I was thinking about energy futures (or forwards) when I was struck by something which I can not understand. Perhaps somebody can offer some insight? If we look at the forward price of gas, then we see that it is a sinusoidial type graph with maturity (i.e. high forward price in winter, low in summer). This makes sense since we know that winter spot prices are higher than summer spots. But, In calculating the future price or forward price of gas for a maturity T, we use (as suggested by Schwartz): F(t,T) = E* [ S(T) ] where * denotes risk free measure. The SDE for S might incorporate mean reversion. My question is this: Does this method take into account that forward prices are sinusoidal? I.e. if we evaluate the expression and do a plot of F versus T, will we get a picture that shows high prices in winter, and lower prices in summer? I don't see how.My point is that, surley, we have to incorporate the costs of storage somewhere, and MAYBE this is what will make the forward prices sinusoidal... There is an article about storage in the wilmott magazine but I have struggled to understand it. Thanks, Sam

Firstly, if F(t,T) = E* [ S(T) ] , then you can still have peaks and troughs - the forwards will rise and fall in just the same way as we expect the spot to rise and fall.What I think you are referring to is the equation F(t,T)=S(t)e^(r(T-t))This only allows the sinusoidal behaviour of which you speak if you generalise to F(t,T)=S(t)e^(r-u+c)(T-t)where r is the interest rate, c is the cost of storage, and u is the convenience yield (defined as the benefit accruing to the holder of the asset but not to the holder of the forward). Although in reality all of r, u and c will be time dependent.In summer, all the gas is being stored, so the cost of storage goes up. Likewise, the convenience of having physical gas is pretty low - it isn't as if there is suddenly going to be a shortage. So c goes up and u goes down, and the forward price falls. So the forward price of gas at the start of winter (after the long summer) is higher than the spot price at the start of summer. Giving the desired result.At an extreme, think of electricity during a spike. The spot price is $10,000, while the forward price for the next day is $40. The reason for this is that the benefit of having electricity now (the convenience yield) is enormous, and the storage cost is virtually zero, as no one in their right mind would want to store it, rather than selling it now.In some ways, the convenience yield isn't quantifiable in itself - it is just a balancing entry to give the desired relationship between spot prices and forward prices at different times. If you take this view - well, then, you always get exactly the shape you want.

Thanks mate,This is exactly what I was looking forRegards,Sam

Since forwards can be found from the market, and should be Martingales, one usually considers models likeF(t,T) = F(0,T) + x(t)H(T) + ...where x(t) is a random (Gaussian?) variable with mean zeroH(T) is the main component of the movement in the gas forwardsOne can add other factors. Alternatively, one can start with F(t,T) = F(0,T)*X(t)*...where X(t) is a log normal model with mean 1.At least in the forward measure, X(t) must be a Martingale

Hi,I was thinking about this some more and came across an inconsistency when pricing the forward price of a commodity:If you assume a portfolio short in the forward contract and long in the underlying then you can easily derive the arbitrage free forward price at time t to be F(t,T) = S(t).exp((r-u+c)(T-t))...where we assume that r is interest, u is convenience, c is storage cost. This comes from a cost of carry analysis...At time = t,Short F(t,T)Long energy ---> finance this through loan at risk free rateStore enegyat time = T,remove energy from storage and use it to meet obligation of F(t,T)repay loan, pay storage, incorporate convenience yield, cost = s(t)exp((r+c-u)(T-t))therefore for no arb, F(t,T)=S(t)exp((r+c-u)(T-t))I was trying to derive the relationship going the other way... Assume that we go long one forward contract and short the underlying.At time = t,Buy F(t,T)Short S(t) ----> invest at risk free rateAt time =TPayment for underlying = F(t,T)S(t) grows at risk free rate, you must also incorporate the convenience yield---> S(t) at time t grows to S(t).exp((r-u)(T-t)) at time TFor no arb, F(t,T) = S(t).exp((r-u)(T-t))No storage cost comes into the analysis!!! Therefore, have we got 2 arb free prices? What have I missed here?Thanks,Sam

These arbitrage arguments only give inequalities - the combination of them will be required to give an equality.So, a short forward, long physical arbitrage analysis, gives:F(t,T)<=S(t)exp((r+c-u)(T-t)) and a long forward, short physical arbitrage analysis gives:F(t,T) >= S(t)exp((r-u)(T-t))so we getS(t)exp((r-u)(T-t)) <= F(t,T) <=S(t)exp((r+c-u)(T-t)) which is not an inconsistency (as the left hand side will be less than the right had side). But basically, no one will sell you forwards for less than it costs them to produce them, so the forward price tends to be the right hand side.Keep in mind, however, that the way that most people look at these equations, c and u are derived from S and F, not the other way round.

I feel that the second arbitrage argument proposed by Sam is incorrect. Even if we invest the money recieved from selling the underlying at the risk free rate, the price of underlying at time T should be equal to S(t)exp(r-u+c)*(T-t), coz any merchant who bought the underlying at time t has incurred storage costs and benefited from convenience. He will charge a price for the period accordingly. Thus, in effect we can have only one arbitrage argumentF(T,t) = S(t) exp(r-c+u)(T-t) if any trader sells futures at a lesser price, BUY IT, coz he is gonna incur the costs of storage and you benefit!!Correct me if I am wrong DareD

gjlipman,There is an arbitrage here:You can write a forward contract and hedge it with a long position on the same forward. But I think you are on the right line... I read somewhere that the energy markets are full of these kind of arbitrages, but they are often difficult to exploit since participants must have a presence in the spot, derivatives and storage markets.... Thanks,Sam

What I meant is, if, for example, F(t,T) = S(t)exp((r+c-u)(T-t)) ,there is no way to arbitrage it, despite your knowledge that S(t)exp((r-u)(T-t)) <= F(t,T)if you short the physical, and go long the forward, you will lose money as S(t)exp((r-u)(T-t))<F(t,T)and if you go long the physical and short the forward, you will have no money.so there is no way to arbitrage and make money.(if you look at the last bit of section 1.1 in Hull 4th edition, they give the standard arbitrage logic for a gold future. note that they need to go both ways to get one equality - one way on its own only gives an inequality for the price).

gjlipman,Agreed. There is no arb if you use the forward and the underlying... I was refering to the arb from using only the forwards. What we have been saying so far (to my understanding) is that there is a buyers price and a sellers price, and for the forward markets these are different due to the cost of storage. If you look carefully at your equations, you see that the long position (buying price):F(t,T) >= S(t)exp((r-u)(T-t))is SOMETIMES (depending on the actual numbers) lower than the selling price (which is the short position analysis):F(t,T)<=S(t)exp((r+c-u)(T-t)) Therefore, you can buy low, sell high. Don't even need to think about the underlying because you can close you position buy going long the forward contract and shorting the SAME maturity contract. As you said this is redundant since c and u are usually unknown and calculated from the forward prices.Thanks,Sam

On second thought... forget what I just said... I got confused!! There is only one forward price as you said... Thanks for all the help,Sam