-
Notifications
You must be signed in to change notification settings - Fork 7
/
二叉树构建.py
133 lines (118 loc) · 5.27 KB
/
二叉树构建.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
# -*- coding: utf-8 -*-
"""
Created on Tue May 16 16:00:45 2017
@author: Unknow
"""
import numpy as np
import seaborn as sns
import math
from matplotlib import pyplot as plt
# 设置基本参数
ttm = 3.0 # 到期时间，单位年
tSteps = 25 # 时间方向步数
r = 0.03 # 无风险利率
d = 0.02 # 标的股息率
sigma = 0.2 # 波动率
strike = 100.0 # 期权行权价
spot = 100.0 # 标的现价
# 二叉树框架（可以通过传入不同的treeTraits类型，设计不同的二叉树结构）
class BinomialTree:
def __init__(self,spot,riskFree,dividend,tSteps,maturity,sigma,treeTraits):
self.dt = maturity/tSteps
self.spot = spot
self.r = riskFree
self.d = dividend
self.tSteps = tSteps
self.discount = math.exp(-self.r*self.dt)
self.v = sigma
self.up = treeTraits.up(self)
self.down = treeTraits.down(self)
self.upProbability = treeTraits.upProbability(self)
self.downProbability = 1.0 - self.upProbability
self._build_lattice()
def _build_lattice(self):
# 完成构造二叉树的工作
self.lattice = np.zeros((self.tSteps+1,self.tSteps+1))
self.lattice[0][0] = self.spot
for i in range(self.tSteps):
for j in range(i+1):
self.lattice[i+1][j+1] = self.up*self.lattice[i][j]
self.lattice[i+1][0] = self.down*self.lattice[i][0]
# 处理欧式期权
def roll_back(self,payOff):
for i in range(self.tSteps,0,-1):
for j in range(i,0,-1):
if i == self.tSteps:
self.lattice[i-1][j-1] = self.discount*(self.upProbability*payOff(self.lattice[i][j])+self.downProbability*payOff(self.lattice[i][j-1]))
else:
self.lattice[i-1][j-1] = self.discount*(self.upProbability*self.lattice[i][j]+self.downProbability*self.lattice[i][j-1])
# 处理美式期权
def roll_back_american(self,payOff):
for i in range(self.tSteps,0,-1):
for j in range(i,0,-1):
if i == self.tSteps:
europeanValue = self.discount*(self.upProbability*payOff(self.lattice[i][j])+self.downProbability*payOff(self.lattice[i][j-1]))
else:
europeanValue = self.discount*(self.upProbability*self.lattice[i][j]+self.downProbability*self.lattice[i][j-1])
exerciseValue = payOff(self.lattice[i-1][j-1])
# ExerciseValue 就是立即行权的价值，EuropeanValue为对应节点的欧式价值。
self.lattice[i-1][j-1] = max(europeanValue,exerciseValue)
# JarrowRudd二叉树描述
class JarrowRuddTraits:
def up(tree):
return math.exp((tree.r-tree.d-0.5*tree.v*tree.v)*tree.dt+tree.v*math.sqrt(tree.dt))
def down(tree):
return math.exp((tree.r-tree.d-0.5*tree.v*tree.v)*tree.dt-tree.v*math.sqrt(tree.dt))
def upProbability(tree):
return 0.5
# Cox-Ross-Rubinstein二叉树描述
class CRRTraits:
def up(tree):
return math.exp(tree.v*math.sqrt(tree.dt))
def down(tree):
return math.exp(-tree.v*math.sqrt(tree.dt))
def upProbability(tree):
return 0.5+0.5*(tree.r-tree.d-0.5*tree.v*tree.v)*tree.dt/(tree.v*math.sqrt(tree.dt))
# 偿付函数
def pay_off(spot):
global strike
return max(spot-strike,0.0)
testTree = BinomialTree(spot,r,d,tSteps,ttm,sigma,JarrowRuddTraits)
testTree.roll_back(pay_off)
print(u"二叉树价格：%.4f"%testTree.lattice[0][0])
stepSizes = range(25,500,25)
jrRes = []
crrRes = []
for tSteps in stepSizes:
testTree = BinomialTree(spot,r,d,tSteps,ttm,sigma,JarrowRuddTraits)
testTree.roll_back(pay_off)
jrRes.append(testTree.lattice[0][0])
testTree = BinomialTree(spot,r,d,tSteps,ttm,sigma,CRRTraits)
testTree.roll_back(pay_off)
crrRes.append(testTree.lattice[0][0])
plt.figure(figsize = (16,8))
plt.plot(stepSizes, jrRes, '-.', marker = 'o', markersize = 10)
plt.plot(stepSizes, crrRes, '-.', marker = 'd', markersize = 10)
plt.legend(['Jarrow - Rudd', 'Cox - Ross - Rubinstein', u'解析解'],prop={'family':'SimHei','size':15})
plt.xlabel(u'二叉树步数',fontproperties='SimHei')
plt.title(u'二叉树算法欧式期权收敛性测试',fontproperties='SimHei')
plt.grid(True)
plt.show()
stepSizes = range(25,500,25)
jrRes = []
crrRes = []
for tSteps in stepSizes:
testTree = BinomialTree(spot,r,d,tSteps,ttm,sigma,JarrowRuddTraits)
testTree.roll_back_american(pay_off)
jrRes.append(testTree.lattice[0][0])
testTree = BinomialTree(spot,r,d,tSteps,ttm,sigma,CRRTraits)
testTree.roll_back_american(pay_off)
crrRes.append(testTree.lattice[0][0])
plt.figure(figsize = (16,8))
plt.plot(stepSizes, jrRes, '-.', marker = 'o', markersize = 10)
plt.plot(stepSizes, crrRes, '-.', marker = 'd', markersize = 10)
plt.legend(['Jarrow - Rudd（美式）', 'Cox - Ross - Rubinstein（美式）', u'解析解'],prop={'family':'SimHei','size':15})
plt.xlabel(u'二叉树步数',fontproperties='SimHei')
plt.title(u'二叉树算法美式期权收敛性测试',fontproperties='SimHei')
plt.grid(True)
plt.show()