Hi guys,I try to decide whether a geometric / multiplicative or an arithmetic / additive model would be better to simulate a price process say of TTF natural gas.If you measure volatility as std of log returns, then a a geometric model would produce a flat line in a price vs volatility plane, while the arithmetic model would have a declining volatility line from low to high prices.Problem is that if I plot that historically it does not proof anything because at least the price is not stationary, hence any relationship could be coincidental because the time series trended.If I just look at the distribution fit of lognormal (geometric) and normal (arithmetic) then very different price level also destroy this idea. How could I decide this?CheersSteilermeier

In the absence of an options market?You would typically study the relationship between dx and x, where x is your variable of interest. I collected a very large number of reference on this a few years ago from a rates perspective. I would need to dig this up. You could do a search for 'backbone' if you want to hit material written by quants, 'level effect' for more financial econometrics related ideas, 'additive vs multiplicative noise' maybe also.Rebonato has one or two papers on this also. There is also a paper by Ho and Goodman.

A very good basic educational problem. It is interesting to read answers.

How might this relate to non-stationarity, as that might be a bigger issue?

When we use say 30 day past period to define average or stdv of the return at a moment t at the moment t + 1 we can conclude time to time that behavior of these estimates of these primary parameters on [ t , t + 1 ] do not corresponds to their estimates. From this fact it might come a feeling that these parameters are not good estimated because of its non stationary nature.

maximum likelihood

But all tests or estimates deal with past data and its remarkable deviations seem be probable in the next future

You want to be a little careful about how you handle non log-normality for price processes. The reason why much of the literature about backbones etc is for interest-rates is that they are not prices. In the long-run, you should not expect, e.g., a gas price to look very different proportionally at $3 from how it does at $2. How is $3 special in a way that $2 is not? It is one thing to have processes that give, say, stochastic volatility dynamics which result in higher volatility for higher price in the short term, but quite another to bake the price level into your longer-term vol dynamics.If you are proposing such a model, why? What's the intuition as to why the price level should matter? And over what time-period are you looking to estimate? And what time-scale do you expect your model to apply to in the future?

I know essentially zero about oil and gas, but still it is easy to imagine why price levels can matter. For example, let's say associated to each price level, there are a certain number of production rigs which are profitable at that level.If the price falls below that level for n days, those rigs may be taken off-line. I'm sure things are not that simple, but my point is that, for numerous reasons, price levels can matter in financial times series. As another example, while price levels probably don't matter much for the broad based equity indices (partly due to how they are maintained), they will matter at the single name stock level, particularly at low levels.I agree that there are short-term and long-term distinctions here, too. And, it is certainly a burden of the modeler to show why particular price levels should be important.

Alan is very modest, he hasn't mentioned that he is an author of the classic paper on modelling asset prices dynamics when dependent on asset price level...P

Obviously, Alan, in the short run I completely agree with you - as I implied. But I still think it's dangerous to bake price level relationships into dynamic models, at least unless you are being much more precise and specific about the sources of the dependence than a crude "how log-normal is it?" level. Certainly, in gas, it's usually more useful to use up modeling degrees of freedom in worrying, say, about seasonality and storage.

The first question I would have is: what are your criteria for the "best model"? Do you want the closest fit on the long-term distribution or a better fit on local distribution? In your application, what is the cost/profit of being wrong/right? Is it worse to get the tails wrong (on one side or the other) or the body of the distribution wrong?The second issue is that the dynamic model would probably be dominated by short-term price inelasticity and long-term price elasticity. In the short-term both suppliers and consumers of gas probably have more fixed commitments or a economic model that lacks volume flexibility. A modest shock to supply (e.g., problem in the delivery network) or demand (e.g., a cold snap) sends prices surging.Third, it seems to me that both Alan and crmorcom are right. Price level is important. But a specific price level is NOT baked into the dynamics for all time. At any given time, both producers and consumers have a elasticity tied to prices. The tricky part about natural gas is that any price level function will have both near-absolute components (e.g., supply-side gas extraction & delivery costs that make it profitable/unprofitable) and relative components (e.g., demand-side substitution with other energy sources that link the price levels of gas to the prices of coal, oil, etc.). Both would vary over time.Finally, it's not clear to me that any fixed model will work that well given broader technological and policy changes. Surely the rise of renewable energy, fracking, flex-fuel vehicles, and carbon emission regulations might have dramatic effects on elasticity over time.

QuoteOriginally posted by: Traden4AlphaThe first question I would have is: what are your criteria for the "best model"? Do you want the closest fit on the long-term distribution or a better fit on local distribution? In your application, what is the cost/profit of being wrong/right? Is it worse to get the tails wrong (on one side or the other) or the body of the distribution wrong?The second issue is that the dynamic model would probably be dominated by short-term price inelasticity and long-term price elasticity. In the short-term both suppliers and consumers of gas probably have more fixed commitments or a economic model that lacks volume flexibility. A modest shock to supply (e.g., problem in the delivery network) or demand (e.g., a cold snap) sends prices surging.Third, it seems to me that both Alan and crmorcom are right. Price level is important. But a specific price level is NOT baked into the dynamics for all time. At any given time, both producers and consumers have a elasticity tied to prices. The tricky part about natural gas is that any price level function will have both near-absolute components (e.g., supply-side gas extraction & delivery costs that make it profitable/unprofitable) and relative components (e.g., demand-side substitution with other energy sources that link the price levels of gas to the prices of coal, oil, etc.). Both would vary over time.Finally, it's not clear to me that any fixed model will work that well given broader technological and policy changes. Surely the rise of renewable energy, fracking, flex-fuel vehicles, and carbon emission regulations might have dramatic effects on elasticity over time.It is a standard [$]\chi ^2[$] test which can used to verify Gaussian distribution implied by geometric or arithmetic models. This test can be used with say the same 30 degrees of freedom to verify which one better macht to Gaussian distribution of Wiener process in the models. If price level is important then drift or variance or both of them should depend on price. It is possible to make a test taking return and volatility when price changes within a particular strips, ie price range can be divided [$] 0 \,= \,S _0 \,<\, .\,. \,. \,< S _n \, = S_{max}[$] and check average return and variance when observed data belongs to each strip.