#!/bin/python
# -*- coding: utf-8 -*-
from __future__ import division, print_function, absolute_import
import numpy as np
import mafipy.function
# ----------------------------------------------------------------------------
# Black payers/receivers swaption
# ----------------------------------------------------------------------------
[docs]def black_payers_swaption_value(
init_swap_rate, option_strike, swap_annuity, option_maturity, vol):
"""black_payers_swaption_value
calculates value of payer's swaptions under black model.
.. math::
\\begin{eqnarray}
V_{\mathrm{payersswap}}(t)
& = &
A(t)
\mathrm{E}_{t}^{A}
\left[
(S(T) - K)^{+}
\\right]
\\\\
& = &
A(t)(S(t)N(d_{1}) - KN(d_{2})),
\\\\
d_{1}
& = &
\\frac{
\ln\left(\\frac{S(t)}{K} \\right)
+ \\frac{1}{2}\sigma^{2}(T - t)
}{
\sigma \sqrt{T - t}
},
\\\\
d_{2}
& = &
\\frac{
\ln\left(\\frac{S(t)}{K} \\right)
- \\frac{1}{2}\sigma^{2}(T - t)
}{
\sigma \sqrt{T - t}
}
\end{eqnarray}
where
:math:`A(t)` is `swap_annuity`,
:math:`S(t)` is `init_swap_rate`,
:math:`K` is `option_strike`,
:math:`\sigma` is `vol`.
:param float init_swap_rate: initial swap rate.
:param float option_strike: swaption strike.
:param float swap_annuity: annuity of referencing swap
:param float option_maturity: swaption maturity.
:param float vol: volatilty. this must be positive.
:raises AssertionError: if volatility is not positive.
"""
assert(vol > 0.0)
option_value = mafipy.function.black_scholes_call_value(
init_swap_rate, option_strike, 0.0, option_maturity, vol)
return swap_annuity * option_value
[docs]def black_receivers_swaption_value(
init_swap_rate, option_strike, swap_annuity, option_maturity, vol):
"""black_receivers_swaption_value
calculates value of receiver's swaptions under black model.
.. math::
\\begin{eqnarray}
V_{\mathrm{receiver}}(t; S(t), K, A(t), T, \sigma)
& = &
A(t)
\mathrm{E}_{t}^{A}
\left[
(K - S(T))^{+}
\\right]
\end{eqnarray}
where
:math:`S(t)` is `init_swap_rate`,
:math:`A(t)` is `swap_annuity`,
:math:`K` is `option_strike`,
:math:`T` is `option_maturity`,
:math:`\sigma` is `vol`.
:math:`d_{1}, d_{2}` are defined in :py:func:`black_payers_swaption_value`.
:param init_swap_rate: initial swap rate.
:param option_strike: strike of swaption.
:param swap_annuity: annuity of referencing swap.
:param option_maturity: maturity of swaption.
:param vol: volatility. This must be non-negative.
:raises AssertionError: if volatility is not positive.
"""
assert(vol > 0.0)
# option is expired
if option_maturity < 0.0 or np.isclose(option_maturity, 0.0):
return 0.0
payer_value = black_payers_swaption_value(
init_swap_rate, option_strike, swap_annuity, option_maturity, vol)
forward_value = swap_annuity * (init_swap_rate - option_strike)
return payer_value - forward_value
[docs]def black_payers_swaption_value_fprime_by_strike(
init_swap_rate,
option_strike,
swap_annuity,
option_maturity,
vol):
"""black_payers_swaption_value_fprime_by_strike
First derivative of value of payer's swaption with respect to strike
under black model.
See :py:func:`black_payers_swaption_value`.
.. math::
\\frac{\partial }{\partial K}
V_{\mathrm{payer}}(K; S, A, T, \sigma)
= - A\Phi(d_{2}(K))
where
:math:`S` is `init_swap_rate`,
:math:`K` is `option_strike`,
:math:`A` is `swap_annuity`,
:math:`T` is `option_maturity`,
:math:`\sigma` is `vol`,
:math:`d_{1}, d_{2}` is defined
in :py:func:`black_payers_swaption_value`,
:math:`\Phi(\cdot)` is c.d.f. of standard normal distribution,
:math:`\phi(\cdot)` is p.d.f. of standard normal distribution.
:param float init_swap_rate:
:param float option_strike:
:param float swap_annuity:
:param float option_maturity:
:param float vol: volatility. must be non-negative.
:return: value of derivative.
:rtype: float
:raises AssertionError: if volatility is not positive.
.. note::
Roughly speaking, this function calculates :math:`A(t) \Phi(d)`.
Percentile of probability 1 is infinity so that
assumption that annuity is equal to 1 is not good assumption
beacuse of :math:`\Phi(d)` returns 1 in some cases.
"""
assert(vol > 0.0)
# option is expired
if option_maturity < 0.0 or np.isclose(option_maturity, 0.0):
return 0.0
value = mafipy.function.black_scholes_call_value_fprime_by_strike(
init_swap_rate, option_strike, 0.0, option_maturity, vol)
return swap_annuity * value
[docs]def black_payers_swaption_value_fhess_by_strike(
init_swap_rate,
option_strike,
swap_annuity,
option_maturity,
vol):
"""black_payers_swaption_value_fhess_by_strike
Second derivative of value of payer's swaption with respect to strike
under black model.
See :py:func:`black_payers_swaption_value`.
.. math::
\\frac{\partial^{2} }{\partial K^{2}}
V_{\mathrm{payer}}(K; S, A, T, \sigma)
= - A\phi(d_{2}(K)) d_{2}^{\prime}(K)
where
:math:`S` is `init_swap_rate`,
:math:`K` is `option_strike`,
:math:`A` is `swap_annuity`,
:math:`T` is `option_maturity`,
:math:`\sigma` is `vol`,
:math:`d_{1}, d_{2}` is defined
in :py:func:`black_payers_swaption_value`,
:math:`\Phi(\cdot)` is c.d.f. of standard normal distribution,
:math:`\phi(\cdot)` is p.d.f. of standard normal distribution.
:param float init_swap_rate: initial swap rate.
:param float option_strike:
:param float swap_annuity:
:param float option_maturity:
:param float vol: volatility. must be non-negative.
:return: value of derivative.
:rtype: float
:raises AssertionError: if volatility is not positive.
"""
assert(vol > 0.0)
value = mafipy.function.black_scholes_call_value_fhess_by_strike(
init_swap_rate, option_strike, 0.0, option_maturity, vol)
return swap_annuity * value
[docs]def black_payers_swaption_value_third_by_strike(
init_swap_rate,
option_strike,
swap_annuity,
option_maturity,
vol):
"""black_payers_swaption_value_third_by_strike
Third derivative of value of payer's swaption with respect to strike
under black model.
See :py:func:`black_payers_swaption_value`.
.. math::
\\frac{\partial^{3} }{\partial K^{3}}
V_{\mathrm{payer}}(K; S, A, T, \sigma)
= - A
\left(
\phi^{\prime}(d_{2}(K)) (d_{2}^{\prime}(K))^{2}
+ \phi(d_{2}(K)) d_{2}^{\prime\prime}(K)
\\right)
where
:math:`S` is `init_swap_rate`,
:math:`K` is `option_strike`,
:math:`A` is `swap_annuity`,
:math:`T` is `option_maturity`,
:math:`\sigma` is `vol`,
:math:`d_{1}, d_{2}` is defined
in :py:func:`black_payers_swaption_value`,
:math:`\Phi(\cdot)` is c.d.f. of standard normal distribution,
:math:`\phi(\cdot)` is p.d.f. of standard normal distribution.
:param float init_swap_rate: initial swap rate.
:param float option_strike: strike of swaption.
:param float swap_annuity: annuity of referencing swap.
:param float option_maturity: maturity of swaption.
:param float vol: volatility. must be non-negative.
:return: value of derivative.
:rtype: float
:raises AssertionError: if volatility is not positive.
"""
assert(vol > 0.0)
# option is expired
if option_maturity < 0.0 or np.isclose(option_maturity, 0.0):
return 0.0
value = mafipy.function.black_scholes_call_value_third_by_strike(
init_swap_rate, option_strike, 0.0, option_maturity, vol)
return swap_annuity * value
# ----------------------------------------------------------------------------
# black payer's/reciever's swaption greeks
# ----------------------------------------------------------------------------
[docs]def black_payers_swaption_delta(
init_swap_rate, option_strike, swap_annuity, option_maturity, vol):
"""black_payers_swaption_delta
calculates delta of payer's swaptions under black model.
:param float init_swap_rate: initial swap rate.
:param float option_strike: swaption strike.
:param float swap_annuity: annuity of referencing swap
:param float option_maturity: swaption maturity.
:param float vol: volatilty. this must be positive.
:return: delta
:rtype: float.
:raises AssertionError: if volatility is not positive.
"""
assert(vol > 0.0)
bs_delta = mafipy.function.black_scholes_call_delta(
init_swap_rate, option_strike, 0.0, option_maturity, vol)
return swap_annuity * bs_delta
[docs]def black_payers_swaption_vega(
init_swap_rate, option_strike, swap_annuity, option_maturity, vol):
"""black_payers_swaption_vega
calculates vega of payer's swaption under black model.
:param float init_swap_rate: initial swap rate.
:param float option_strike: swaption strike.
:param float swap_annuity: annuity of referencing swap
:param float option_maturity: swaption maturity.
:param float vol: volatilty. this must be positive.
:return: vega.
:rtype: float.
:raises AssertionError: if volatility is not positive.
"""
assert(vol > 0.0)
bs_vega = mafipy.function.black_scholes_call_vega(
init_swap_rate, option_strike, 0.0, option_maturity, vol)
return swap_annuity * bs_vega
[docs]def black_payers_swaption_volga(
init_swap_rate, option_strike, swap_annuity, option_maturity, vol):
"""black_payers_swaption_volga
calculates volga of payer's swaption under black model.
:param float init_swap_rate: initial swap rate.
:param float option_strike: swaption strike.
:param float swap_annuity: annuity of referencing swap
:param float option_maturity: swaption maturity.
:param float vol: volatilty. this must be positive.
:return: volga.
:rtype: float.
:raises AssertionError: if volatility is not positive.
"""
assert(vol > 0.0)
bs_volga = mafipy.function.black_scholes_call_volga(
init_swap_rate, option_strike, 0.0, option_maturity, vol)
return swap_annuity * bs_volga
[docs]def black_payers_swaption_vega_fprime_by_strike(
init_swap_rate, option_strike, swap_annuity, option_maturity, vol):
"""black_payers_swaption_vega_fprime_by_strike
calculates derivative of vega with respect to strike under black model.
This is required for :py:func:`sabr_pdf`.
:param float init_swap_rate: initial swap rate.
:param float option_strike: swaption strike.
:param float swap_annuity: annuity of referencing swap
:param float option_maturity: swaption maturity.
:param float vol: volatilty. this must be positive.
:return: derivative of vega w.r.t. strike.
:rtype: float.
:raises AssertionError: if volatility is not positive.
"""
assert(vol > 0.0)
bs_vega_fprime = mafipy.function.black_scholes_call_vega_fprime_by_strike(
init_swap_rate, option_strike, 0.0, option_maturity, vol)
return swap_annuity * bs_vega_fprime
# ----------------------------------------------------------------------------
# black payer's/reciever's swaption distribution
# ----------------------------------------------------------------------------
[docs]def black_swaption_cdf(
init_swap_rate, option_strike, swap_annuity, option_maturity, vol):
"""black_swaption_cdf
calculates value of c.d.f. of black swaption.
:py:func:`black_payers_swaption_value_fprime_by_strike`.
:param float init_swap_rate: initial swap rate.
:param float option_strike: option strike.
:param float swap_annuity: swap annuity.
:param float option_maturity: maturity of swaption.
:param float vol: volatility. non-negative.
:return: value of c.d.f. of black swaption model.
:rtype: float.
"""
assert(vol > 0.0)
return (1.0
+ black_payers_swaption_value_fprime_by_strike(
init_swap_rate,
option_strike,
swap_annuity,
option_maturity,
vol) / swap_annuity)
[docs]def black_swaption_pdf(
init_swap_rate, option_strike, swap_annuity, option_maturity, vol):
"""black_swaption_pdf
calculates value of p.d.f. of black swaption.
:py:func:`black_payers_swaption_value_fhess_by_strike`.
:param float init_swap_rate: initial swap rate.
:param float option_strike: option strike.
:param float swap_annuity: swap annuity.
:param float option_maturity: maturity of swaption.
:param float vol: volatility. non-negative.
:return: value of p.d.f. of black swaption model.
:rtype: float.
"""
assert(vol > 0.0)
return (black_payers_swaption_value_fhess_by_strike(
init_swap_rate,
option_strike,
swap_annuity,
option_maturity,
vol) / swap_annuity)