def price_CDS(FV, c, T, rate, intensities, R, freq=4, t=0, freq2=0):
import numpy as np
import scipy.integrate as integrate
if isinstance(rate, (float, int)):
discount = np.exp(-rate * (T - t))
f = lambda u: np.exp(-rate * (u - t))
if isinstance(intensities, (float, int)):
surv = np.exp(-intensities * (T - t))
g = lambda u: np.exp(-intensities * (u - t)) * intensities
h = lambda u: f(u) * g(u)
elif isinstance(intensities, tuple) and all(
isinstance(x, list) for x in intensities):
model = piecewise_exponential(intensities[0], intensities[1])
surv = model.survival2(T - t)
g = lambda u: model.pdf2(u - t)
h = lambda u: f(u) * g(u)
elif isinstance(rate, tuple) and all(isinstance(x, list) for x in rate):
app_rate = np.interp(T - t, rate[0], rate[1])
discount = np.exp(-app_rate * (T - t))
f = lambda u: np.exp(-np.interp(u - t, rate[0], rate[1]) * (u - t))
if isinstance(intensities, (float, int)):
surv = np.exp(-intensities * (T - t))
g = lambda u: np.exp(-(u - t) * intensities) * intensities
h = lambda u: f(u) * g(u)
elif isinstance(intensities, tuple) and all(
isinstance(x, list) for x in intensities):
model = piecewise_exponential(intensities[0], intensities[1])
surv = model.survival2(T - t)
g = lambda u: model.pdf2(u - t)
h = lambda u: f(u) * g(u)
return FV*(discount*surv + integrate.quad(h,t,T)[0]) - \
risky_bond_price(FV,c,T,rate,intensities,R,freq,t,freq2)
def piece_expo(self,
maturities,
intensities,
r=0,
t0=0,
RPV01=False,
prob_dist=False):
import numpy as np
from piecewise_expo import piecewise_exponential
T, R, freq = self.T, self.R, self.freq
pe = piecewise_exponential(maturities, intensities)
f = lambda t: pe.pdf2(t)
S = lambda t: pe.survival2(t)
D = lambda t: np.exp(-r * t)
import scipy.integrate as integ
q1 = integ.quad(lambda s: f(s) * D(s - t0) / S(t0) * (s > t0), t0,
T)[0]
if freq == 0:
q2 = integ.quad(lambda s: S(s) * D(s - t0) / S(t0) * (s > t0), t0,
T)[0]
else:
dt = 1 / freq
times = [
x for x in np.linspace(1 / freq, T, int(T * freq)) if x > t0
]
q2 = dt * sum(
[S(x) / S(t0) * np.exp(-r * (x - t0)) for x in times])
return (1 - R) * q1 / q2 if RPV01 == False else ((1 - R) * q1 / q2, q2)
def risky_bond_price4(FV,c,T,rates,intensities,R,freq=4,t=0,freq2=0):
def times(t,T,freq = 4):
r"The above function gives the times of cash-flows posterior to time t"
if freq*(T-t)==int(freq*(T-t)):
k = freq*(T-t)-1
else:
k = int(freq*(T-t))
import numpy as np
return np.linspace(T-k/freq,T,(k+1),endpoint = True)
import numpy as np
if t==T:
return FV + (c/freq+1)*FV
elif t>T or t<0:
raise TypeError('the time when you want to evaluate the bond surpasses\
the lifetime of the bond or it is negative')
else:
import scipy.integrate as integrate
time_points = times(t,T,freq)
k = len(time_points)
app_rates = [np.interp(time_points[i]-t,rates[0],rates[1])\
for i in range(k)]
if freq2==0:
disc_rates = [np.exp(-app_rates[i]*(time_points[i]-t)) for i in range(k)]
else:
disc_rates = [1/(1+app_rates[i]/freq2)**(freq2*(time_points[i]-t)) for i in range(k)]
cash_flows = [c/freq*FV]*k
cash_flows[-1]=cash_flows[-1]+FV
model = piecewise_exponential(intensities[0],intensities[1])
survivs = [model.survival(time_points[i]-t) for i in range(len(time_points))]
a = np.dot(np.array(disc_rates)*np.array(survivs),np.array(cash_flows))
f = lambda u: np.exp(-(u-t)*np.interp(u-t,rates[0],rates[1]))
g = lambda u: model.pdf2(u-t)
h = lambda u: f(u)*g(u)
b = R*FV*integrate.quad(h,t,T,args=(),full_output = 1)[0]
return a+b
def piece_expo_defer(self,maturities,intensities,r=0,t0=0,RPV01 = False\
,T_reg_pmt=None,Tp = None):
import numpy as np
from piecewise_expo import piecewise_exponential
FV, c, T, R, freq, B = self.FV, self.c, self.T, self.R, self.freq, self.B
pe = piecewise_exponential(maturities, intensities)
N = outstanding_notional(FV, c, T, freq, B)
f = lambda t: pe.pdf2(t)
S = lambda t: pe.survival2(t)
D = lambda t: np.exp(-r * t)
import scipy.integrate as integ
q1 = integ.quad(lambda s: f(s) * D(s - t0) / S(t0) * (s > t0) * N(s),
t0, T)[0]
if freq == 0:
q2 = integ.quad(lambda s:S(s)*D(s-t0)/S(t0)*(s>t0)*N(s),t0,T)[0] \
if T_reg_pmt==None and Tp == None else \
integ.quad(lambda s:S(s)*D(s-t0)/S(t0)*(s>t0)*N(s),t0,T_reg_pmt)[0]\
+S(Tp)/S(t0)*np.exp(-r*(Tp-t0))
else:
dt = 1 / freq
times = [x for x in np.linspace(1/freq,T_reg_pmt,\
int(T_reg_pmt*freq)) if x>t0]+[Tp] if Tp!=None else \
[x for x in np.linspace(1/freq,T_reg_pmt,int(T_reg_pmt*freq))\
if x>t0]
q2 = dt * sum(
[S(x) / S(t0) * np.exp(-r * (x - t0)) * N(x) for x in times])
return (1 - R) * q1 / q2
def pdf_exp(u):
if isinstance(intensities, (float, int)):
return intensities * np.exp(-intensities * (u - t))
elif isinstance(intensities, tuple) and all(
isinstance(x, list) for x in intensities):
model = piecewise_exponential(intensities[0], intensities[1])
return model.pdf2(u - t)
def test_class_piece():
b1 = piecewise_exponential([0.2, 0.3, 0.4], [1, 2])
x = b1.rvs2(10000)
count = sum([(x[i] < 2) * 1 for i in range(len(x))]) / len(x)
count2 = sum([(x[i] < 1) * 1 for i in range(len(x))]) / len(x)
print(count)
import numpy as np
print(1 - np.exp(-0.5))
print(count2, 1 - np.exp(-0.2))
def price_LtD_swap3(FV1,FV2,c,T,rate,intensities1,intensities2,R1,R2,freq = 4,t=0,freq2 = 0):
import numpy as np
import scipy.integrate as integrate
def times(t,T,freq):
if freq*(T-t)==int(freq*(T-t)):
k = freq*(T-t)-1
else:
k = int(freq*(T-t))
return np.linspace(T-k/freq,T,(k+1),endpoint = True)
h = 1/freq
time_points = times(t,T,freq)
from piecewise_expo import piecewise_exponential
model1 = piecewise_exponential(intensities1[0],intensities1[1])
model2 = piecewise_exponential(intensities2[0],intensities2[1])
def surv_ltd(u):
return 1-model1.cdf2(u-t)*model2.cdf(u-t)
def pdf_ltd(u):
return model1.pdf2(u-t)*model2.cdf(u-t)+model1.cdf2(u-t)*model2.pdf2(u-t)
def discount(u):
if isinstance(rate,(float,int)):
if freq2==0:
return np.exp(-rate*(u-t))
else:
return 1/(1+rate/freq2)**(freq2*(u-t))
elif isinstance(rate,tuple) and all(isinstance(x,list) for x in rate):
app_rate = np.interp(u-t,rate[0],rate[1])
if freq2==0:
return np.exp(-app_rate*(u-t))
else:
return 1/(1+app_rate/freq2)**(freq2*(u-t))
def RPV01():
survivs = [surv_ltd(u) for u in time_points]
discounts = [discount(u) for u in time_points]
return h*np.dot(survivs,discounts)
fct1 = lambda y: model1.cdf2(y) * model1.pdf2(y)
proba = integrate.quad(fct1,0,np.Inf)[0]
fct2 = lambda u:discount(u)*pdf_ltd(u)
discounted_leg = ((1-R2)*FV2*proba + (1-R1)*FV1*(1-proba))*integrate.quad(fct2,t,T)[0]
premium_leg = c*(FV2*proba + FV1*(1-proba))*RPV01()
return discounted_leg - premium_leg
def test_expo_dist():
f = stats.expon
print(f.cdf(1, loc=0, scale=1 / 1.1))
g = piecewise_exponential([0.5, 1], [0.2, 0.3, 0.4])
print(g.cdf(1))
def test_class_piece2():
b1 = piecewise_exponential([1, 2, 3, 4], [0.2, 0.3, 0.4, 0.7, 0.9])
print(b1.survival(1), b1.survival(2))
print(b1.survival2(1), b1.survival2(2))
print(b1.survival(4.1))
print(b1.survival2(4.1))
