def SABR_beta_d(f, t, k, CallPut, alpha, beta, rho, volvol):
vol_1 = lognormal_vol(k, f, t, alpha, beta, rho, volvol) * 100
vol_2 = lognormal_vol(k, f, t, alpha, beta + 0.01, rho, volvol) * 100
b_s = BlackScholes(f, 0, 0, t)
if CallPut == 'C':
vanna = (b_s.BS_call(k, vol_2) - b_s.BS_call(k, vol_1)) / (0.01 * rho)
else:
vanna = (b_s.BS_put(k, vol_2) - b_s.BS_put(k, vol_1)) / (0.01 * rho)
return vanna
def SABR_alpha_deri(f, t, k, CallPut, alpha, beta, rho, volvol):
vol_1 = lognormal_vol(k, f, t, alpha * 0.99, beta, rho, volvol) * 100
vol_2 = lognormal_vol(k, f, t, alpha * 1.01, beta, rho, volvol) * 100
b_s = BlackScholes(f, 0, 0, t)
if CallPut == 'C':
volga = (b_s.BS_call(k, vol_2) - b_s.BS_call(k, vol_1)) / (0.02 *
alpha)
else:
volga = (b_s.BS_put(k, vol_2) - b_s.BS_put(k, vol_1)) / (0.02 * alpha)
return volga
def SABR_theta(f, t, k, CallPut, alpha, beta, rho, volvol):
t2 = t * 1.01
b_s = BlackScholes(f, 0, 0, t)
b_s_2 = BlackScholes(f, 0, 0, t2)
if CallPut == 'C':
theta = -(b_s.BS_call(
k, 100 * lognormal_vol(k, f, t, alpha, beta, rho, volvol)
) - b_s_2.BS_call(
k, 100 * lognormal_vol(k, f, t2, alpha, beta, rho, volvol))) / (t -
t2)
else:
theta = -(b_s.BS_put(
k, 100 * lognormal_vol(k, f, t, alpha, beta, rho, volvol)
) - b_s_2.BS_put(
k, 100 * lognormal_vol(k, f, t2, alpha, beta, rho, volvol))) / (t -
t2)
return theta
def SABR_delta(f, t, k, CallPut, alpha, beta, rho, volvol):
f2 = f * 1.01
b_s = BlackScholes(f * 1.01, 0, 0, t)
b_s_2 = BlackScholes(f * 0.99, 0, 0, t)
if CallPut == 'C':
delta_i = (b_s.BS_call(
k, 100 * lognormal_vol(k, f * 1.01, t, alpha, beta, rho, volvol)) -
b_s_2.BS_call(
k, 100 * lognormal_vol(k, f * 0.99, t, alpha, beta, rho,
volvol))) / (0.02 * f)
else:
delta_i = (b_s.BS_put(
k, 100 * lognormal_vol(k, f * 1.01, t, alpha, beta, rho, volvol)) -
b_s_2.BS_put(
k, 100 * lognormal_vol(k, f * 0.99, t, alpha, beta, rho,
volvol))) / (0.02 * f)
return delta_i
def fit_beta(f, f_new, t, t2, k, v_sln_t1, C_t1, C_t2, C_P_list, v_sln_atm_t1,
v_sln_atm_t2):
beta_list = np.arange(0, 11, 1) / 10
f_diff = f_new - f
C_diff = C_t2 - C_t1
min_hedge_error = 1e10
best_beta = beta_list[0]
hedge_error_list = []
for beta in beta_list:
hedge_error = 0
param = fit(f, t, beta, k, v_sln_t1)
alpha_t1 = alpha(v_sln_atm_t1 / 100, f, t, beta, param[1], param[2])
alpha_t2 = alpha(v_sln_atm_t2 / 100, f_new, t2, beta, param[1],
param[2])
f2 = f * 1.01
b_s = BlackScholes(f, 0, 0, t)
b_s_2 = BlackScholes(f2, 0, 0, t)
delta_list = []
vega_list = []
for num in range(len(k)):
s_k = k[num]
c_p = C_P_list[num]
if c_p == 'C':
delta_i = (b_s.BS_call(
s_k, 100 * lognormal_vol(s_k, f, t, param[0], beta,
param[1], param[2])) -
b_s_2.BS_call(
s_k, 100 *
lognormal_vol(s_k, f2, t, param[0], beta,
param[1], param[2]))) / (f - f2)
vega_i = (b_s.BS_call(s_k, 100*lognormal_vol(s_k, f, t, alpha_t1, beta, param[1], param[2])) -
b_s.BS_call(s_k, 100*lognormal_vol(s_k, f, t, alpha_t1 *1.01, beta, param[1], param[2]))) /\
(alpha_t1 - 1.01 * alpha_t1)
else:
delta_i = (b_s.BS_put(
s_k, 100 * lognormal_vol(s_k, f, t, param[0], beta,
param[1], param[2])) -
b_s_2.BS_put(
s_k, 100 *
lognormal_vol(s_k, f2, t, param[0], beta,
param[1], param[2]))) / (f - f2)
vega_i = (b_s.BS_put(
s_k, 100 * lognormal_vol(s_k, f, t, alpha_t1, beta,
param[1], param[2])) -
b_s.BS_put(
s_k, 100 * lognormal_vol(
s_k, f, t, alpha_t1 * 1.01, beta, param[1],
param[2]))) / (alpha_t1 - 1.01 * alpha_t1)
delta_list.append(delta_i)
vega_list.append(vega_i)
for i in range(len(C_diff)):
#sig = (C_diff[i] - delta_list[i] * f_diff ) / C_t2[i]
sig = (C_diff[i] - delta_list[i] * f_diff - vega_list[i] *
(alpha_t2 - alpha_t1)) / C_t2[i]
hedge_error += sig**2
hedge_error_list.append(hedge_error)
if hedge_error < min_hedge_error:
min_hedge_error = hedge_error
best_beta = beta
return best_beta