[huuan26]

Summary

Implements shifted short-rate support in BDTTree (financepy/models/bdt_tree.py) to allow calibration and derivative valuation under negative interest rates, resolving #20.

Mathematical Formulation & Changes

• Shifted Lognormal Short-Rate Process:
  Extends the constant-volatility Black–Derman–Toy short-rate dynamics to:

$$d\ln(r_t + s) = \theta(t)\,dt + \sigma\,dW_t, \quad r_t + s > 0$$

where $s$ (shift: float = 0.0 by default) uses the same sign convention as BlackShifted and SABRShifted for full backward compatibility.

• Tree Calibration (f, search_root, build_tree_fast, BDTTree):
  • Propagates the positive shifted short rate $x_{m,i} = x_0 \exp\big(2(i - m_0)\sigma\sqrt{\Delta t}\big)$ across nodes at step $m$ (with middle node $m_0 = \lfloor m/2 \rfloor$) and stores the actual short rate $r_{m,i} = x_{m,i} - s$ in rt[m, i], so Arrow–Debreu state price updates and backward-induction pricers (bond_option, bermudan_swaption, callable_puttable_bond_tree) discount at the true (possibly negative) short rate without modification.
  • Validates that discount factors are positive and that $r_t + s > 0$, i.e. $DF(t_{m+1}) < \left(\sum_{i=0}^m Q_{m,i}\right) e^{s \Delta t}$, at every time step, raising FinError when the shift is insufficient for a negative-rate curve.

Testing

• Added unit tests in tests/unit/test_FinModelRatesBDT.py:
  • test_shifted_bdt_negative_rates_calibration_and_spacing: Verifies Arrow–Debreu state price calibration $\sum_{i=0}^m Q_{m,i} = DF(t_m)$, log-spacing of $r_{m,i} + s$, and European bond option put–call parity on a negative-rate discount curve with $r = -1.5%$ and $s = 4%$.
  • test_shifted_bdt_zero_vol_shift_invariance_and_pricing: Verifies that at $\sigma = 0$, calibrated short rates and option/swaption values under negative rates are invariant to the choice of valid shift.
  • test_shifted_bdt_rejects_rates_below_negative_shift: Verifies that FinError is raised when $r_t + s \le 0$ either initially or at later tree steps.

Fixes #20