Take Black-Sholes, for example

 

V is option price in the future, S is current price. r is the safe rate, the safe rate from whom? 

What is t? t really is the number of coin tosses in total before one has to start again, in the target. Log(t) because this will be lognormal for N, the total number of transactions. The total number of transactions is what we need to renormalize the system, scale it properly with the value added nets so we are not pretending anything about safe anybody; risk is shared. This is hidden Markov model from one angle. N is uncertain, it is a consensus with bound variation that all parties are aware of.  Check me on the t thing, it was off the top.

The solution is logbinomial, finite bound positive variation. But if you have a central bank then you need the central bank in your portfolio. You will fat model bank taxes. Or you need a balancing model for government dead weight, which is a value net, and can cover tax distortions by direct balance. A government value net model includes anything the Fed does, since you government model assumes those taxes directly and they are paid in instability of the value net.  This is the best Markov model for government.

r is interest collected per round of interest swaps which happen as needed, another random, but bound,  variable. 

If you want the true value of a corporation, it is their liquidity. In pure sandbox, that is the Loan/Savings second derivative, relative to coin toss count. The ability of the company to respond and stay at maximum entropy. That second derivative means the corporation is capturing gans and avoiding losses almost as fast as the pit boss. 

But in pure sandbox, stock markets are redundant, meaningless because the market queues have no relation to their business. In pure sandbox model, corporations just keep and S/L account at the automated S/L.

A corporation is really, in the Markov model, a common standard of accounting and thus will have a constant transaction rate which really helps to maintain maximum entropy. So in the model we just collapse the balance sheet into a savings vs loans account, i which deposits and loans are synchronously traded at auction .  This is the neutral money model, the assumption being that the banker is nearly instantaneous relative to physical transactions, in the automated version, it requires trading bots.

The stock market as we know it exists because no one had a clue about fair banking until sometime this century.  The coefficients on the Black-Sholes all represent distortions by a central banker. Use the sandbox model, move time up to the insurance level, a posterior estimate, ex post.  Sandbox is semi Martingale, it only remembers two step back.

When you fit the model, you will find he automated pit boss only counts once per game, and that forces the imbalance,  to account for bounded variation. That is spin, that is Pauli exclusion it happens because the entropy of spin is zero, and allocation of coin tosses goes according to entropy.  That pit boss is the cause, and one can see it in the market since the pit boss must serialize bids.  So the pit boss intervenes 'rarely', intervening means, confirm a bid.  If the pit boss intervenes too much then N is very low and the binomials are out of balance, you gonna wobble. That means you are off your Markov point and on to another.

Sandbox is the Lucas model, all paths connected and closed, entropy maximized. It is the adaptive Shannon model. Time is not excluded, it is passed up.