def main():
S = np.linspace(0, 200, 200 * 8 + 1)
Sl = 0
Su = 250
N = 128 * T
K = 8 * (Su - Sl)
Sk = (K * (S - Sl) / (Su - Sl)).astype(int)
legend = []
label = "$\\Omega^v = [%i, 5]$"
model = FDEModel(N, dS, payoff)
fig = plt.figure()
fig.set_figwidth(1.8 * fig.get_figwidth())
ax = fig.add_subplot(1, 2, 1)
for i in range(1, 5):
Conv.times = [(i, 5)]
ax.plot(S, model.price(Sl, Su, K).V[0][Sk])
legend.append(label % i)
plt.xlabel("Stock Price")
plt.ylabel("Convertible Bond Price")
plt.legend(legend)
ax = fig.add_subplot(1, 2, 2, projection="3d")
plot_model(ax, dS, payoff)
def test_table20_5(self):
"""Test Example 20.1 using explicit finite difference method."""
model = FDEModel(10, self.dS, self.V)
P = model.price(0, 100, 20, scheme=ExplicitScheme, boundary="ignore")
for i in reversed(range(len(P.V))):
self.assertTrue(
(np.abs(self.table20_5[::-1, i] - P.V[i]) < 0.01).all())
def main():
S = np.linspace(0, 200, 200 * 8 + 1)
Sl = 0
Su = 250
N = 128 * T
K = 8 * (Su - Sl)
Sk = (K * (S - Sl) / (Su - Sl)).astype(int)
legend = []
label = "$\\Omega^v = [%i, 5]$"
model = FDEModel(N, dS, payoff)
fig = plt.figure()
fig.set_figwidth(1.8 * fig.get_figwidth())
ax = fig.add_subplot(1, 2, 1)
for i in range(1, 5):
Conv.times = [(i, 5)]
ax.plot(S, model.price(Sl, Su, K).V[0][Sk])
legend.append(label % i)
plt.xlabel("Stock Price")
plt.ylabel("Convertible Bond Price")
plt.legend(legend)
ax = fig.add_subplot(1, 2, 2, projection="3d")
plot_model(ax, dS, payoff)
def plot_model(ax, dS, payoff):
N = 40
model = FDEModel(N, dS, payoff)
P = model.price(0, 250, 125, scheme=CrankNicolsonScheme)
colours = choices(P)
plot_strips(ax, P.t, P.S[:100], np.array(P.V)[:, :100].T, facecolors=colours[:100])
ax.set_xlabel("Time")
ax.set_ylabel("Stock Price")
ax.set_zlabel("Portfolio Value")
legend(ax)
def main():
S = np.arange(24, 121)
Sl = 0
Su = 200
N = 128 * T
K = 8 * (Su - Sl)
Sk = K * (S - Sl) / (Su - Sl)
dS1 = copy.copy(dS)
dS1.lambd_ = lambda S: 0.02 * (S / 100)**-1.2
dS1.cap_lambda = True
dS2 = copy.copy(dS)
dS2.lambd_ = lambda S: 0.02 * (S / 100)**-2.0
dS2.cap_lambda = True
model1 = FDEModel(N, dS, payoff)
model2 = FDEModel(N, dS1, payoff)
model3 = FDEModel(N, dS2, payoff)
plt.plot(S, model1.price(Sl, Su, K).V[0][Sk])
plt.plot(S, model2.price(Sl + 1, Su, K - 8).V[0][Sk - 8])
plt.plot(S, model3.price(Sl + 1, Su, K - 8).V[0][Sk - 8])
plt.xlim(S[0], S[-1])
plt.ylim(50, 150)
plt.xlabel("Stock Price")
plt.ylabel("Convertible Bond Price")
plt.legend(
["Constant hazard rate", "$\\alpha = -1.2$", "$\\alpha = -2.0$"],
loc=2)
def main():
N = 128 * T
S = np.arange(24, 121)
model1 = FDEModel(N, dS, payoff)
model2 = FDEModel(N, dS_var12, payoff)
model3 = FDEModel(N, dS_var20, payoff)
plt.plot(S, delta(S, model3))
plt.plot(S, delta(S, model2))
plt.plot(S, delta(S, model1))
plt.xlim(S[0], S[-1])
plt.ylim(-1, 1)
plt.xlabel("Stock Price")
plt.ylabel("Convertible Bond Price")
plt.legend(["$\\alpha = -2.0$", "$\\alpha = -1.2$", "Constant hazard rate"], loc=4)
def plot_model(ax, dS, payoff):
N = 40
model = FDEModel(N, dS, payoff)
P = model.price(0, 250, 125, scheme=CrankNicolsonScheme)
colours = choices(P)
plot_strips(ax,
P.t,
P.S[:100],
np.array(P.V)[:, :100].T,
facecolors=colours[:100])
ax.set_xlabel("Time")
ax.set_ylabel("Stock Price")
ax.set_zlabel("Portfolio Value")
legend(ax)
def main():
S = np.linspace(0, 120, 120 * 8 + 1)
Sl = 0
Su = 200
N = 128 * T
K = 8 * (Su - Sl)
Sk = (K * (S - Sl) / (Su - Sl)).astype(int)
model1 = FDEModel(N, dS, payoff)
model2 = FDEModel(N, dS_var, payoff)
plt.plot(S, model1.price(Sl, Su, K).V[0][Sk])
plt.plot(S, np.append(A.R * A.N, model2.price(Sl + 1, Su, K - 8).V[0][Sk[:-1]]))
plt.ylim([40, 160])
plt.xlabel("Stock Price")
plt.ylabel("Convertible Bond Price")
plt.legend(["Constant $\\lambda$", "Synthesis $\\lambda$"], loc=2)
def main():
S = np.arange(80, 121)
Sl = 0
Su = 200
N = 128 * T
K = 8 * (Su - Sl)
Sk = K * (S - Sl) / (Su - Sl)
model1 = FDEModel(N, dS, payoff)
model2 = FDEModel(N, dS_partial, payoff)
model3 = FDEModel(N, dS_total, payoff)
plt.plot(S, model1.price(Sl, Su, K).V[0][Sk])
plt.plot(S, model2.price(Sl, Su, K).V[0][Sk])
plt.plot(S, model3.price(Sl, Su, K).V[0][Sk])
plt.ylim(100, 150)
plt.xlabel("Stock Price")
plt.ylabel("Convertible Bond Price")
plt.legend(["No default", "Partial default", "Total Default"], loc=2)
def main():
S = np.linspace(0, 100, 100 * 8 + 1)
Sl = 0
Su = 150
N = 128 * T
K = 8 * (Su - Sl)
Sk = (K * (S - Sl) / (Su - Sl)).astype(int)
legend = []
label = "$\\Omega^p = \\{%i\\}$"
model = FDEModel(N, dS, payoff)
fig = plt.figure()
ax = fig.add_subplot(111)
for i in range(1, 5):
P.times = [i]
ax.plot(S, model.price(Sl, Su, K).V[0][Sk])
legend.append(label % i)
plt.xlabel("Stock Price")
plt.ylabel("Convertible Bond Price")
plt.legend(legend)
def main():
S = np.linspace(0, 200, 200 * 8 + 1)
Sl = 0
Su = 250
N = 128 * T
K = 8 * (Su - Sl)
Sk = (K * (S - Sl) / (Su - Sl)).astype(int)
legend = []
label = "$\\Omega^c = \\{%i\\}$"
model = FDEModel(N, dS, payoff)
fig = plt.figure()
ax = fig.add_subplot(111)
for i in range(1, 5):
C.times = [i]
ax.plot(S, model.price(Sl, Su, K).V[0][Sk])
legend.append(label % i)
plt.xlabel("Stock Price")
plt.ylabel("Convertible Bond Price")
plt.legend(legend)
def test_call(self):
"""Test that the finite difference model correctly prices a call option."""
K = 100
def price(r, S):
d1 = (np.log(S / K) + r + 0.0005) / 0.1
d2 = d1 - 0.1
price = stats.norm.cdf(d1) * S
price -= stats.norm.cdf(d2) * K * np.exp(-r)
return price
dS = WienerJumpProcess(0.1, 0.1)
dSdq = WienerJumpProcess(0.1, 0.1, 0.1)
S = np.linspace(0, 200, 41)
V = CallE(1, K)
model = FDEModel(64, dS, V)
model2 = FDEModel(64, dSdq, V)
accuracy = np.array((15, 15, 15, 14, 12, 11, 9, 8, 6, 5, 4, 3, 2, 2, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2))
accuracy2 = np.array((15, 15, 11, 10, 9, 8, 7, 5, 4, 3, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1))
for scheme in SCHEMES:
P = model.price(0, 200, 40, scheme=scheme)
self.assertTrue((abs(P.V[0] - price(0.1, S)) < 10.0**-accuracy).all())
P = model2.price(0, 200, 40, scheme=scheme)
self.assertTrue((abs(P.V[0] - price(0.2, S)) < 10.0**-accuracy2).all())
return
def main():
Sl = 0
Su = 200
N = 128 * T
K = 8 * (Su - Sl)
S = 100
Sk = K * (S - Sl) / (Su - Sl)
ETA = np.linspace(0, 1, 65)
name = ["No default"]
fde = FDEModel(N, dS, payoff)
plt.plot(ETA, [fde.price(Sl, Su, K).V[0][Sk]] * len(ETA))
for lambd_ in (0.01, 0.02, 0.03):
dS.lambd_ = lambd_
V = []
for eta in ETA:
dS.eta = eta
V.append(fde.price(Sl, Su, K).V[0][Sk])
plt.plot(ETA, V)
name.append("$\\lambda = %i\\%%$" % (lambd_ * 100))
plt.xlabel("$\\eta$")
plt.ylabel("Price at $S=100$")
plt.legend(name, loc=3)
def test_value(self):
"""Test the value object for the finite difference model."""
dS = WienerJumpProcess(0.1, 0.1, 0.02, 0)
N = 16
T = 1
ct = np.linspace(0, T, N // 2 + 1)
c = 1
Sl = 0
Su = 200
K = 40
model = FDEModel(N, dS, Stack([CallA(T, 100), Annuity(T, ct, c)]))
P = model.price(Sl, Su, K)
S = P.S
# Test N
self.assertEqual(P.N, N)
# Test time series
self.assertEqual(P.t[0], 0)
self.assertEqual(P.t[-1], T)
self.assertEqual(len(P.t), N + 1)
# Test Coupon sequence
self.assertTrue((P.C[::2] == c).all())
self.assertTrue((P.C[1::2] == 0).all())
# Test K
self.assertEqual(P.K, K)
# Test Stock series
self.assertEqual(P.S[0], Sl)
self.assertEqual(P.S[-1], Su)
self.assertEqual(len(P.S), K + 1)
# Test default sequence
self.assertEqual(len(P.V), N + 1)
for i in range(1, N + 1):
self.assertLessEqual(P.V[i][0] - P.C[i], P.V[i - 1][0])
self.assertLessEqual(P.V[i][-1] - P.C[i], P.V[i - 1][-1])
self.assertTrue((P.V[i][:-1] - 1e14 <= P.V[i][1:]).all())
self.assertEqual(len(P.X), N)
for i in range(1, N):
self.assertGreaterEqual(P.X[i][0], P.X[i - 1][0])
self.assertLessEqual(P.X[i][-1], P.X[i - 1][-1])
self.assertTrue((P.X[i][:-1] <= P.X[i][1:]).all())
def main():
S = np.linspace(0, 200, 200 * 8 + 1)
Sl = 0
Su = 250
N = 128 * T
K = 8 * (Su - Sl)
Sk = (K * (S - Sl) / (Su - Sl)).astype(int)
model = FDEModel(N, dS, payoff)
fig = plt.figure()
fig.set_figwidth(1.8 * fig.get_figwidth())
ax = fig.add_subplot(1, 2, 1)
ax.plot(S, model.price(Sl, Su, K).V[0][Sk])
C.times = [2]
ax.plot(S, model.price(Sl, Su, K).V[0][Sk])
plt.xlabel("Stock Price")
plt.ylabel("Convertible Bond Price")
plt.legend(["$\\Omega^c = [2, 5)$", "$\\Omega^c = \\{2\\}$"])
ax = fig.add_subplot(1, 2, 2, projection="3d")
plot_model(ax, dS, payoff)
def main():
S = np.linspace(0, 200, 200 * 8 + 1)
Sl = 0
Su = 250
N = 128 * T
K = 8 * (Su - Sl)
Sk = (K * (S - Sl) / (Su - Sl)).astype(int)
model = FDEModel(N, dS, payoff)
fig = plt.figure()
fig.set_figwidth(1.8 * fig.get_figwidth())
ax = fig.add_subplot(1, 2, 1)
ax.plot(S, model.price(Sl, Su, K).V[0][Sk])
A.N = -A.C
ax.plot(S, model.price(Sl, Su, K).V[0][Sk])
plt.xlabel("Stock Price")
plt.ylabel("Convertible Bond Price")
plt.legend(["$R = 104$", "$R = 0$"])
ax = fig.add_subplot(1, 2, 2, projection="3d")
plot_model(ax, dS, payoff)
def main():
Sl = 0
Su = 200
N = 128 * T
K = 8 * (Su - Sl)
S = 100
Sk = K * (S - Sl) / (Su - Sl)
ETA = np.linspace(0, 1, 65)
name = ["No default"]
fde = FDEModel(N, dS, payoff)
plt.plot(ETA, [fde.price(Sl, Su, K).V[0][Sk]] * len(ETA))
for lambd_ in (0.01, 0.02, 0.03):
dS.lambd_ = lambd_
V = []
for eta in ETA:
dS.eta = eta
V.append(fde.price(Sl, Su, K).V[0][Sk])
plt.plot(ETA, V)
name.append("$\\lambda = %i\\%%$" % (lambd_ * 100))
plt.xlabel("$\\eta$")
plt.ylabel("Price at $S=100$")
plt.legend(name, loc=3)
def main():
S0 = 100
Sl = 0
Su = 200
X = range(11)
N = lambda x: 2**(x + 1) * T
K = lambda x: 2**((x + 1) // 2) * (Su - Sl)
fig = plt.figure()
fig.set_figwidth(1.8 * fig.get_figwidth())
plt_x = fig.add_subplot(1, 2, 1)
plt_t = fig.add_subplot(1, 2, 2)
p = []
t = []
for x in X:
start = time.time()
p.append(float(BinomialModel(N(x), dS, payoff).price(S0)))
t.append(time.time() - start)
plt_x.plot(X, p)
plt_t.plot(t, p)
print p[-1]
for scheme in (ImplicitScheme, CrankNicolsonScheme, PenaltyScheme):
p = []
t = []
for x in X:
k = K(x)
Sk = k * (S0 - Sl) / (Su - Sl)
start = time.time()
p.append(FDEModel(N(x), dS, payoff).price(Sl, Su, k, scheme=scheme).V[0][Sk])
t.append(time.time() - start)
plt_x.plot(X, p[:len(X)])
plt_t.plot(t, p)
print p[-1]
label(plt_x, "$\\delta_t = 2^{-(x + 1)}$; $\\delta_S = 2^{-\\frac{x + 1}{2}}$")
label(plt_t, "Computational Time (seconds)")
#plt_t.set_xlim(0, 2.5)
plt_t.set_xscale('log')
def main():
S = np.linspace(0, 120, 120 * 8 + 1)
Sl = 0
Su = 200
N = 128 * T
K = 8 * (Su - Sl)
Sk = (K * (S - Sl) / (Su - Sl)).astype(int)
model1 = FDEModel(N, dS, payoff)
model2 = FDEModel(N, dS_var, payoff)
plt.plot(S, model1.price(Sl, Su, K).V[0][Sk])
plt.plot(
S, np.append(A.R * A.N,
model2.price(Sl + 1, Su, K - 8).V[0][Sk[:-1]]))
plt.ylim([40, 160])
plt.xlabel("Stock Price")
plt.ylabel("Convertible Bond Price")
plt.legend(["Constant $\\lambda$", "Synthesis $\\lambda$"], loc=2)
def main():
S = np.arange(80, 121)
Sl = 0
Su = 200
N = 128 * T
K = 8 * (Su - Sl)
Sk = K * (S - Sl) / (Su - Sl)
model1 = FDEModel(N, dS, payoff)
model2 = FDEModel(N, dS_partial, payoff)
model3 = FDEModel(N, dS_total, payoff)
plt.plot(S, model1.price(Sl, Su, K).V[0][Sk])
plt.plot(S, model2.price(Sl, Su, K).V[0][Sk])
plt.plot(S, model3.price(Sl, Su, K).V[0][Sk])
plt.ylim(100, 150)
plt.xlabel("Stock Price")
plt.ylabel("Convertible Bond Price")
plt.legend(["No default", "Partial default", "Total Default"], loc=2)
def test_forward(self):
"""Test that the finite difference model correctly prices a forward contract."""
dS = WienerJumpProcess(0.1, 0.1)
dSdq = WienerJumpProcess(0.1, 0.1, 0.02)
S = np.linspace(0, 200, 41)
V = Forward(1, 100)
model = FDEModel(64, dS, V)
model2 = FDEModel(64, dSdq, V)
for scheme in SCHEMES:
if isinstance(scheme, CrankNicolsonScheme):
accuracy = 5
else:
accuracy = 2
P = model.price(0, 200, 40, scheme=scheme)
self.assertTrue((abs(P.V[0] - (S - 100 * np.exp(-0.1))) < 10.0**-accuracy).all())
P = model2.price(0, 200, 40, scheme=scheme)
self.assertTrue((abs(P.V[0] - (S - 100 * np.exp(-0.12))) < 10.0**-accuracy).all())
def test_annuity(self):
"""Test the pricing of a series of payments."""
def price(r):
erdt = np.exp(-r)
i = 1 / erdt - 1
return (1 - erdt**(T * 2)) / i + 10 * erdt**(T * 2)
dS = WienerJumpProcess(0.1, 0.1)
dSdq = WienerJumpProcess(0.1, 0.1, 0.02)
# Semi-annual payments of 1
T = 10
V = Annuity(T, np.arange(0.5, T + 0.5, 0.5), 1, 10)
model = FDEModel(T * 8, dS, V)
model2 = FDEModel(T * 8, dSdq, V)
for scheme in SCHEMES:
P = model.price(0, 200, 25, scheme=scheme)
self.assertTrue((np.abs(P.V[0] - price(0.05)) < 1e-1).all())
P = model2.price(0, 200, 25, scheme=scheme)
self.assertTrue((np.abs(P.V[0] - price(0.06)) < 1e-1).all())
def main():
S = np.arange(24, 121)
Sl = 0
Su = 200
N = 128 * T
K = 8 * (Su - Sl)
Sk = K * (S - Sl) / (Su - Sl)
dS1 = copy.copy(dS)
dS1.lambd_ = lambda S: 0.02 * (S / 100)**-1.2
dS1.cap_lambda = True
dS2 = copy.copy(dS)
dS2.lambd_ = lambda S: 0.02 * (S / 100)**-2.0
dS2.cap_lambda = True
model1 = FDEModel(N, dS, payoff)
model2 = FDEModel(N, dS1, payoff)
model3 = FDEModel(N, dS2, payoff)
plt.plot(S, model1.price(Sl, Su, K).V[0][Sk])
plt.plot(S, model2.price(Sl + 1, Su, K - 8).V[0][Sk - 8])
plt.plot(S, model3.price(Sl + 1, Su, K - 8).V[0][Sk - 8])
plt.xlim(S[0], S[-1])
plt.ylim(50, 150)
plt.xlabel("Stock Price")
plt.ylabel("Convertible Bond Price")
plt.legend(["Constant hazard rate", "$\\alpha = -1.2$", "$\\alpha = -2.0$"], loc=2)
def test_table20_5(self):
"""Test Example 20.1 using explicit finite difference method."""
model = FDEModel(10, self.dS, self.V)
P = model.price(0, 100, 20, scheme=ExplicitScheme, boundary="ignore")
for i in reversed(range(len(P.V))):
self.assertTrue((np.abs(self.table20_5[::-1, i] - P.V[i]) < 0.01).all())
