Gradients#

class rateslib.solver.Gradients#

Bases: object

A catalogue of all the gradients used in optimisation routines and risk sensitivties.

Attributes Summary

J

2d Jacobian array of calibrating instrument rates with respect to curve variables, of size (n, m);

J2

3d array of second derivatives of calibrating instrument rates with respect to curve variables, of size (n, n, m);

J2_pre

3d array of second derivatives of calibrating instrument rates with respect to curve variables for all Solvers including pre_solvers, of size (pre_n, pre_n, pre_m);

grad_s_s_vT

3d array of second derivatives of curve variables with respect to calibrating instruments, of size (m, m, n);

grad_s_s_vT_pre

3d array of second derivatives of curve variables with respect to calibrating instruments, of size (pre_m, pre_m, pre_n);

grad_s_vT

2d Jacobian array of curve variables with respect to calibrating instruments, of size (m, n);

grad_s_vT_pre

2d Jacobian array of curve variables with respect to calibrating instruments including all pre solvers attached to the Solver, of size (pre_m, pre_n).

grad_v_rT

Alias of J.

grad_v_v_rT

Alias of J2.

grad_v_v_rT_pre

Alias of J2_pre.

Methods Summary

grad_f_Pbase(npv, grad_f_P, f, fx_vars)

1d array of derivatives of base currency PV with respect to FX rate variables, of size (len(fx_vars)).

grad_f_Ploc(npv, fx_vars)

1d array of derivatives of local currency PV with respect to FX rate variable, of size (len(fx_vars)).

grad_f_f(f, fx_vars)

1d array of total derivatives of FX conversion rate with respect to FX rate variables, of size (len(fx_vars));

grad_f_fT_Pbase(npv, grad_f_fT_P, f, fx_vars)

2d array of derivatives of base currency PV with respect to calibrating instrument rate variables, of size (pre_m, pre_m).

grad_f_fT_Ploc(npv, fx_vars)

2d array of derivatives of local currency PV with respect to FX rate variables, of size (len(fx_vars), len(fx_vars)).

grad_f_fT_f_pre(f, fx_vars)

2d array of derivatives of FX conversion rate with respect to calibrating instruments, of size (pre_m, pre_m);

grad_f_f_rT_pre(fx_vars)

3d array of second derivatives of calibrating instrument rates with respect to FX rates, of size (len(fx_vars), len(fx_vars), pre_m);

grad_f_f_vT_pre(fx_vars)

3d array of second derivatives of curve variables with respect to FX rates, of size (len(fx_vars), len(fx_vars), pre_n);

grad_f_rT_pre(fx_vars)

2d Jacobian array of calibrating instrument rates with respect to FX rate variables, of size (len(fx_vars), pre_m);

grad_f_sT_Pbase(npv, grad_f_sT_P, f, fx_vars)

2d array of derivatives of base currency PV with respect to FX variables and calibrating instrument rate variables, of size (len(fx_vars), pre_m).

grad_f_sT_Ploc(npv, fx_vars)

2d array of derivatives of local currency PV with respect to calibrating instruments, of size (pre_m, pre_m).

grad_f_sT_f_pre(f, fx_vars)

2d array of derivatives of FX conversion rate with respect to calibrating instruments, of size (pre_m, pre_m);

grad_f_s_vT_pre(fx_vars)

3d array of second derivatives of curve variables with respect to FX rates and calibrating instrument rates, of size (len(fx_vars), pre_m, pre_n);

grad_f_vT_pre(fx_vars)

2d array of the derivatives of curve variables with respect to FX rates, of size (len(fx_vars), pre_n).

grad_f_v_rT_pre(fx_vars)

3d array of second derivatives of calibrating instrument rates with respect to FX rates and curve variables, of size (len(fx_vars), pre_n, pre_m);

grad_s_Pbase(npv, grad_s_P, f)

1d array of derivatives of base currency PV with respect to calibrating instruments, of size (pre_m).

grad_s_Ploc(npv)

1d array of derivatives of local currency PV with respect to calibrating instruments, of size (pre_m).

grad_s_f_pre(f)

1d array of FX conversion rate with respect to calibrating instruments, of size (pre_m);

grad_s_sT_Pbase(npv, grad_s_sT_P, f)

2d array of derivatives of base currency PV with respect to calibrating instrument rate variables, of size (pre_m, pre_m).

grad_s_sT_Ploc(npv)

2d array of derivatives of local currency PV with respect to calibrating instruments, of size (pre_m, pre_m).

grad_s_sT_f_pre(f)

2d array of derivatives of FX conversion rate with respect to calibrating instruments, of size (pre_m, pre_m);

gradp_f_vT_Ploc(npv, fx_vars)

2d array of (partial) derivatives of local currency PV with respect to FX rate variables and curve variables, of size (len(fx_vars), pre_n).

Attributes Documentation

J#

2d Jacobian array of calibrating instrument rates with respect to curve variables, of size (n, m);

\[[J]_{i,j} = [\nabla_\mathbf{v} \mathbf{r^T}]_{i,j} = \frac{\partial r_j}{\partial v_i}\]

Depends on self.r.

J2#

3d array of second derivatives of calibrating instrument rates with respect to curve variables, of size (n, n, m);

\[[J2]_{i,j,k} = [\nabla_\mathbf{v} \nabla_\mathbf{v} \mathbf{r^T}]_{i,j,k} = \frac{\partial^2 r_k}{\partial v_i \partial v_j}\]

Depends on self.r.

J2_pre#

3d array of second derivatives of calibrating instrument rates with respect to curve variables for all Solvers including pre_solvers, of size (pre_n, pre_n, pre_m);

\[[J2]_{i,j,k} = [\nabla_\mathbf{v} \nabla_\mathbf{v} \mathbf{r^T}]_{i,j,k} = \frac{\partial^2 r_k}{\partial v_i \partial v_j}\]

Depends on self.r and pre_solvers.J2.

grad_s_s_vT#

3d array of second derivatives of curve variables with respect to calibrating instruments, of size (m, m, n);

\[[\nabla_\mathbf{s} \nabla_\mathbf{s} \mathbf{v^T}]_{i,j,k} = \frac{\partial^2 v_k}{\partial s_i \partial s_j}\]
grad_s_s_vT_pre#

3d array of second derivatives of curve variables with respect to calibrating instruments, of size (pre_m, pre_m, pre_n);

\[[\nabla_\mathbf{s} \nabla_\mathbf{s} \mathbf{v^T}]_{i,j,k} = \frac{\partial^2 v_k}{\partial s_i \partial s_j}\]
grad_s_vT#

2d Jacobian array of curve variables with respect to calibrating instruments, of size (m, n);

\[[\nabla_\mathbf{s}\mathbf{v^T}]_{i,j} = \frac{\partial v_j}{\partial s_i} = \mathbf{J^+}\]
grad_s_vT_pre#

2d Jacobian array of curve variables with respect to calibrating instruments including all pre solvers attached to the Solver, of size (pre_m, pre_n).

\[[\nabla_\mathbf{s}\mathbf{v^T}]_{i,j} = \frac{\partial v_j}{\partial s_i} = \mathbf{J^+}\]
grad_v_rT#

Alias of J.

grad_v_v_rT#

Alias of J2.

grad_v_v_rT_pre#

Alias of J2_pre.

Methods Documentation

grad_f_Pbase(npv, grad_f_P, f, fx_vars)#

1d array of derivatives of base currency PV with respect to FX rate variables, of size (len(fx_vars)).

\[\nabla_\mathbf{s} P^{bas}(\mathbf{v(s, f)}) = \nabla_\mathbf{s} P^{loc}(\mathbf{v(s, f)}) f_{loc:bas} + P^{loc} \nabla_\mathbf{s} f_{loc:bas}\]
Parameters:

  • npv – Dual or Dual2 A local currency NPV of a period of a leg.

  • grad_f_P – ndarray The local currency delta risks w.r.t. FX pair variables.

  • f – Dual or Dual2 The local:base FX rate.

  • fx_vars – list or tuple of str The variable tags for automatic differentiation of FX rate sensitivity

grad_f_Ploc(npv, fx_vars)#

1d array of derivatives of local currency PV with respect to FX rate variable, of size (len(fx_vars)).

\[\begin{split}\\nabla_\\mathbf{f} P^{loc}(\\mathbf{v(s, f), f}) = \\frac{\\partial P^{loc}}{\\partial f_i}+ \\frac{\partial v_z}{\\partial f_i} \\frac{\\partial P^{loc}}{\\partial v_z}\end{split}\]
Parameters:

  • npv – Dual or Dual2 A local currency NPV of a period of a leg.

  • fx_vars – list or tuple of str The variable tags for automatic differentiation of FX rate sensitivity

grad_f_f(f, fx_vars)#

1d array of total derivatives of FX conversion rate with respect to FX rate variables, of size (len(fx_vars));

\[[\nabla_\mathbf{f} f_{loc:bas}]_{i} = \frac{d f}{d f_i}\]
Parameters:

  • f (Dual or Dual2) – The value of the local to base FX conversion rate.

  • fx_vars (list or tuple of str) – The variable name tags for the FX rate sensitivities

grad_f_fT_Pbase(npv, grad_f_fT_P, f, fx_vars)#

2d array of derivatives of base currency PV with respect to calibrating instrument rate variables, of size (pre_m, pre_m).

\[\nabla_\mathbf{s} \nabla_\mathbf{s}^\mathbf{T} P^{bas}(\mathbf{v(s, f), f})\]
Parameters:

  • npv – Dual or Dual2 A local currency NPV of a period of a leg.

  • grad_f_fT_P – ndarray The local currency gamma risks w.r.t. FX rate variables.

  • f – Dual or Dual2 The local:base FX rate.

  • fx_vars – list or tuple of str The variable tags for automatic differentiation of FX rate sensitivity

grad_f_fT_Ploc(npv, fx_vars)#

2d array of derivatives of local currency PV with respect to FX rate variables, of size (len(fx_vars), len(fx_vars)).

\[\nabla_\mathbf{f} \nabla_\mathbf{s}^\mathbf{T} P^{loc}(\mathbf{v(s, f), f}) = \frac{ d^2 P^{loc}(\mathbf{v(s, f), f)}) }{d f_i d f_j}\]
Parameters:

  • npv – Dual2 A local currency NPV of a period of a leg.

  • fx_vars – list or tuple of str The variable tags for automatic differentiation of FX rate sensitivity

grad_f_fT_f_pre(f, fx_vars)#

2d array of derivatives of FX conversion rate with respect to calibrating instruments, of size (pre_m, pre_m);

\[[\nabla_\mathbf{f} \nabla_\mathbf{f}^\mathbf{T} f_{loc:bas}(\mathbf{v(s, f), f)})]_{i,j} = \frac{d^2 f}{d f_i d f_j}\]
Parameters:

  • f (Dual or Dual2) – The value of the local to base FX conversion rate.

  • fx_vars (list or tuple of str) – The variable name tags for the FX rate sensitivities

grad_f_f_rT_pre(fx_vars)#

3d array of second derivatives of calibrating instrument rates with respect to FX rates, of size (len(fx_vars), len(fx_vars), pre_m);

\[[\nabla_\mathbf{f} \nabla_\mathbf{f} \mathbf{r^T}]_{i,j,k} = \frac{\partial^2 r_k}{\partial f_i \partial f_j}\]
Parameters:

fx_vars (list or tuple of str) – The variable name tags for the FX rate sensitivities.

grad_f_f_vT_pre(fx_vars)#

3d array of second derivatives of curve variables with respect to FX rates, of size (len(fx_vars), len(fx_vars), pre_n);

\[[\nabla_\mathbf{f} \nabla_\mathbf{f} \mathbf{v^T}]_{i,j,k} = \frac{\partial^2 v_k}{\partial f_i \partial f_j}\]
Parameters:

fx_vars (list or tuple of str) – The variable name tags for the FX rate sensitivities.

grad_f_rT_pre(fx_vars)#

2d Jacobian array of calibrating instrument rates with respect to FX rate variables, of size (len(fx_vars), pre_m);

\[[\nabla_\mathbf{f}\mathbf{r^T}]_{i,j} = \frac{\partial r_j}{\partial f_i}\]
Parameters:

fx_vars (list or tuple of str) – The variable name tags for the FX rate sensitivities.

grad_f_sT_Pbase(npv, grad_f_sT_P, f, fx_vars)#

2d array of derivatives of base currency PV with respect to FX variables and calibrating instrument rate variables, of size (len(fx_vars), pre_m).

\[\nabla_\mathbf{f} \nabla_\mathbf{s}^\mathbf{T} P^{bas}(\mathbf{v(s, f), f})\]
Parameters:

  • npv – Dual or Dual2 A local currency NPV of a period of a leg.

  • grad_f_sT_P – ndarray The local currency gamma risks w.r.t. FX rate variables and calibrating instrument variables.

  • f – Dual or Dual2 The local:base FX rate.

  • fx_vars – list or tuple of str The variable tags for automatic differentiation of FX rate sensitivity

grad_f_sT_Ploc(npv, fx_vars)#

2d array of derivatives of local currency PV with respect to calibrating instruments, of size (pre_m, pre_m).

\[\nabla_\mathbf{f} \nabla_\mathbf{s}^\mathbf{T} P^{loc}(\mathbf{v(s, f), f}) = \frac{ d^2 P^{loc}(\mathbf{v(s, f), f)}) }{d f_i \partial s_j}\]
Parameters:

  • npv – Dual2 A local currency NPV of a period of a leg.

  • fx_vars – list or tuple of str The variable tags for automatic differentiation of FX rate sensitivity

grad_f_sT_f_pre(f, fx_vars)#

2d array of derivatives of FX conversion rate with respect to calibrating instruments, of size (pre_m, pre_m);

\[[\nabla_\mathbf{f} \nabla_\mathbf{s}^\mathbf{T} f_{loc:bas}(\mathbf{v(s, f), f)})]_{i,j} = \frac{d^2 f}{d f_i \partial s_j}\]
Parameters:

  • f (Dual or Dual2) – The value of the local to base FX conversion rate.

  • fx_vars (list or tuple of str) – The variable name tags for the FX rate sensitivities

grad_f_s_vT_pre(fx_vars)#

3d array of second derivatives of curve variables with respect to FX rates and calibrating instrument rates, of size (len(fx_vars), pre_m, pre_n);

\[[\nabla_\mathbf{f} \nabla_\mathbf{s} \mathbf{v^T}]_{i,j,k} = \frac{\partial^2 v_k}{\partial f_i \partial s_j}\]
Parameters:

fx_vars (list or tuple of str) – The variable name tags for the FX rate sensitivities.

grad_f_vT_pre(fx_vars)#

2d array of the derivatives of curve variables with respect to FX rates, of size (len(fx_vars), pre_n).

\[[\nabla_\mathbf{f}\mathbf{v^T}]_{i,j} = \frac{\partial v_j}{\partial f_i} = -\frac{\partial r_z}{\partial f_i} \frac{\partial v_j}{\partial s_z}\]
Parameters:

fx_vars (list or tuple of str) – The variable name tags for the FX rate sensitivities

grad_f_v_rT_pre(fx_vars)#

3d array of second derivatives of calibrating instrument rates with respect to FX rates and curve variables, of size (len(fx_vars), pre_n, pre_m);

\[[\nabla_\mathbf{f} \nabla_\mathbf{v} \mathbf{r^T}]_{i,j,k} = \frac{\partial^2 r_k}{\partial f_i \partial v_j}\]
Parameters:

fx_vars (list or tuple of str) – The variable name tags for the FX rate sensitivities.

grad_s_Pbase(npv, grad_s_P, f)#

1d array of derivatives of base currency PV with respect to calibrating instruments, of size (pre_m).

\[\nabla_\mathbf{s} P^{bas}(\mathbf{v(s, f)}) = \nabla_\mathbf{s} P^{loc}(\mathbf{v(s, f)}) f_{loc:bas} + P^{loc} \nabla_\mathbf{s} f_{loc:bas}\]
Parameters:

  • npv – Dual or Dual2 A local currency NPV of a period of a leg.

  • grad_s_P – ndarray The local currency delta risks w.r.t. calibrating instruments.

  • f – Dual or Dual2 The local:base FX rate.

grad_s_Ploc(npv)#

1d array of derivatives of local currency PV with respect to calibrating instruments, of size (pre_m).

\[\nabla_\mathbf{s} P^{loc} = \frac{\partial P^{loc}}{\partial s_i}\]
Parameters:

npv – Dual or Dual2 A local currency NPV of a period of a leg.

grad_s_f_pre(f)#

1d array of FX conversion rate with respect to calibrating instruments, of size (pre_m);

\[[\nabla_\mathbf{s} f_{loc:bas}]_{i} = \frac{\partial f}{\partial s_i}\]
Parameters:

f (Dual or Dual2) – The value of the local to base FX conversion rate.

grad_s_sT_Pbase(npv, grad_s_sT_P, f)#

2d array of derivatives of base currency PV with respect to calibrating instrument rate variables, of size (pre_m, pre_m).

\[\nabla_\mathbf{s} \nabla_\mathbf{s}^\mathbf{T} P^{bas}(\mathbf{v(s, f), f})\]
Parameters:

  • npv – Dual or Dual2 A local currency NPV of a period of a leg.

  • grad_s_sT_P – ndarray The local currency gamma risks w.r.t. calibrating instrument variables.

  • f – Dual or Dual2 The local:base FX rate.

grad_s_sT_Ploc(npv)#

2d array of derivatives of local currency PV with respect to calibrating instruments, of size (pre_m, pre_m).

\[\nabla_\mathbf{s} \nabla_\mathbf{s}^\mathbf{T} P^{loc}(\mathbf{v, f}) = \frac{ \partial^2 P^{loc}(\mathbf{v(s, f)}) }{\partial s_i \partial s_j}\]
Parameters:

npv – Dual2 A local currency NPV of a period of a leg.

grad_s_sT_f_pre(f)#

2d array of derivatives of FX conversion rate with respect to calibrating instruments, of size (pre_m, pre_m);

\[[\nabla_\mathbf{s} \nabla_\mathbf{s}^\mathbf{T} f_{loc:bas}]_{i,j} = \frac{\partial^2 f}{\partial s_i \partial s_j}\]
Parameters:

f (Dual or Dual2) – The value of the local to base FX conversion rate.

gradp_f_vT_Ploc(npv, fx_vars)#

2d array of (partial) derivatives of local currency PV with respect to FX rate variables and curve variables, of size (len(fx_vars), pre_n).

\[\nabla_\mathbf{f} \nabla_\mathbf{v}^\mathbf{T} P^{loc}(\mathbf{v, f}) = \frac{ \partial ^2 P^{loc}(\mathbf{v, f)}) }{\partial f_i \partial v_j}\]
Parameters:

  • npv – Dual2 A local currency NPV of a period of a leg.

  • fx_vars – list or tuple of str The variable tags for automatic differentiation of FX rate sensitivity