Why Time-Decay SABR Effective Parameters Require Piecewise Formulas
Summary
The document asks whether the effective parameters in a time-decay SABR model can be expressed by one closed-form formula across all values of the decay start time. It presents two results from Sander Willems’s backward-looking SABR work: one for decay already active at the initial time and another for a flat period followed by decay. The formulas use different parameter expressions, including a separate quantity in the flat-then-decay case.
The author reports implementing each theorem and checking the algebra and code against the paper’s appendices, but does not provide a unifying derivation, numerical comparison, or follow-up reference. The central issue is therefore an open implementation question rather than a demonstrated result: whether analytic continuation or approximation can replace a sign-based branch. The document supplies no evidence that a single formula exists or that either approximation is reliable, so the stated piecewise regimes remain the only formulas described.
Key ideas
- The backward-looking SABR results use different effective-parameter formulas for two decay-start regimes.
- The branch depends on whether decay is active at the initial time or begins later.
- The author reports checking both implementations against the paper’s appendices.
- The document does not establish a unified formula or validate an analytic continuation or approximation.
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# Generalizing Sander Willems’s Time-Decay SABR Effective-Parameter Formula
# Generalizing Sander Willems’s Time-Decay SABR Effective-Parameter Formula
I’m implementing Sander Willems’s “Backward-looking SABR” (arXiv:2004.04501v4) and have run into a bit of confusion around the two closed-form theorems:
- Theorem 4.1 covers the case $T_s\le0$ (i.e. we’re already in the decay regime at $t=0$), and gives $$ \hat\nu^2 = \frac{\nu^2\,\zeta}{2q+1},\quad \hat\rho = \frac{2\rho}{\sqrt{\zeta\,(3q+2)}},\quad \hat\alpha^2 = \frac{\alpha^2}{2q+1}\Bigl(\frac{\tau_1}{\tau_1-\tau_0}\Bigr)^{2q} e^{\tfrac12(\nu^2/(2q+1)\,\zeta)\,\tau_1}. $$
- Theorem 4.2 covers $T_s\ge0$ (flat-then-decay) and uses a totally different $\gamma$-based formula: $$ \hat\nu^2 = \nu^2\,\gamma\frac{2q+1}{T^3\tau_1},\quad \hat\rho = \rho\,\frac{3T^2 + 2q\,\tau_0^2 + \tau_1^2}{\sqrt{\gamma}(6q+4)},\quad \hat\alpha^2 = \frac{\alpha^2}{(2q+1)\,T\tau_1}e^{\tfrac12\,H\,\tau_1}. $$
In my Python/C++ code I’ve faithfully implemented both branches and switch based on the sign of $T_s$. However, I hoped there might be a single “master” formula (like the $\gamma$–form) that simply continues through $T_s=0$ and automatically reduces to Theorem 4.1 when $T_s<0$.
Is there any known way to unify these two theorems into one closed-form expression valid for all $T_s\in\mathbb{R}$? Or must one always use the piecewise approach (Thm 4.1 for $T_s\le0$, Thm 4.2 for $T_s\ge0$)? Have others tried an analytic continuation or approximation to avoid the sign-check in code?
I’ve double-checked the algebra and code against Willems’s appendices but so far haven’t found a single-shot formula. Any insight (or references to follow-up work) would be much appreciated!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.