Valuation of European volatility call options

in Gruenbichler-Longstaff (1996) model

square-root diffusion framework

— semianalytical formula

from scipy.stats import ncx2
import numpy as np

Semianalytical option pricing formula of GL96

def calculate_option_value(V0, kappa, theta, sigma, zeta, T, r, K):
”’ Calculation of European call option price in GL96 model.

            Parameters

            ==========

            V0  :   float

                            current volatility  level

            kappa   :   float

                            mean    reversion   factor

            theta   :   float

                            long-run    mean    of  volatility

            sigma   :   float

                            volatility  of  volatility

            zeta    :

                            volatility  risk    premium

            T   :   float

                            time-to-maturity

            r   :   float

                            risk-free   short   rate

            K   :   float

                            strike  price   of  the option



            Returns

            =======

            value   :   float

                            net present value   of  volatility  call    option

            ”’

D = np.exp(-r * T) # discount factor

variables

alpha = kappa theta
beta = kappa + zeta
gamma = 4 beta / (sigma * 2 ( 1 - np.exp(-beta T)))
nu = 4 alpha / sigma * 2
lamb = gamma np.exp(-beta T) V0
cx1 = 1 - ncx2.cdf(gamma K, nu + 4 , lamb)
cx2 = 1 - ncx2.cdf(gamma K, nu + 2 , lamb)
cx3 = 1 - ncx2.cdf(gamma * K, nu, lamb)

formula for European call price

value = (D np.exp(-beta T) V0 cx1

• D (alpha / beta) ( 1 - np.exp(-beta * T))

                    *   cx2 -   D   *   K   *   cx3)

return value

To simplify the implementation of the web service we write a convenience function,

get_option_value, which will check for the provision of all needed parameters to

calculate a call option value. The function is stored in a Python module called

vol_pricing_service.py, the code of which is shown in Example 14-9. This script also

contains a dictionary with all the necessary parameters and brief descriptions of these

parameters. The function will return an error message detailing what is missing whenever

one or more parameters are missing. If all necessary parameters are provided during the

web service call, the function calls the pricing function calculate_option_value from

the vol_pricing_formula.py script.

Example 14-9. Python script for volatility option valuation and web service helper

function