tf_quant_finance.models.heston.approximations.european_option_price

Last updated: 2023-03-16.

tf_quant_finance.models.heston.approximations.european_option_price#

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Calculates European option prices under the Heston model.

tf_quant_finance.models.heston.approximations.european_option_price(
    *, strikes, expiries, spots=None, forwards=None, is_call_options=None,
    discount_rates=None, dividend_rates=None, discount_factors=None, variances,
    mean_reversion, theta, volvol, rho=None, integration_method=None, dtype=None,
    name=None, **kwargs
)

Heston originally published in 1993 his eponymous model [3]. He provided a semi- analytical formula for pricing European option via Fourier transform under his model. However, as noted by Albrecher [1], the characteristic function used in Heston paper can suffer numerical issues because of the discontinuous nature of the square root function in the complex plane, and a second version of the characteric function which doesn’t suffer this shortcoming should be used instead. Attari [2] further refined the numerical method by reducing the number of numerical integrations (only one Fourier transform instead of two) and with an integrand function decaying quadratically instead of linearly. Attari’s numerical method is implemented here.

Heston model:#

  dF/F = sqrt(V) * dW_1
  dV = mean_reversion * (theta - V) * dt * sigma * sqrt(V) * dW_2
  <dW_1,dW_2> = rho *dt

The variance V follows a square root process.

Example#

import tf_quant_finance as tff
import numpy as np
prices = tff.models.heston.approximations.european_option_price(
    variances=0.11,
    strikes=102.0,
    expiries=1.2,
    forwards=100.0,
    is_call_options=True,
    mean_reversion=2.0,
    theta=0.5,
    volvol=0.15,
    rho=0.3,
    discount_factors=1.0,
    dtype=np.float64)
# Expected print output of prices:
# 24.82219619

References#

[1] Hansjorg Albrecher, The Little Heston Trap https://perswww.kuleuven.be/~u0009713/HestonTrap.pdf [2] Mukarram Attari, Option Pricing Using Fourier Transforms: A Numerically Efficient Simplification https://papers.ssrn.com/sol3/papers.cfm?abstract_id=520042 [3] Steven L. Heston, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options http://faculty.baruch.cuny.edu/lwu/890/Heston93.pdf Args: strikes: A real Tensor of any shape and dtype. The strikes of the options to be priced. expiries: A real Tensor of the same dtype and compatible shape as strikes. The expiry of each option. spots: A real Tensor of any shape that broadcasts to the shape of the strikes. The current spot price of the underlying. Either this argument or the forwards (but not both) must be supplied. forwards: A real Tensor of any shape that broadcasts to the shape of strikes. The forwards to maturity. Either this argument or the spots must be supplied but both must not be supplied. is_call_options: A boolean Tensor of a shape compatible with strikes. Indicates whether the option is a call (if True) or a put (if False). If not supplied, call options are assumed. discount_rates: An optional real Tensor of same dtype as the strikes and of the shape that broadcasts with strikes. If not None, discount factors are calculated as e^(-rT), where r are the discount rates, or risk free rates. At most one of discount_rates and discount_factors can be supplied. Default value: None, equivalent to r = 0 and discount factors = 1 when discount_factors also not given. dividend_rates: An optional real Tensor of same dtype as the strikes and of the shape that broadcasts with strikes. Default value: None, equivalent to q = 0. discount_factors: An optional real Tensor of same dtype as the strikes. If not None, these are the discount factors to expiry (i.e. e^(-rT)). Mutually exclusive with discount_rates. If neither is given, no discounting is applied (i.e. the undiscounted option price is returned). If spots is supplied and discount_factors is not None then this is also used to compute the forwards to expiry. At most one of discount_rates and discount_factors can be supplied. Default value: None, which maps to e^(-rT) calculated from discount_rates. variances: A real Tensor of the same dtype and compatible shape as strikes. The initial value of the variance. mean_reversion: A real Tensor of the same dtype and compatible shape as strikes. The mean reversion strength of the variance square root process. theta: A real Tensor of the same dtype and compatible shape as strikes. The mean reversion level of the variance square root process. volvol: A real Tensor of the same dtype and compatible shape as strikes. The volatility of the variance square root process (volatility of volatility) rho: A real Tensor of the same dtype and compatible shape as strikes. The correlation between spot and variance. integration_method: An instance of math.integration.IntegrationMethod. Default value: None which maps to the Simpsons integration rule. dtype: Optional tf.DType. If supplied, the dtype to be used for conversion of any supplied non-Tensor arguments to Tensor. Default value: None which maps to the default dtype inferred by TensorFlow. name: str. The name for the ops created by this function. Default value: None which is mapped to the default name heston_price. **kwargs: Additional parameters for the underlying integration method. If not supplied and integration_method is Simpson, then uses IntegrationMethod.COMPOSITE_SIMPSONS_RULE with num_points=1001, and bounds lower=1e-9, upper=100. Returns: A Tensor of the same shape as the input data which is the price of European options under the Heston model.

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