing (although I see it much clearer now after the above discussion with bearish and Alan). One consequence is that the policy (schedule of injections and withdrawals) you get out of the respective optimization problem using the risk neutral measure are not necessarily optimal or rational for all the agents in the market.This was the reason for my question: Is it not better to calibrate a physical process (not easy, I know) and then determine the optimal decisions based on your risk preferences and overall situations in the company?I agree that the answer depends what the application is. I see two possible applications for my purposes:(1) Give guidance to portfolio management on how to use existing storages in an optimal way. Following the above I would rather use the physical measure for this.(2) Tell traders what market prices they can expect, when trading (virtual) storage. In this case I would rather use the risk neutral measure.

Yes - extrinsic value is another way to describe it.In another few hints as to why which measure you use is not your biggest problem (and also, indirectly, why the answer is hard to answer even in principle), think about this:For Natural Gas (and I'm only going to talk about US-traded instruments here, for the sake of clarity), monthly futures are traded. These have zero drift in the risk-neutral measure (they're futures). How do you get to the real-world measure? Well, that should be the actual drift of the "spot price" plus the "convenience yield". What is the spot price? Well there isn't one in a conventional sense. The gas is deliverable (if you're trading CME NG contracts) daily over a whole month after the last trade date: the futures price is the expectation of the forward average of the "spot" price. So you need to think quite hard about which drift you're measuring. You might think you can look at different months to tell, but that's even worse because you know the drift is going to be not just seasonal, but is going to depend on the price spreads. Why? Because the drift (convenience yield) is going to depend on the injection/withdrawal policies of the storage operators (as well as demand, which is seasonal).There's this, too: assuming you can measure a "convenience yield" of some sort, what is your drift risk-premium going to mean? In equities, it's all quite easy in principle. You have very good reasons (beyond Modigliani-Miller) to expect an equity risk-premium, and that it should be strictly positive. That being said, arguments have been raging for decades about how big it is and how constant it is. But for gas prices? Why would you expect commodity prices systematically to drift one way or another? Your risk-free rate is nominal, so you've dealt with inflation already. Why would gas prices systematically drift up (or down) relative to the general price level?So, along with my earlier comments, this should make you suspect that being very precise about your choice of measure *DOESN'T MATTER VERY MUCH*. By all means put in risk-premium parameters (there will be more than one - quite a few, actually, given seasonality and that vol is non-constant and not completely tradeable). If you are careful/honest about measuring/calibrating them, you will find that you have such enormous standard errors associated with them, that you might as well assume they are zero. You are going to have a very complex model, already: you need to be very judicious about how many extra free parameters you add to it.

I agree to your general perception that calibrating/estimating a stochastic model for the forward curve of gas is not easy. However, I disagree with your assessment that if it is difficult to estimate a parameter the best way to go about it is to assume it is zero.To respond to your arguments in detail:(1) To my knowledge there is a spot price on the Henry Hub which is for delivery next day (not the whole month). This could be used to calculate a convenience yield, isn't it?(2) There are multi-factor spot price models that yield both a risk free and a physical measure, i.e., in particular estimate the convenience yield. See for example Short-Term Variations and Long-TermDynamics in Commodity Prices by Schartz&Smith. If this model is augmented with a seasonality component (deterministic or stochastic, both is possible) it should capture most of the features of gas markets.(3) It does not seem very absurd to me that gas or oil prices have a (long-term) upwards trend. Theory of finite resources gives a theoretical foundation for this (Hotelling's rule) and the common sense realization that as cheap sources deplete more expensive sources (shale gas/oil, deep sea drilling, oil-sands, ...) are tapped driving up the price also backs this view. I admit that these fundamental factors are not the only drivers of price and have a rather slow effect that is frequently overshadowed by short-term market sentiments as well as political and macro-economic developments.(4) Of course, (3) says nothing about why the price of one specific future contract, say Dec 2015, should have an upward trend. This in my opinion can only be explained by convenience yield and the price of storing the commodity.

I'm not saying at all that you should assume the parameter is zero. I'm trying to explain why worrying overly at this stage about which measure you will use is a little beside the point, which was your original question. As I posted earlier, I agree with bearish's point about simultaneously estimating/calibrating/fitting risk-premia along with other parameters. And I am also trying to explain why, when you have gone through all this, using classical tests or fairly uninformative priors you are likely to find that you have very imprecise estimates for price drift risk-premia. Trust me on this. Vol risk-premia are another matter entirely, but you've not yet mentioned variable volatility (see (2))(2) I quite like Schwartz and Smith, by the way, and have used modified versions of it myself - it's a useful starting point. There are a few features of gas markets it is missing which you might expect to make a large difference in pricing storage: - volatility is *highly* non-constant, and not just seasonally - even beyond that, march/april and oct/nov prices/volatilities have a more complex (and rather important relative to injection/withdrawal decisions) dependence on price spread levels(3) I think what you're trying to say, in the context of a model which cares about the next year and will be fitted against no more than a decade of data, is that you agree with me (4) Did you just sneak in an extra 12 parameters per year? How are you going to reliably identify a different drift for an individual futures price? If you find that your drift for, say Z15 is different from the one you get for Z16, how are you going to explain the difference? Are you going to believe that they're different? Why? We're talking the drift here, of course, not the individual path realization.