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BS_FDM.py
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BS_FDM.py
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from numpy.core import empty, clip, zeros, exp, sqrt, ceil
from numpy import linspace
import scipy.sparse.linalg.dsolve as linsolve
from scipy import sparse
class BlackScholesSolver:
"""Finite-difference solver for the Black-Scholes PDE in its most basic form.
The problem to solve is given by:
Function
f = f(t,x) over the rectangle 0 <= t <= T, Bl <= x <= Bu.
PDE
rf = df/dt + rx df/dx + (1/2)(sigma x)^2 d^f/dx^2
Boundary conditions
given on the three sides of the rectangle
t = T; x = Bl; x = Bu
where r, sigma, T, Bl, Bu are given parameters.
"""
def __init__(self, r, sigma, T, Bl, Bu, Fl, Fu, Fp, m, n):
"""Initialize the finite-difference solver.
Parameters:
r - interest rate
sigma - volatility
T - maturity time
Bl - stock price on lower boundary
Bu - stock price on upper boundary
Fl - value of option on lower boundary
Fu - value of option on upper boundary
Fp - pay-off at maturity
m - number of time steps to take when discretizing PDE
n - number of points in x (stock price) domain
when discretizing PDE; does not include the boundary points
"""
self.r = r; self.sigma = sigma; self.T = T
self.Bl = Bl; self.Bu = Bu
self.Fl = Fl; self.Fu = Fu
self.m = m; self.n = n
# Step sizes
self.dt = float(T)/m
self.dx = float(Bu-Bl)/(n+1)
self.xs = Bl/self.dx
# Grid that will eventually contain the finite-difference solution
self.u = empty((m+1, n))
self.u[0,:] = Fp # initial condition
def build_sparse_explicit(s):
"""(internal) Set up the sparse matrix system for the explicit method."""
A = sparse.lil_matrix((s.n, s.n))
for j in xrange(0, s.n):
xd = j+1+s.xs
ssxx = (s.sigma * xd) ** 2
A[j,j] = 1.0 - s.dt*(ssxx + s.r)
if j > 0:
A[j,j-1] = 0.5*s.dt*(ssxx - s.r*xd)
if j < s.n-1:
A[j,j+1] = 0.5*s.dt*(ssxx + s.r*xd)
s.A = A.tocsr()
def build_sparse_implicit(s):
"""(internal) Set up the sparse matrix system for the implicit method."""
C = sparse.lil_matrix((s.n, s.n))
for j in xrange(0, s.n):
xd = j+1+s.xs
ssxx = (s.sigma * xd) ** 2
C[j,j] = 1.0 + s.dt*(ssxx + s.r)
if j > 0:
C[j,j-1] = 0.5*s.dt*(-ssxx + s.r*xd)
if j < s.n-1:
C[j,j+1] = 0.5*s.dt*(-ssxx - s.r*xd)
# Store matrix with sparse LU decomposition already performed
s.C = linsolve.splu(C)
# Buffer to store right-hand side of the linear system Cu = v
s.v = empty((n, ))
def build_sparse_crank_nicolson(s):
"""(internal) Set up the sparse matrices for the Crank-Nicolson method. """
A = sparse.lil_matrix((s.n, s.n))
C = sparse.lil_matrix((s.n, s.n))
for j in xrange(0, s.n):
xd = j+1+s.xs
ssxx = (s.sigma * xd) ** 2
A[j,j] = 1.0 - 0.5*s.dt*(ssxx + s.r)
C[j,j] = 1.0 + 0.5*s.dt*(ssxx + s.r)
if j > 0:
A[j,j-1] = 0.25*s.dt*(+ssxx - s.r*xd)
C[j,j-1] = 0.25*s.dt*(-ssxx + s.r*xd)
if j < s.n-1:
A[j,j+1] = 0.25*s.dt*(+ssxx + s.r*xd)
C[j,j+1] = 0.25*s.dt*(-ssxx - s.r*xd)
s.A = A.tocsr()
s.C = linsolve.splu(C) # perform sparse LU decomposition
# Buffer to store right-hand side of the linear system Cu = v
s.v = empty((n, ))
def time_step_explicit(s, i):
"""(internal) Solve the PDE for one time step using the explicit method."""
# Perform the next time step
s.u[i+1,:] = s.A * s.u[i,:]
# and mix in the two other boundary conditions not accounted for above
xdl = 1+s.xs; xdu = s.n+s.xs
s.u[i+1,0] += s.Fl[i] * 0.5*s.dt*((s.sigma*xdl)**2 - s.r*xdl)
s.u[i+1,s.n-1] += s.Fu[i] * 0.5*s.dt*((s.sigma*xdu)**2 + s.r*xdu)
def time_step_implicit(s, i):
"""(internal) Solve the PDE for one time step using the implicit method."""
s.v[:] = s.u[i,:]
# Add in the two other boundary conditions
xdl = 1+s.xs; xdu = s.n+s.xs
s.v[0] -= s.Fl[i+1] * 0.5*s.dt*(-(s.sigma*xdl)**2 + s.r*xdl)
s.v[s.n-1] -= s.Fu[i+1] * 0.5*s.dt*(-(s.sigma*xdu)**2 - s.r*xdu)
# Perform the next time step
s.u[i+1,:] = s.C.solve(s.v)
def time_step_crank_nicolson(s, i):
"""(internal) Solve the PDE for one time step using the Crank-Nicolson method."""
# Perform explicit part of time step
s.v[:] = s.A * s.u[i,:]
# Add in the two other boundary conditions
xdl = 1+s.xs; xdu = s.n+s.xs
s.v[0] += s.Fl[i] * 0.25*s.dt*(+(s.sigma*xdl)**2 - s.r*xdl)
s.v[s.n-1] += s.Fu[i] * 0.25*s.dt*(+(s.sigma*xdu)**2 + s.r*xdu)
s.v[0] -= s.Fl[i+1] * 0.25*s.dt*(-(s.sigma*xdl)**2 + s.r*xdl)
s.v[s.n-1] -= s.Fu[i+1] * 0.25*s.dt*(-(s.sigma*xdu)**2 - s.r*xdu)
# Perform implicit part of time step
s.u[i+1,:] = s.C.solve(s.v)
def solve(self, method='crank-nicolson'):
"""Solve the Black-Scholes PDE with the parameters given at initialization.
Arguments:
method - Indicates which finite-difference method to use. Choices:
'explicit': explicit method
'implicit': implicit method
'crank-nicolson': Crank-Nicolson method
'smoothed-crank-nicolson':
Crank-Nicolson method with initial smoothing
by the implicit method
"""
i_start = 0
if method == 'implicit':
self.build_sparse_implicit()
time_step = self.time_step_implicit
elif method == 'explicit':
self.build_sparse_explicit()
time_step = self.time_step_explicit
elif method == 'crank-nicolson' or method is None:
self.build_sparse_crank_nicolson()
time_step = self.time_step_crank_nicolson
elif method == 'smoothed-crank-nicolson':
self.build_sparse_implicit()
for i in range(0, 10):
self.time_step_implicit(i)
i_start = 10
self.build_sparse_crank_nicolson()
time_step = self.time_step_crank_nicolson
else:
raise ValueError('incorrect value for method argument')
for i in xrange(i_start, m):
time_step(i)
return self.u
def european_call(r, sigma, T, Bu, m, n, Bl=0.0, barrier=None, method=None):
"""Compute prices for a European-style call option."""
X = linspace(0.0, B, n+2)
X = X[1:-1]
Fp = clip(X-K, 0.0, 1e600)
if barrier is None:
Fu = B - K*exp(-r * linspace(0.0, T, m+1))
Fl = zeros((m+1, ))
elif barrier == 'up-and-out':
Fu = Fl = zeros((m+1,))
bss = BlackScholesSolver(r, sigma, T, Bl, Bu, Fl, Fu, Fp, m, n)
return X, bss.solve(method)
def european_put(r, sigma, T, Bu, m, n, Bl=0.0, barrier=None, method=None):
"""Compute prices for a European-style put option."""
X = linspace(0.0, B, n+2)
X = X[1:-1]
Fp = clip(K-X, 0.0, 1e600)
if barrier is None:
Fu = zeros((m+1,))
Fl = K*exp(-r * linspace(0.0, T, m+1))
elif barrier == 'up-and-out':
Fu = Fl = zeros((m+1,))
bss = BlackScholesSolver(r, sigma, T, Bl, Bu, Fl, Fu, Fp, m, n)
return X, bss.solve(method)
def plot_solution(T, X, u):
# The surface plot
'''
Xm, Tm = pylab.meshgrid(X, linspace(T, 0.0, u.shape[0]))
fig_surface = pylab.figure()
# ax = matplotlib.axes3d.Axes3D(fig_surface)
ax = Axes3D(fig_surface)
ax.plot_surface(Xm, Tm, u)
ax.set_ylabel('Time $t$')
ax.set_xlabel('Stock price $x$')
ax.set_zlabel('Option value $f(t,x)$')
'''
# The color temperature plot
fig_color = pylab.figure()
ax = pylab.gca()
ax.set_xlabel('Time $t$')
ax.set_ylabel('Stock price $x$')
ax.imshow(u.T, interpolation='bilinear', origin='lower',
cmap=matplotlib.cm.hot, aspect='auto', extent=(T,0.0, X[0],X[-1]))
# Plot of price function at time 0
fig_zero = pylab.figure()
pylab.plot(X, u[m-1,:])
ax = pylab.gca()
ax.set_xlabel('Stock price $x$')
ax.set_ylabel('Option value $f(0,x)$')
return fig_color, fig_zero
def parse_options():
from optparse import OptionParser
parser = OptionParser()
parser.add_option("-r", "--interest", dest="r", default="0.10",
help="interest rate [default: %default]")
parser.add_option("-v", "--volatility", dest="sigma", default="0.40",
help="volatility [default: %default]")
parser.add_option("-K", "--strike", dest="K", default="50.00",
help="strike price [default: %default]")
parser.add_option("-T", "--maturity", dest="T", default="0.5",
help="maturity time [default: %default]")
parser.add_option("-B", "--bound", dest="B", default="100.00",
help="upper bound on stock price [default: %default]")
parser.add_option("-m", "--time-steps", dest="m", default="100",
help="number of time steps [default: %default]")
parser.add_option("-n", "--space-steps", dest="n", default="200",
help="number of steps in stock-price space [default: %default]")
parser.add_option("--dt", dest="dt", help="time step size")
parser.add_option("--dx", dest="dx", help="stock-price step size")
parser.add_option("--method", dest="method", help="finite-difference method")
parser.add_option("-C", "--call", dest="is_call", action="store_true",
help="value a European-style call option")
parser.add_option("-P", "--put", dest="is_put", action="store_true",
help="value a European-style put option")
parser.add_option("--barrier", dest="barrier",
help="value a barrier option")
parser.add_option("--plot", dest="plot", action="store_true",
help="plot results")
parser.add_option("--save-plot", dest="save_plot", action="store_true",
help="save plots to EPS files")
(options, args) = parser.parse_args()
return options
if __name__ == "__main__":
options = parse_options()
# Parameters
r = float(options.r)
sigma = float(options.sigma)
K = float(options.K)
T = float(options.T)
B = float(options.B)
m = int(options.m)
n = int(options.n)
if options.dt is not None:
m = ceil(T/float(options.dt))
if options.dx is not None:
n = ceil(B/float(options.dx)) - 1
if options.is_put:
X, u = european_put(r, sigma, T, B, m, n,
barrier=options.barrier,
method=options.method)
else:
X, u = european_call(r, sigma, T, B, m, n,
barrier=options.barrier,
method=options.method)
# Print out results at time 0
print "Stock price x Option price f(0,x)"
for i in xrange(0, n):
print "%10.4f %10.4f " % (X[i], u[m,i])
# Generate plots if user requests
if options.plot or options.save_plot:
import pylab
import matplotlib
from mpl_toolkits.mplot3d import Axes3D
golden_mean = (sqrt(5.0)-1.0)/2.0
pylab.rcParams.update( \
{'backend': 'ps',
'ps.usedistiller': 'xpdf',
'axes.labelsize': 10,
'text.fontsize': 10,
'xtick.labelsize': 8,
'ytick.labelsize': 8,
'figure.figsize': [ 7.0, golden_mean*7.0 ],
'text.usetex': True })
fig_color, fig_zero = plot_solution(T, X, u)
# Show figures
if options.plot:
pylab.show()
# Save figures to EPS format
if options.save_plot:
# fig_surface.savefig('bs-surface.eps')
#fig_color.savefig('bs-color.eps')
fig_zero.savefig('bs-zero.eps')