Calibrating SABR Volatility with Shifted Strikes

# Interpreting SABR calibration model output


# Interpreting SABR calibration model output












Calibrate a SABR model?

Following on from this question, I have used the same market data they attached but am unsure on interpreting the output.

When I plot the SABR probabilities against strike for data below, no smile is displayed, only a skew/downward slope as all the output volatilities are decreasing.

How do I use this data to produce the volatility smile graphs? or does this data just not display smile?

The first line of market data Travaglini uses is;

and the first line of the output SABR volatilities is;

market data and code from https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2725485

## Answer by StackG (score 2)

https://quant.stackexchange.com/a/58594

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'])
```