tf_quant_finance.models.sabr.approximations.implied_volatility

Last updated: 2023-03-16.

tf_quant_finance.models.sabr.approximations.implied_volatility#

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Computes the implied volatility under the SABR model.

tf_quant_finance.models.sabr.approximations.implied_volatility(
    *, strikes, expiries, forwards, alpha, beta, volvol, rho, shift=0.0, volatility_
    type=tf_quant_finance.models.sabr.approximations.SabrImpliedVolatilityType.LOGNO
    RMAL, approximation_type=tf_quant_finance.models.sabr.approximations.SabrApproxi
    mationType.HAGAN, dtype=None, name=None
)

The SABR model specifies the risk neutral dynamics of the underlying as the following set of stochastic differential equations:

  dF = sigma F^beta dW_1
  dsigma = volvol sigma dW_2
  dW1 dW2 = rho dt

  F(0) = f
  sigma(0) = alpha

where F(t) represents the value of the forward price as a function of time, and sigma(t) is the volatility.

Here, we implement an approximate solution as proposed by Hagan [1], and back out the equivalent implied volatility that would’ve been obtained under either the normal model or the Black model.

Example#

import tf_quant_finance as tff
import tensorflow as tf

equiv_vol = tff.models.sabr.approximations.implied_volatility(
    strikes=np.array([106.0, 11.0]),
    expiries=np.array([17.0 / 365.0, 400.0 / 365.0]),
    forwards=np.array([120.0, 20.0]),
    alpha=1.63,
    beta=0.6,
    rho=0.00002,
    volvol=3.3,
    dtype=tf.float64)
# Expected: [0.33284656705268817, 1.9828728139982792]

# Running this inside a unit test passes:
# equiv_vol = self.evaluate(equiv_vol)
# self.assertAllClose(equiv_vol, 0.33284656705268817)

References#

[1] Hagan et al, Managing Smile Risk, Wilmott (2002), 1:84-108

Args:#

  • strikes: Real Tensor of arbitrary shape, specifying the strike prices. Values must be strictly positive.

  • expiries: Real Tensor of shape compatible with that of strikes, specifying the corresponding time-to-expiries of the options. Values must be strictly positive.

  • forwards: Real Tensor of shape compatible with that of strikes, specifying the observed forward prices of the underlying. Values must be strictly positive.

  • alpha: Real Tensor of shape compatible with that of strikes, specifying the initial values of the stochastic volatility. Values must be strictly positive.

  • beta: Real Tensor of shape compatible with that of strikes, specifying the model exponent beta. Values must satisfy 0 <= beta <= 1.

  • volvol: Real Tensor of shape compatible with that of strikes, specifying the model vol-vol multipliers. Values of volvol must be non-negative.

  • rho: Real Tensor of shape compatible with that of strikes, specifying the correlation factors between the Wiener processes modeling the forward and the volatility. Values must satisfy -1 < rho < 1.

  • shift: Optional Tensor of shape compatible with that of strkies, specifying the shift parameter(s). In the shifted model, the process modeling the forward is modified as: dF = sigma * (F + shift) ^ beta * dW. With this modification, negative forward rates are valid as long as F > -shift. Default value: 0.0

  • volatility_type: Either SabrImpliedVolatility.NORMAL or LOGNORMAL. Default value: LOGNORMAL.

  • approximation_type: Instance of SabrApproxmationScheme. Default value: HAGAN.

  • dtype: Optional: tf.DType. If supplied, the dtype to be used for converting values to Tensors. Default value: None, which means that the default dtypes inferred from strikes is used.

  • name: str. The name for the ops created by this function. Default value: ‘sabr_approx_implied_volatility’.

Returns:#

A real Tensor of the same shape as strikes, containing the corresponding equivalent implied volatilities.

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