tf_quant_finance.experimental.svi.total_variance_from_raw_svi_parameters

Last updated: 2023-03-16.

tf_quant_finance.experimental.svi.total_variance_from_raw_svi_parameters#

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Computes modeled total variance using raw SVI parameters.

tf_quant_finance.experimental.svi.total_variance_from_raw_svi_parameters(
    *, svi_parameters, log_moneyness=None, forwards=None, strikes=None, dtype=None,
    name=None
)

The SVI volatility model parameterizes an option’s total implied variance, defined as w(k,t) := sigmaBS(k,t)^2 * t, where k := log(K/F) is the options’s log-moneyness, t is the time to expiry, and sigmaBS(k,t) is the Black-Scholes market implied volatility. For a fixed timeslice (i.e. given expiry t), the raw SVI parameterization consists of 5 parameters (a,b,rho,m,sigma), and the model approximation formula for w(k,t) as a function of k is (cf.[1]):

w(k) = a + b * (rho * (k - m) + sqrt{(k - m)^2 + sigma^2)}

The raw parameters have the following interpretations (cf.[2]): a vertically shifts the variance graph b controls the angle between the left and right asymptotes rho controls the rotation of the variance graph m horizontally shifts the variance graph sigma controls the graph smoothness at the vertex (ATM)

Example#

import numpy as np
import tensorflow as tf
import tf_quant_finance as tff

svi_parameters = np.array([-0.1825, 0.3306, -0.0988, 0.0368, 0.6011])

forwards = np.array([2402.])
strikes = np.array([[1800., 2000., 2200., 2400., 2600., 2800., 3000.]])

total_var = tff.experimental.svi.total_variance_from_raw_svi_parameters(
    svi_parameters=svi_parameters, forwards=forwards, strikes=strikes)

# Expected: total_var tensor (rounded to 4 decimal places) should contain
# [[0.0541, 0.0363, 0.02452, 0.0178, 0.0153, 0.0161, 0.0194]]

References:#

[1] Gatheral J., Jaquier A., Arbitrage-free SVI volatility surfaces. https://arxiv.org/pdf/1204.0646.pdf [2] Gatheral J, A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives. http://faculty.baruch.cuny.edu/jgatheral/madrid2004.pdf

Args:#

  • svi_parameters: A rank 2 real Tensor of shape [batch_size, 5]. The raw SVI parameters for each volatility skew.

  • log_moneyness: A rank 2 real Tensor of shape [batch_size, num_strikes]. The log-moneyness of the option.

  • forwards: A rank 2 real Tensor of shape [batch_size, num_strikes]. The forward price of the option at expiry.

  • strikes: A rank 2 real Tensor of shape [batch_size, num_strikes]. The option’s strike price.

  • dtype: Optional tf.Dtype. If supplied, the dtype for the input and output Tensors will be converted to this. Default value: None which maps to the dtype inferred from log_moneyness.

  • name: Python str. The name to give to the ops created by this function. Default value: None which maps to svi_total_variance.

Returns:#

A rank 2 real Tensor of shape [batch_size, num_strikes].

Raises:#

  • ValueError: If exactly one of forwards and strikes is supplied.

  • ValueError: If both log_moneyness' and forwards` are supplied or if neither is supplied.

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