Sergius


Total Posts: 6 
Joined: Jan 2020 


Can you help me with pricing theory?
There are three assets: A, B and C with prices P_A, P_B and P_C respectively. There is a production or transportation process that transforms A to B and B to C.
Let P_B and P_C be connected: P_B=P_A⋅x, P_C=P_B⋅x=P_A⋅x^2.
For example, some product is being transported from point A to point B and then from point B to point C. Its price rises because of expenses (let them be proportional to price for simplicity), risks, interest rate, etc.
So if x changes then P_B react linearly and P_C nonlinearly. It is possible to create a portfolio that always wins. Am I understand right that such model has to be impossible on an efficient market? Maybe possible, but not stable because P_B or P_C is mispriced.
Thanks! 




nikol


Total Posts: 1126 
Joined: Jun 2005 


1) It's difficult to imagine production or transportation costs which are proportional to the price of the initial product.
2) "It is possible to create a portfolio that always wins". Show it. Do you mean that you want to: a) buy/sell products themselves or b) lock into contracts with buy/sell commitments ? There might be the case such that arbitrage possibility might exist in theory but never comes on surface. 



ahd


Total Posts: 30 
Joined: May 2017 


@sergius, do you have an example in mind? As you say, the cost of a unit of B could be proportional to the cost of a unit of A if B is made from A and demand for B is stable, i.e. P_B / P_A = x as P_A moves around. But normally if C is made from B then P_C/P_B = y where y!=x. You could have y=x at a given moment but it still wouldn't be a structural relationship. Do you have an example in mind where y=x structurally?
edit: i just signed in to check things out and i noticed that my post said "...P_C/P_B = y where yx". i think i used sidebyside angle brackets between y and x as a notequals sign and they got dropped. i just tried it and i'm right: the angle brackets show up fine in preview mode but must be interpreted as html tags and get dropped in the post. live and learn... 




Sergius


Total Posts: 6 
Joined: Jan 2020 


@ahd, I've chosen y=x for simplicity, as a starting point. It is theoretical only. But if it is "not right" in theory then it may be worth searching for alike inefficiencies in real life. Prices depend on a lot of parameters. Some of them (like interest rates, risks, etc.) are included in price models as z^n multipliers (P_C=P_A*x^2). For example, someone buys goods at point A, transport to B and sells there. He or she invests price P_A and expenses E and want to make some profit. Some expenses are price proportional. For example, insurance. Profit depends on riskfree interest rate and risks. Profit is also price proportional. The same is with B>C. P_B=(P_A+E1)*z and P_C=(P_B+E2)*z=(P_A+E1)*z^2+E2*z.
I'm not trying to find a profitable strategy that is ready to use. If this kind of inefficiency really exist it is hard to use. If the idea is right in theory then finding out ways how to use it would be the next step. 



gaj


Total Posts: 103 
Joined: Apr 2018 


you can only conclude P_B <= P_A * x. equality does not have to hold.
> It is possible to create a portfolio that always wins.
how exactly?
> Am I understand right that such model has to be impossible on an efficient market?
efficient market does not rule out non linear relationships, e.g., options. 




Sergius


Total Posts: 6 
Joined: Jan 2020 


@nikol 1) You are right. What I've described is very simple theoretical case. Real cases are much more complicated. Expenses are more likely not to be proportional to the price (except insurance and maybe some others). But interest rate and risks are proportional. I have described this case in my previous answer. 2) Let P_B price be proportional to x and P_C  to x^2. You can create portfolio V=a*P_B +b*P_C. If you choose a and b that dV/dx=0 then V rises from every change of x. 



Sergius


Total Posts: 6 
Joined: Jan 2020 


@gaj, thanks! > how exactly? Another strategy in addition to described in my previous reply. Borrow two units of B, exchange one to A, one to C, wait until x changes and then exchange back to B. Amount of B rises if x changes. 




gaj


Total Posts: 103 
Joined: Apr 2018 


> If you choose a and b that dV/dx=0 ...
a and b depends on x. when x moves, you have to adjust the portfolio. what is the cost of rebalancing?
> dV/dx=0 then V rises from every change of x
those two statements are literally the opposite of each other 



Sergius


Total Posts: 6 
Joined: Jan 2020 


@gaj
you choose a and b for current value of x=x0. There is no need to rebalance it. dV/dx=0 in x0, but d^2V/dx^2 in x0 > 0. It's a parabolic function with minimum in x0. 




gaj


Total Posts: 103 
Joined: Apr 2018 


ok i understand now. i believe your argument is correct, it is indeed an arbitrage within this model.
several points already mentioned above that make this model unrealistic in the real world: 1) production cost doesn't give you the exact pricing, only a bound. 2) costs aren't usually multiplicative. 3) the cost from A>B and B>C have to be the same at all times, it's hard to imagine how this would happen in the real world. 



Sergius


Total Posts: 6 
Joined: Jan 2020 


@gaj thanks! This is what I wanted to know. I agree with all three points, but this is only a special case. Actually, to make this idea work it is not necessary to be P_B/P_A=P_C/P_B. My next step is to show that one multiplicative parameter (interest rate, risk, etc) among many additive in complex chain models (A>B>C) is enough to make prices slightly inefficient (and contain opportunities, which may be hidden, hard to use, but still opportunities). It would mean that some normal processes (economical, technological,etc.) contain inefficiency in their nature. I find this idea interesting. Who knows, maybe I'll get interesting results in the end. 



