La calibración de Sabr no se ajusta a la volatilidad del mercado

Question

Estoy intentando calibrar SABR pero no me cuadra la volatilidad dada.

   import QuantLib as ql
import matplotlib.pyplot as plt
import numpy as np
from scipy.optimize import minimize
from scipy.optimize import  differential_evolution
moneyness=np.array([    120.00, 115.00, 114.00,     113.00,     112.00,     111.00 ,
     110.00, 109.00 ,108.00,    107.00, 106.00, 105.00, 104.00, 103.00, 102.50, 102.00,
    101.50, 101.00, 100.50, 100.00,     99.50,  99.00,  98.50,  98.00,  97.50,  97.00,
    96.50,  96.00,  95.50,95.00,    94.50,  94.00,  93.50,  93.00,  92.00,  91.00,90.00 ])
moneyness=moneyness/100
fwd = 1.361794
strikes=fwd*moneyness
expiryTime = 30/365
marketVols = np.array([0.0302,  0.08827,    0.10019,    0.11206,    0.12343,    0.13305,    0.13968,
              0.14325,  0.14431,    0.14337,    0.14077,    0.13706,    0.1325, 0.12664,
              0.12355,  0.12024,    0.11694,    0.11371,    0.11062,    0.10772,    0.10526,
              0.10307,  0.10118 ,0.09953,   0.09811,    0.0967, 0.09487,    0.09313,    0.09096,
              0.08795,  0.08359,    0.07751,    0.07003,    0.06203,    0.04591,    0.02975,    0.01347 ])
marketVols=1-marketVols
marketVols[20]=1-10.97/100

def f(params):
    params[0] = max(params[0], 1e-8)  # Avoid alpha going negative
    params[1] = max(params[1], 1e-8)  # Avoid beta going negative
    params[2] = max(params[2], 1e-8)  # Avoid nu going negative
    params[3] = max(params[3], -0.999)  # Avoid rhp going < -1.0
    params[3] = min(params[3], 0.999)  # Avoid rho going > 1.0

    vols = np.array([
        ql.sabrVolatility(strike, fwd, expiryTime, params[0],params[1],params[2],params[3])
        #(Rate strike, Rate forward,  Time expiryTime,Real alpha,Real beta,Real nu,Real rho,

        for strike in strikes
    ])
    return ((vols - np.array(marketVols))**2 ).mean() **.5

bounds = [(0.000001,1000000),(0.000001,0.99) ,(0.00001, 1000000), (-0.99, 0.99)]
result = differential_evolution(f, bounds,tol=0.0000001)

params=result.x

print(1-ql.sabrVolatility(1.279, fwd, expiryTime, params[0],params[1],params[2],params[3]))
newVols = [ql.sabrVolatility(strike, fwd, expiryTime, params[0],params[1],params[2],params[3]) for strike in strikes]
plt.plot(strikes, marketVols, marker='o', label="market")
plt.plot(strikes, newVols, marker='o', label="SABR")
plt.legend()
plt.show()

Answer 1

Pues a mí me parece muy bien. El modelo SABR sólo tiene 4 parámetros y no se puede hacer mucho con ellos.

Si tienes muchas volatilidades, especialmente si tienen una distribución bastante irregular, como en la imagen, esto es lo que puedes esperar al intentar ajustar el modelo SABR.