Daisuke Oyama
Faculty of Economics, University of Tokyo
From Miranda and Fackler, Applied Computational Economics and Finance, 2002, Section 7.6.4
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%matplotlib inline
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import numpy as np
from scipy import sparse
import matplotlib.pyplot as plt
import quantecon as qe
from quantecon.markov import DiscreteDP, backward_induction, sa_indices
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T = 0.5 # Time expiration (years)
vol = 0.2 # Annual volatility
r = 0.05 # Annual interest rate
strike = 2.1 # Strike price
p0 = 2 # Current price
N = 100 # Number of periods to expiration
# Time length of a period
tau = T/N
# Discount factor
beta = np.exp(-r*tau)
# Up-jump factor
u = np.exp(vol*np.sqrt(tau))
# Up-jump probability
q = 1/2 + np.sqrt(tau)*(r - (vol**2)/2)/(2*vol)
We follow the state-action pairs formulation approach. We let the state space consist of the possible values of the asset price and the state indicating that "the option has been exercised".
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# Possible price values
ps = u**np.arange(-N, N+1) * p0
# Number of states
n = len(ps) + 1 # State n-1: "the option has been exercised"
# Number of actions
m = 2 # 0: hold, 1: exercise
# Number of feasible state-action pairs
L = n*m - 1 # At state n-1, there is only one action "do nothing"
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# Arrays of state and action indices
s_indices, a_indices = sa_indices(n, m)
s_indices, a_indices = s_indices[:-1], a_indices[:-1]
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# Reward vector
R = np.empty((n, m))
R[:, 0] = 0
R[:-1, 1] = strike - ps
R = R.ravel()[:-1]
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# Transition probability array
Q = sparse.lil_matrix((L, n))
for i in range(L-1):
if a_indices[i] == 0:
Q[i, min(s_indices[i]+1, len(ps)-1)] = q
Q[i, max(s_indices[i]-1, 0)] = 1 - q
else:
Q[i, n-1] = 1
Q[L-1, n-1] = 1
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# Create a DiscreteDP
ddp = DiscreteDP(R, Q, beta, s_indices, a_indices)
The backward induction algorithm for finite horizon dynamic programs is offered
as the function backward_induction.
(By default, the terminal value function is set to the vector of zeros.)
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vs, sigmas = backward_induction(ddp, N)
In the returns, vs is an $(N+1) \times n$ array that contains the optimal value functions,
where vs[0] is the value vector at the current period (i.e., period $0$)
for different prices
(with the value vs[0, -1] = 0 for the state "the option has been exercised" included),
and sigmas is an $N \times n$ array that contins the optimal policy functions.
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v = vs[0]
max_exercise_price = ps[sigmas[::-1].sum(-1)-1]
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fig, axes = plt.subplots(1, 2, figsize=(12, 4))
axes[0].plot([0, strike], [strike, 0], 'k--')
axes[0].plot(ps, v[:-1])
axes[0].set_xlim(0, strike*2)
axes[0].set_xticks(np.linspace(0, 4, 5, endpoint=True))
axes[0].set_ylim(0, strike)
axes[0].set_yticks(np.linspace(0, 2, 5, endpoint=True))
axes[0].set_xlabel('Asset Price')
axes[0].set_ylabel('Premium')
axes[0].set_title('Put Option Value')
axes[1].plot(np.linspace(0, T, N), max_exercise_price)
axes[1].set_xlim(0, T)
axes[1].set_ylim(1.6, strike)
axes[1].set_xlabel('Time to Maturity')
axes[1].set_ylabel('Asset Price')
axes[1].set_title('Put Option Optimal Exercise Boundary')
axes[1].tick_params(right='on')
plt.show()
The figure in the right panel looks different from the corresponding figure in Miranda and Fackler (Figure 7.4(b), p.180).
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