The paper takes each observed smile, bumps all of the strikes by a shift term to make them positive, and the fits a SABR smile to them.
When I do the same thing with the dataset you've attached above (I remove the -150 point because it's vol of 0.0 breaks things) I get the following 'smiley' fit, which looks similar to the results presented above:
This was generated like this:
import numpy as np
import pandas as pd
import QuantLib as ql
from matplotlib import pyplot as plt
from scipy import optimize
# Parameters from the post
strikes = [-100, -50, -25, 0, 25, 50, 100, 150]
vols = [1.047, 0.4812, 0.4327, 0.4268, 0.4148, 0.4253, 0.4322, 0.4495]
fwd = 0.01076
expiry = 0.25
shift = 0.0110 # Shift chosen to make first strike positive
# params are sigma_0, beta, vol_vol, rho
params = [0.4, 0.6, 0.1, -0.4]
# Optimise SABR least squares using python's minimize function
def f(params):
alpha, beta, nu, rho = params[0], params[1], params[2], params[3]
alpha = max(alpha, 1e-8) # Avoid alpha going negative
beta = max(beta, 1e-8) # Avoid beta going negative
nu = max(nu, 1e-8) # Avoid nu going negative
rho = max(rho, -0.999) # Avoid rhp going < -1.0
rho = min(rho, 0.999) # Avoid rho going > 1.0
calc_vols = np.array([
ql.sabrVolatility(strike*1e-4 + shift, fwd + shift, expiry, alpha, beta, nu, rho)
for strike in strikes
])
error = ((calc_vols - np.array(vols))**2 ).mean() **.5
return error
cons = (
{'type': 'ineq', 'fun': lambda x: x[0]},
{'type': 'ineq', 'fun': lambda x: 0.99 - x[1]},
{'type': 'ineq', 'fun': lambda x: x[1]},
{'type': 'ineq', 'fun': lambda x: x[2]},
{'type': 'ineq', 'fun': lambda x: 1. - x[3]**2}
)
result = optimize.minimize(f, params, constraints=cons, options={'eps': 1e-5})
alpha, beta, nu, rho = result['x'][0], result['x'][1], result['x'][2], result['x'][3]
calc_vols = np.array([
ql.sabrVolatility(strike*1e-4 + shift, fwd + shift, expiry, alpha, beta, nu, rho)
for strike in strikes
])
results = pd.DataFrame([vols, calc_vols], columns=strikes, index=['market', 'SABR'])
