CIR利子計算の理論

ヴァシチェックの計算式を理解してからコックス、ロス、ルビンスタインの方法をかんがえはじめた。While the derivation of the time derivatives Bt and At follow from the term structure, the solutions A and B obtained from the Riccati equations represented by these time derivatives are not so obvious. Strangely, Riccati equations, where y' is written in terms of y^2 involve the determination of a "general" solution based on a known "specific" solution. It is unclear why the specific solution cannot be taken as the final solution. Strangely, in many cases, the general solution and specific solutions are incompatible, namely substituting for y in the general solution with the specific solution leads to a singularity, ie infinity.
A Riccati equation seems to be roughly equivalent to a Cauchy problem, if any sort of specific solution or initial value is given. A Cauchy problem is a PDE for which the solution is constrained to a hypersurface. the Riccati equation can be simplified to a Bernoulli equation, y' + p(x)y=q(x)y^n via change of variable.
Interestingly, the Bt formula in the CIR model is a Riccati equation with constant coefficients, apparently meaning that it's solution is well-defined, even without a given specific solution. Interestingly, my integration of B'=ab +(1/2)(sigma^2)B^2-1, which expresses B in terms of a tangent, does not agree with the solution in print, which uses an exponential in the numerator and the denominator.
Interestingly, another solution to the Riccati equation for B which converts it to a first-order ODE also seems incorrect, as the arithmetic discards a "v" solution in the denominator.
The A solutions, based on direct integration, also seem byzantine. For now, work on the Cox-Ingersoll-Ross bond problem will be suspended, and the bond pricing formula will be considered unvalidated. The simplest cross-check would be to differentiate the solutions with respect to t.