# Intertemporal pricing theory for futures

(Last Updated On: July 7, 2015)

Intertemporal pricing theory for futures

This set of notes will definitely screw your mind if you are not strong in math. I would suggest watching this video. I am still trying to wrap my head around this but this will most likely explain pricing dynamics in the futures market for any storable commodity.

Images attached so refer to these as well

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Theory of normal backwardation (keynes)
Theory of price Storage ( h working)

This involves the theory of pricing of different delivery methods and how they relate to the spot price

Theory of backwardation argues futures markets are set up for hedging purposes. So the market is set up for hedgers. Hedgers are always net short. So speculators are not long.
If you take all the hedges in the market, the number of shorts will exceed the number of long positions. It means that hedgers are always selling let me buy for price protection. Speculators are not long which we assume speculators a risk reverse. It also means you are investing for profit so you’re not expecting a negative return.
Is also seen that speculators cannot assume or forecast prices. His theory explains that speculators are selling insurance to hedgers. Hedgers are trying to protect themselves well speculators are providing insurance for hedgers. They charge something called a risk premium.

If you have 100 days to maturity, but your cash price is \$2.50. As demonstrated, we expect the cash price to rise because of storage costs. If the cost of storage rises from \$2.50 to \$3 we can expect the cost of storage to be \$.50. If the speculator did not charge a risk premium
Some may buy at the future price of \$2.90 because it could still make it \$.10 profit at time of maturity but the hedger will need to lower the future Price for more speculators. If you’re expected spot price at maturity is going to be three dollars, the difference will become the risk premium. This is what the specular will need to pay for insurance on the future price. The hedger will not be able to sell at the expected spot price. Hedger needs to sell at a lower price of the expected spot price. The future price is expected to rise as you approach maturity which will rise relative to the expected spot price.
This backwardation argues that the cash price is determined by supply and demand in the market. It also argues that the future price is equal to the expected cash price minus the risk premium. This also becomes the theory of price storage.
There is a relationship between The cash price and the future price has a downward bias estimate of the forthcoming cash price. This is why it is called the theory of normal backwardation which is a downward biased relationship. You can view the futures price as a prediction of the forthcoming cash price. People still debate about the risk premium. Speculators are willing to sell the insurance for a very low risk premium.

Theory of price storage
The cash and the futures price are not Autonomous as they are determined together. Rather the futures market existing for The purposes of allowing hedgers to buy insurance. It is said that intertemporal prices reflect supply and demand for storage. Are you focus on inventory of stock.

Ft,t t Price of today while T is delivery time of the future.
See attached equation for future price. St is spot price today plus the opportunity cost in money investested in in the spot commodity (r). Wt,T his warehouse cost of between current time and time of. Delivery plus convenience yield (Ct,T)
Warehouse cost is the total cost including overhead like insurance
Convenience yield is is a negative cost or a return associated with holding a physical inventory.
This equation is the standard arbitrage equation for a storable commodity, if the arbitrage condition holds then you have an equilibrium situation. This is one the futures price is the same as the spot price plus cost of holding the inventory minus the convenience yield.

If the futures price is greater then spot price plus the cost of money in the spot market Plus the warehouse cost plus the convenience yield, we have an arbitrage opportunity. See attached image for equation
In this case you buy the spot sell the future
For instance, Ct,T=0 spot price is \$2.90 r=.05 W=.10 your future price equals 3.05 there is no arbitrage opportunity. If the future price is equal to \$3.20 while all other cause stay the same, you will have an opportunity for arbitrage. Buy the spot at \$2.90 which will cost you \$.15 to hold it, so the future price of \$3.20 you will earn a riskless return. You deliver on the contract over the hundred days, you’ll earn a \$.15 per bushel. If people do this, over time people will increase the spot price but reduce the future price so it will hit a equilibrium. You keep doing this until there is no longer an arbitrage opportunity. In essence, this arbitrage opportunity works because there is a lot of warehouse for a storable commodity. Overtime the market will hit its equilibrium, the prices will come in the line according to the arbitrage equation.

Convenience yield is a controversial term since it has a negative cost.

If Ft,T < St(1+rt,T)+wt,T this implies that convenient yield exist
Ct,T = S(1+rt,T).+Wt,T-Ft,T see attached image for equation

If spot price is equal to \$2.90 are is equal to five cents, W equals \$.10, he is equal to five cents
2.9+.05+.1-3 if future price is three dollars C equals .05 see attached example image
Convenience field is a returnas a cost for holding the inventory. There are various reasons for this return as a cause for holding inventory. Simplest reason is like keeping money in your pocket versus keeping it in the bank.
The money in your pocket is like a yield for the convenience of having it. It is like food process is having an inventory in case there is a lock out of stock.Beer company will keep keep an inventory of barley on hand to retain quality of taste. Barley is only harvested once a year. They will keep a one year supply on hand.

St volume of stocks at a current time.
Pt+1 – Pt as an intertemporal time spread. t+1 could be a futures contract as in March or January. T could also be a spot price. St is volume of stock that is carried from one period To the next period. The inter-temporal price spread or a future spread depends on supply and demand of the stocks or St. You have to drive the supply and demand of the stock. See attached chart

Pt=f(Ct) see attached equation
This is an inverse demand curve
If you differentiate of function at t/with respect to consumption is less than zero. See attached image
This is your simple demand curve. The law of demand says if you lower the price the demand goes up. Is a downward sloping curve.
Ct

Consumption in a current time is equal to stocks coming in Carried from a previous Time plus production at t (x) minus stock carried out
Ct=St-1 + Xt – St see attached equation
Ct is consumption
St-1 is incoming stock
Xt is production
St is outgoing stocks

Price spread of the theory of storage

Pt+1 – Pt = ft+1(St+xt+1 – St+1) -ft(St-1 + xt -St) see attached equation
Remember that Ct and Ct+1 or equal to certain terms in above equation

We are interested in the relationship between the price spread and the stocks. If you differentiate (Pt+[)/2St. See attached equation. This equals using the chain rule The differential of ft+1/2Ct+1 * differential ofCt+1/St – differential of ft /Delta Ct * 2 Ct /delta St see attached equation
Because the definition of the demand curve is negative certain terms will also be negative in the above equation
Based on the equation if inventory consumption of tea plus one, if inventory is carried out from T to T plus one if everything else stays constant, consumption should go up. This is because you’re carrying inventory into the next period. The above equation that is attached, by definition it needs to rise.
Similarly if the current consumption of the current time is equal to incoming stocks plus production minus outgoing stock, The consumption must fall
If we need to differentiate the price spread regarding outgoing stock, it is a downward sloping relationship. The attached chart for D
D is that demand for inventory substituted the definition of consumption
In the marketplace for stocks, like any market there is a supply and demand.why are inventories carried from one period To the next? The answer is there are market participants to have a demand for those commodities for the next period. There’s always a demand for that storage as in heating oil.
If the outgoing stock falls, the price spread will Pt Will fall because as of lower consumption. If less inventory is carried into next period, Pt+1 Will go up so spread becomes wider. In attached image your spread will go from point a to point B on demand curve D.
Demand curve D is unstable as it will shift depending on various shocks to production, outgoing stocks, Etc.
So as an example, corn could have a bountiful crop harvest. Or a new oil well has been discovered so production will increase.so if production goes up, demand for storage will shift to the right to the D prime. See attached image.
If expected production will go down, the demand Curve will also shift to the right.
Is expected carry out is to rise, the demand will also shift to the right.

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