Implement pricing analytics and Monte Carlo simulations for stochastic volatility models including log-normal SV model, Heston

ArturSepp Last update: Jan 14, 2024

StochVolModels

Implementation of pricing analytics and Monte Carlo simulations for modeling of options and implied volatilities.

The StochVolPackage provides:

Analytics for Black-Scholes and Normal vols
Interfaces and implementation for stochastic volatility models including log-normal SV model and Heston SV model
Visualization of model implied volatilities

For the analytic implementation of stochastic volatility models, the package provides interfaces for a generic volatility model with the following features.

Interface for analytical pricing of vanilla options using Fourier transform with closed-form solution for moment generating function
Interface for Monte-Carlo simulations of model dynamics

Illustrations

As illustrations of different analytics, this packadge includes the computations and visualisations for

Log-normal Stochastic Volatility Model with Quadratic Drift by Sepp A and Rakhmonov P, SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2522425

stochvolmodels/my_papers/logsv_model_wtih_quadratic_drift

What is a robust stochastic volatility model by Sepp A and Rakhmonov P, SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4647027

stochvolmodels/my_papers/volatility_models

Installation

pip install stochvolmodels

Model Interface
1. Log-normal stochastic volatility model
2. Heston stochastic volatility model
Running log-normal SV pricer
Running Heston SV pricer

Running model calibration to sample Bitcoin options data

Implemented Stochastic Volatility models

The package provides interfaces for a generic volatility model with the following features.

Interface for analytical pricing of vanilla options using Fourier transform with closed-form solution for moment generating function
Interface for Monte-Carlo simulations of model dynamics
Interface for visualization of model implied volatilities

The model interface is in stochvolmodels/pricers/model_pricer.py

Log-normal stochastic volatility model

The analytics for the log-normal stochastic volatility model is based on the paper

Log-normal Stochastic Volatility Model with Quadratic Drift by Artur Sepp and Parviz Rakhmonov

The dynamics of the log-normal stochastic volatility model:

$$dS_{t}=r(t)S_{t}dt+\sigma_{t}S_{t}dW^{(0)}_{t}$$

$$d\sigma_{t}=\left(\kappa_{1} + \kappa_{2}\sigma_{t} \right)(\theta - \sigma_{t})dt+ \beta \sigma_{t}dW^{(0)}{t} + \varepsilon \sigma{t} dW^{(1)}_{t}$$

$$dI_{t}=\sigma^{2}_{t}dt$$

where $r(t)$ is the deterministic risk-free rate; $W^{(0)}_{t}$ and $W^{(1)}_t$ are uncorrelated Brownian motions, $\beta\in\mathbb{R}$ is the volatility beta which measures the sensitivity of the volatility to changes in the spot price, and $\varepsilon>0$ is the volatility of residual volatility. We denote by $\vartheta^{2}$, $\vartheta^{2}=\beta^{2}+\varepsilon^{2}$, the total instantaneous variance of the volatility process.

Implementation of Lognormal SV model is contained in

stochvolmodels/pricers/logsv_pricer.py

Heston stochastic volatility model

The dynamics of Heston stochastic volatility model:

$$dS_{t}=r(t)S_{t}dt+\sqrt{V_{t}}S_{t}dW^{(S)}_{t}$$

$$dV_{t}=\kappa (\theta - V_{t})dt+ \vartheta \sqrt{V_{t}}dW^{(V)}_{t}$$

where $W^{(S)}$ and $W^{(V)}$ are correlated Brownian motions with correlation parameter $\rho$

Implementation of Heston SV model is contained in

stochvolmodels/pricers/heston_pricer.py

Running log-normal SV pricer

Basic features are implemented in

examples/run_lognormal_sv_pricer.py

Imports:

import stochvolmodels as sv
from stochvolmodels import LogSVPricer, LogSvParams, OptionChain

Computing model prices and vols

# instance of pricer
logsv_pricer = LogSVPricer()

# define model params    
params = LogSvParams(sigma0=1.0, theta=1.0, kappa1=5.0, kappa2=5.0, beta=0.2, volvol=2.0)

# 1. compute ne price
model_price, vol = logsv_pricer.price_vanilla(params=params,
                                             ttm=0.25,
                                             forward=1.0,
                                             strike=1.0,
                                             optiontype='C')
print(f"price={model_price:0.4f}, implied vol={vol: 0.2%}")

# 2. prices for slices
model_prices, vols = logsv_pricer.price_slice(params=params,
                                             ttm=0.25,
                                             forward=1.0,
                                             strikes=np.array([0.9, 1.0, 1.1]),
                                             optiontypes=np.array(['P', 'C', 'C']))
print([f"{p:0.4f}, implied vol={v: 0.2%}" for p, v in zip(model_prices, vols)])

# 3. prices for option chain with uniform strikes
option_chain = OptionChain.get_uniform_chain(ttms=np.array([0.083, 0.25]),
                                            ids=np.array(['1m', '3m']),
                                            strikes=np.linspace(0.9, 1.1, 3))
model_prices, vols = logsv_pricer.compute_chain_prices_with_vols(option_chain=option_chain, params=params)
print(model_prices)
print(vols)

Running model calibration to sample Bitcoin options data

btc_option_chain = chains.get_btc_test_chain_data()
params0 = LogSvParams(sigma0=0.8, theta=1.0, kappa1=5.0, kappa2=None, beta=0.15, volvol=2.0)
btc_calibrated_params = logsv_pricer.calibrate_model_params_to_chain(option_chain=btc_option_chain,
                                                                    params0=params0,
                                                                    constraints_type=ConstraintsType.INVERSE_MARTINGALE)
print(btc_calibrated_params)

logsv_pricer.plot_model_ivols_vs_bid_ask(option_chain=btc_option_chain,
                               params=btc_calibrated_params)

Comparison of model prices vs MC

btc_option_chain = chains.get_btc_test_chain_data()
uniform_chain_data = OptionChain.to_uniform_strikes(obj=btc_option_chain, num_strikes=31)
btc_calibrated_params = LogSvParams(sigma0=0.8327, theta=1.0139, kappa1=4.8609, kappa2=4.7940, beta=0.1988, volvol=2.3694)
logsv_pricer.plot_comp_mma_inverse_options_with_mc(option_chain=uniform_chain_data,
                                                  params=btc_calibrated_params,
                                                  nb_path=400000)

Analysis and figures for the paper

All figures shown in the paper can be reproduced using py scripts in

examples/plots_for_paper

Running Heston SV pricer

Examples are implemented here

examples/run_heston_sv_pricer.py
examples/run_heston.py

Content of run_heston.py

import numpy as np
import matplotlib.pyplot as plt
from stochvolmodels import HestonPricer, HestonParams, OptionChain

# define parameters for bootstrap
params_dict = {'rho=0.0': HestonParams(v0=0.2**2, theta=0.2**2, kappa=4.0, volvol=0.75, rho=0.0),
               'rho=-0.4': HestonParams(v0=0.2**2, theta=0.2**2, kappa=4.0, volvol=0.75, rho=-0.4),
               'rho=-0.8': HestonParams(v0=0.2**2, theta=0.2**2, kappa=4.0, volvol=0.75, rho=-0.8)}

# get uniform slice
option_chain = OptionChain.get_uniform_chain(ttms=np.array([0.25]), ids=np.array(['3m']), strikes=np.linspace(0.8, 1.15, 20))
option_slice = option_chain.get_slice(id='3m')

# run pricer
pricer = HestonPricer()
pricer.plot_model_slices_in_params(option_slice=option_slice, params_dict=params_dict)

plt.show()