Calibration of Stoch Vol Jump Model via Numerical Integration Comp Model

1. # Calibration of Complete Model of BCC97
3. # Calibration of Stoch Vol Jump Model to EURO STOXX Option Quotes via Numerical Integration
4. # Data Source: www.eurexchange.com
5. # Bakshi, Cao and Chen (1997)
6. #
7. import sys
8. sys.path.append('09_gmm')
9. import math
10. import numpy as np
11. np.set_printoptions(suppress=True,
12. formatter={'all': lambda x: '%5.3f' % x})
13. import pandas as pd
14. from scipy.optimize import brute, fmin, minimize
15. import matplotlib as mpl
16. mpl.rcParams['font.family'] = 'serif'
17. from BCC_option_valuation import BCC_call_value
18. from CIR_calibration import CIR_calibration, r_list
19. from CIR_zcb_valuation import B
20. from H93_calibration import options
21. #
22. # Calibrate Short Rate Model
23. #
24. kappa_r, theta_r, sigma_r = CIR_calibration()
25. #
26. # Market Data from www.eurexchange.com
27. # as of 30. September 2014
28. #
29. S0 = 3225.93 # EURO STOXX 50 level
30. r0 = r_list[0] # initial short rate
31. #
32. # Parameters from H93 & jump calibrations
33. #
34. kappa_v, theta_v, sigma_v, rho, v0 = np.load('11_cal/opt_sv.npy')
35. lamb, mu, delta = np.load('11_cal/opt_jump.npy')
36. p0 = [kappa_v, theta_v, sigma_v, rho, v0, lamb, mu, delta]
38. #
39. # Calibration Functions
40. #
41. i = 0
42. min_MSE = 5000.0
44. def BCC_error_function(p0):
45. ''' Error function for parameter calibration in BCC97 model via
46. Lewis (2001) Fourier approach.
47. Parameters
48. ==========
49. kappa_v: float
50. mean-reversion factor
51. theta_v: float
52. long-run mean of variance
53. sigma_v: float
54. volatility of variance
55. rho: float
56. correlation between variance and stock/index level
57. v0: float
58. initial, instantaneous variance
59. lamb: float
60. jump intensity
61. mu: float
62. expected jump size
63. delta: float
64. standard deviation of jump
65. Returns
66. =======
67. MSE: float
68. mean squared error
69. '''
70. global i, min_MSE
71. kappa_v, theta_v, sigma_v, rho, v0, lamb, mu, delta = p0
72. if kappa_v < 0.0 or theta_v < 0.005 or sigma_v < 0.0 or \
73. rho < -1.0 or rho > 1.0 or v0 < 0.0 or lamb < 0.0 or \
74. mu < -.6 or mu > 0.0 or delta < 0.0:
75. return 5000.0
76. if 2 * kappa_v * theta_v < sigma_v ** 2:
77. return 5000.0
78. se = []
79. for row, option in options.iterrows():
80. model_value = BCC_call_value(S0, option['Strike'], option['T'],
81. option['r'], kappa_v, theta_v, sigma_v, rho, v0,
82. lamb, mu, delta)
83. se.append((model_value - option['Call']) ** 2)
84. MSE = sum(se) / len(se)
85. min_MSE = min(min_MSE, MSE)
86. if i % 25 == 0:
87. print '%4d |' % i, np.array(p0), '| %7.3f | %7.3f' % (MSE, min_MSE)
88. i += 1
89. return MSE
91. def BCC_calibration_full():
92. ''' Calibrates complete BCC97 model to market quotes. '''
93. # local, convex minimization for all parameters
94. opt = fmin(BCC_error_function, p0,
95. xtol=0.000001, ftol=0.000001,
96. maxiter=450, maxfun=650)
97. np.save('11_cal/opt_full', np.array(opt))
98. return opt
100. def BCC_calculate_model_values(p0):
101. ''' Calculates all model values given parameter vector p0. '''
102. kappa_v, theta_v, sigma_v, rho, v0, lamb, mu, delta = p0
103. values = []
104. for row, option in options.iterrows():
105. model_value = BCC_call_value(S0, option['Strike'], option['T'],
106. option['r'], kappa_v, theta_v, sigma_v, rho, v0,
107. lamb, mu, delta)
108. values.append(model_value)
109. return np.array(values)

# Calibration of Complete Model of BCC97

# Calibration of Stoch Vol Jump Model to EURO STOXX Option Quotes via Numerical Integration
# Data Source: www.eurexchange.com
# Bakshi, Cao and Chen (1997)
#
import sys
sys.path.append('09_gmm')
import math
import numpy as np
np.set_printoptions(suppress=True,
formatter={'all': lambda x: '%5.3f' % x})
import pandas as pd
from scipy.optimize import brute, fmin, minimize
import matplotlib as mpl
mpl.rcParams['font.family'] = 'serif'
from BCC_option_valuation import BCC_call_value
from CIR_calibration import CIR_calibration, r_list
from CIR_zcb_valuation import B
from H93_calibration import options
#
# Calibrate Short Rate Model
#
kappa_r, theta_r, sigma_r = CIR_calibration()
#
# Market Data from www.eurexchange.com
# as of 30. September 2014
#
S0 = 3225.93 # EURO STOXX 50 level
r0 = r_list[0] # initial short rate
#
# Parameters from H93 & jump calibrations
#
kappa_v, theta_v, sigma_v, rho, v0 = np.load('11_cal/opt_sv.npy')
lamb, mu, delta = np.load('11_cal/opt_jump.npy')
p0 = [kappa_v, theta_v, sigma_v, rho, v0, lamb, mu, delta]

#
# Calibration Functions
#
i = 0
min_MSE = 5000.0

def BCC_error_function(p0):
    ''' Error function for parameter calibration in BCC97 model via
    Lewis (2001) Fourier approach.
    Parameters
    ==========
    kappa_v: float
    mean-reversion factor
    theta_v: float
    long-run mean of variance
    sigma_v: float
    volatility of variance
    rho: float
    correlation between variance and stock/index level
    v0: float
    initial, instantaneous variance
    lamb: float
    jump intensity
    mu: float
    expected jump size
    delta: float
    standard deviation of jump
    Returns
    =======
    MSE: float
    mean squared error
    '''
    global i, min_MSE
    kappa_v, theta_v, sigma_v, rho, v0, lamb, mu, delta = p0
    if kappa_v < 0.0 or theta_v < 0.005 or sigma_v < 0.0 or \
                rho < -1.0 or rho > 1.0 or v0 < 0.0 or lamb < 0.0 or \
                    mu < -.6 or mu > 0.0 or delta < 0.0:
        return 5000.0
    if 2 * kappa_v * theta_v < sigma_v ** 2:
        return 5000.0
    se = []
    for row, option in options.iterrows():
        model_value = BCC_call_value(S0, option['Strike'], option['T'],
                                     option['r'], kappa_v, theta_v, sigma_v, rho, v0,
                                     lamb, mu, delta)
        se.append((model_value - option['Call']) ** 2)
    MSE = sum(se) / len(se)
    min_MSE = min(min_MSE, MSE)
    if i % 25 == 0:
        print '%4d |' % i, np.array(p0), '| %7.3f | %7.3f' % (MSE, min_MSE)
    i += 1
    return MSE

def BCC_calibration_full():
    ''' Calibrates complete BCC97 model to market quotes. '''
    # local, convex minimization for all parameters
    opt = fmin(BCC_error_function, p0,
               xtol=0.000001, ftol=0.000001,
               maxiter=450, maxfun=650)
    np.save('11_cal/opt_full', np.array(opt))
    return opt

def BCC_calculate_model_values(p0):
    ''' Calculates all model values given parameter vector p0. '''
    kappa_v, theta_v, sigma_v, rho, v0, lamb, mu, delta = p0
    values = []
    for row, option in options.iterrows():
        model_value = BCC_call_value(S0, option['Strike'], option['T'],
                                     option['r'], kappa_v, theta_v, sigma_v, rho, v0,
                                     lamb, mu, delta)
        values.append(model_value)
    return np.array(values)