Product and Sum Category Theory Models of Financial Contracts
Product Model: Pay or Not-Pay
Sum Model: Financial Legal Remedies
1. Recall the Arrow-Debreu-Sharpe model of t0 to t1 corresponding to the state of initial financial contract in a world of infinite contingent states and the state of pay, respectively.
2. We improved the ADS Model to a Default Invariance Model where the t1 state is now bi-valued to include (a) Pay and (b) Not-Pay, i.e. P and -P.
3. If t1 results in -P then the infinite contingent states continues at t1. This is equivalent (or isomorphic) to the infinite-contingent-states being multiplied by 1.
4. If t1 results in P then the infinite contingent states is annihilated and the certainty of payment makes the financial contract certain and therefore, immediately disengages from the infinitely contingent states of world. This is equivalent (or isomorphic) to the inifinite-contingent-states being multiplied by 0.
5. The 2-state at t1 default invariance model can be further specified in terms of Product and Sum. [This is going to get a bit technical, I warn you.]
Definition of Product
An object P together with a pair of maps P1:P->B1, and P2:P->B2 is called a product of B1 and B2 if for each object X and each pair of maps f1:X->B1, f2:X->B2, there is exactly one map f:X->P for which both f1=P1f and f2=P2f. [See Lawvere & Schanuel, 2009, p. 217.]
[I'll insert a diagram later. Hint: it looks like a chevron with X on the left and an arrow from X to P which is
6. Given this definition of Product, we can now apply what we stated in an earlier blog that payment of a financial contract makes it certain and therefore, takes it out of the realm of uncertainty and is no longer part of ("resides in") a world of infinite contingent states. Thus, an occurrence of payment is equivalent to the value of 0 in the infinite contingent world. In our Product Diagram B1 = (Infinite-Contigency) x (0). Another way of saying this is B1 = Uncertainty x 0, which means, no more uncertainty. This valuation is not what Sharpe and others had supposed, and had in fact given payment the value of 1, which leads to inconsistent and contradictory results.
7. Also, we can see that non-payment or not-paying at t1 means that the infinite-contingent-states of the world continues at t1. This is equivalent to: (infinite-contingent-states) x (1). So, non-payment is actually an identity morphism. In our Default Invariance Product Diagram, B2 = 1. Strangely, f2 will have to be equivalent to infinite contingency divided by infinite contingency.
8. As a sum, we have the following definition:
A pair j1: B1->S, j2; B2->S, of maps in a category makes S a sum of B1 and B2 if for ach object Y and each pair g1:B1-Y, g2:B2-Y, there is exactly one map g:S->Y for which both g1 = gj1 and g2 = gj2. [See Lawvere & Schanuel (2009) Conceptual Mathematics, p. 222]
It is my contention that a legal remedy to a financial contract has the form of a sum as above, where Y is a legal remedy and B1 and B2 to breach and not-breach situations of the contract. This is of course a first approximation of a legal risk theory.
9. The virtue of a Product and a Sum Model for the fundamental and universal unit of law and finance is that by putting them together THROUGH TIME (that is, from t0, t1, t2,...tn) we begin to have a view of how law and finance may be seen under one perspective that allows for both (1) simplification and anticipation of direct results -- i.e., "rough and ready" calculations that border on immediate insight through very complex legal and financial phenomena; and (2) a very detailed and rigorous, bookkeeping or tracking methodology to ensure that our predictions make sense and are grounded on facts.
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