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from pyisda.flat_hazard import strike_vec
from quantlib.time.api import Date
import array
from scipy.optimize import brentq
from scipy.integrate import simps
import datetime
import numpy as np
import pandas as pd
from pandas.tseries.offsets import BDay
from tranche_functions import GHquad
from yieldcurve import roll_yc
from scipy.stats import norm
def year_frac(d1, d2, day_count_conv = "Actual/365"):
""" compute the year fraction between two dates """
if day_count_conv.lower() in ["actual/365", "act/365"]:
return (d2-d1).days/365
elif day_count_conv.lower() in ["actual/360", "act/360"]:
return (d2-d1).days/360
def calib(S0, fp, exercise_date, exercise_date_settle, index,
rolled_curve, tilt, w):
S = S0 * tilt * 1e-4
a, b = strike_vec(S, rolled_curve, exercise_date, exercise_date_settle,
index.start_date, index.end_date, index.recovery)
vec = a - index.fixed_rate * b * 1e-4
return np.inner(vec, w) - fp
def g(index, spread, exercise_date, use_rolled_curve = True):
""" computes the strike clean price using the expected forward yield curve """
if use_rolled_curve:
rolled_curve = roll_yc(index._yc, exercise_date)
else:
rolled_curve = index._yc
step_in_date = exercise_date + datetime.timedelta(days=1)
exercise_date_settle = (pd.Timestamp(exercise_date) + 3* BDay()).date()
sc = SpreadCurve(exercise_date, rolled_curve, index.start_date,
step_in_date, exercise_date_settle,
[index.end_date], array.array('d', [spread * 1e-4]),
index.recovery)
a = index._fee_leg.pv(exercise_date, step_in_date, exercise_date_settle,
rolled_curve, sc, True)
return (spread - index.fixed_rate) * a *1e-4
def ATMstrike(index, exercise_date):
exercise_date_settle = (pd.Timestamp(exercise_date) + 3* BDay()).date()
fp = index.forward_pv(exercise_date)
closure = lambda S: g(index, S, exercise_date) - fp
eta = 1.1
a = index.spread
b = index.spread * eta
while True:
if closure(b) > 0:
break
b *= eta
return brentq(closure, a, b)
class Option:
def __init__(self, index, exercise_date, strike, option_type="payer"):
self.index = index
self._exercise_date = exercise_date
self._forward_yc = roll_yc(self.index._yc, self.exercise_date)
self.exercise_date_settle = (pd.Timestamp(self.exercise_date) + 3* BDay()).date()
self._T = None
self.strike = strike
self.option_type = option_type.lower()
self._Z, self._w = GHquad(50)
self.notional = 1
@property
def exercise_date(self):
return self._exercise_date
@exercise_date.setter
def exercise_date(self, d : datetime.date):
self._exercise_date = d
self.exercise_date_settle = (pd.Timestamp(d) + 3* BDay()).date()
self._forward_yc = roll_yc(self.index._yc, self.exercise_date)
@property
def pv(self):
fp = self.index.forward_pv(self.exercise_date) / self.index.notional
T = self.T
tilt = np.exp(-self.sigma**2/2 * T + self.sigma * self._Z * math.sqrt(T))
rolled_curve = roll_yc(self.index._yc, self.exercise_date)
args = (fp, self.exercise_date, self.exercise_date_settle,
self.index, self._forward_yc, tilt, self._w)
eta = 1.1
a = self.index.spread
b = self.index.spread * eta
while True:
if calib(*((b,) + args)) > 0:
break
b *= eta
S0 = brentq(calib, a, b, args)
G = g(self.index, self.strike, self.exercise_date)
if T == 0:
pv = self.notional * (g(self.index, self.index.spread, self.exercise_date) - G)
if self.option_type == "payer":
return pv if self.index.spread > self.strike else 0
else:
return - pv if self.index.spread < self.strike else 0
Zstar = (math.log(self.strike/S0) + self.sigma**2/2 * T) / \
(self.sigma * math.sqrt(T))
if self.option_type == "payer":
Z = Zstar + np.logspace(0, 1.5, 300) - 1
elif self.option_type == "receiver":
Z = Zstar - np.logspace(0, 1.5, 300) + 1
else:
raise ValueError("option_type needs to be either 'payer' or 'receiver'")
S = S0 * np.exp(-self.sigma**2/2 * T + self.sigma * Z * math.sqrt(T))
a, b = strike_vec(S * 1e-4, rolled_curve, self.exercise_date,
self.exercise_date_settle,
self.index.start_date, self.index.end_date, self.index.recovery)
val = ((a - b * self.index.fixed_rate*1e-4) - G) * 1/math.sqrt(2*math.pi) * np.exp(-Z**2/2)
df_scale = self.index._yc.discount_factor(self.exercise_date_settle)
return self.notional * simps(val, Z) * df_scale
@property
def pv2(self):
G = g(self.index, self.strike, self.exercise_date)
fp = self.index.forward_pv(self.exercise_date) / self.index.notional
forward_annuity = self.index.forward_annuity(self.exercise_date)
DA_forward_spread = fp / forward_annuity + self.index.fixed_rate * 1e-4
strike_tilde = self.index.fixed_rate * 1e-4 + G / forward_annuity
return forward_annuity * black(DA_forward_spread,
strike_tilde,
self.T,
self.sigma,
self.option_type) * self.notional
@property
def delta(self):
old_index_pv = self.index.pv
old_pv = self.pv
self.index.spread += 0.1
notional_ratio = self.index.notional/self.notional
delta = (self.pv - old_pv)/(self.index.pv - old_index_pv) * notional_ratio
self.index.spread -= 0.1
return delta
@property
def T(self):
if self._T:
return self._T
else:
return year_frac(self.index.trade_date, self.exercise_date) + 1/365
@property
def gamma(self):
pass
@property
def theta(self):
old_pv = self.pv
self._T = self.T - 1/365
theta = self.pv - old_pv
self._T = None
return theta
@property
def vega(self):
old_pv = self.pv
self.sigma += 0.01
vega = self.pv - old_pv
self.sigma -= 0.01
return vega
def d1(F, K, sigma, T):
return (np.log(F / K) + sigma**2 * T / 2) / (sigma * np.sqrt(T))
def d2(F, K, sigma, T):
return d1(F, K, sigma, T) - sigma * np.sqrt(T)
def black(F, K, T, sigma, option_type = "payer"):
chi = 1 if option_type == "payer" else -1
if option_type == "payer":
return F * norm.cdf(d1(F, K, sigma, T)) - K * norm.cdf(d2(F, K, sigma, T))
else:
return K * norm.cdf(- d2(F, K, sigma, T)) - F * norm.cdf(- d1(F, K, sigma, T))
def option(index, exercise_date, sigma, K, option_type="payer"):
""" computes the pv of an option using Pedersen's model """
fp = index.forward_pv(exercise_date)/index.notional
#forward_yc = yield_curve.expected_forward_curve(exercise_date)
#expiry is end of day (not sure if this is right)
T = year_frac(index.trade_date, exercise_date)
Z, w = GHquad(50)
tilt = np.exp(-sigma**2/2 * T + sigma * Z * math.sqrt(T))
exercise_date_settle = (pd.Timestamp(exercise_date) + 3* BDay()).date()
args = (fp, exercise_date, exercise_date_settle, index, tilt, w)
## atm forward is greater than spread
eta = 1.1
a = index.spread
b = index.spread * eta
while True:
if calib(*((b,) + args)) > 0:
break
b *= eta
S0 = brentq(calib, a, b, args)
S = S0 * tilt
G = g(index, K, exercise_date)
handle = lambda Z: g(index, S0 * math.exp(-sigma**2/2 * T + sigma * Z * math.sqrt(T)),
exercise_date) - G
Zstar = brentq(handle, -3, 3)
if option_type.lower() == "payer":
Z = Zstar + np.logspace(0, 1.1, 300) - 1
elif option_type.lower() == "receiver":
Z = Zstar - np.logspace(0, 1.1, 300) + 1
else:
raise ValueError("option_type needs to be either 'payer' or 'receiver'")
S = S0 * np.exp(-sigma**2/2 * T + sigma * Z * math.sqrt(T))
a, b = strike_vec(S, index._yc, exercise_date, exercise_date_settle,
index.start_date, index.end_date, index.recovery)
val = ((a - b * index.fixed_rate)/df - G) * 1/math.sqrt(2*math.pi) * np.exp(-Z**2/2)
return simps(val, Z) * yield_curve.discount_factor(exercise_date_settle)
if __name__ == "__main__":
import datetime
from analytics import Index
from swaption import Option
ig27_5yr = Index.from_name('ig', 27, '5yr', datetime.date(2016, 10, 24))
ig27_5yr.spread = 74
payer = Option(ig27_5yr, datetime.date(2016, 12, 21), 65)
payer.sigma = 0.428
payer.notional = 10e6
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