🚀 StochVolModels Package: stochvolmodels

stochvolmodels package implements pricing analytics and Monte Carlo simulations for valuation of European call and put options and implied volatilities of different stochastic volatility models including Karasinski-Sepp long-normal stochastic volatility model and Heston stochastic volatility model.

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StochVolModels

The StochVol package provides:

1. Analytics for Black-Scholes and Normal vols
2. Interfaces and implementation for stochastic volatility models, including Karasinski-Sepp log-normal SV model and Heston SV model using analytical method with Fourier transform and Monte Carlo simulations
3. 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.

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

Illustrations of using package analytics for research work is provided in top-level package my_papers which contains computations and visualisations for several papers

Installation

Install using

pip install stochvolmodels

Upgrade using

pip install --upgrade stochvolmodels

Close using

git clone https://github.com/ArturSepp/StochVolModels.git

Core Dependencies

• python >= 3.8
• numba >= 0.56.4
• numpy >= 1.22.4
• scipy >= 1.10
• pandas >= 2.2.0
• matplotlib >= 3.2.2
• seaborn >= 0.12.2

Optional dependencies: qis ">=2.3.1" (for running code in my_papers)

Table of contents

1. Model Interface
   1. Log-normal stochastic volatility model
   2. Heston stochastic volatility model
2. Running log-normal SV pricer
   1. Computing model prices and vols
   2. Running model calibration to sample Bitcoin options data
   3. Comparison of model prices vs MC
   4. Analysis and figures for the paper
3. Running Heston SV pricer
4. Supporting Illustrations for Public Papers

Running model calibration to sample Bitcoin options data

Implemented Stochastic Volatility models <a name="introduction"></a>

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

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

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

Log-normal stochastic volatility model <a name="logsv"></a>

The analytics for Karasinki-Sepp 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 <a name="hestonsv"></a>

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 <a name="paragraph1"></a>

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 <a name="subparagraph1"></a>

# 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 <a name="subparagraph2"></a>

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 <a name="subparagraph3"></a>

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 <a name="subparagraph4"></a>

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

examples/plots_for_paper

Running Heston SV pricer <a name="heston"></a>

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()

Supporting Illustrations for Public Papers <a name="papers"></a>

As illustrations of different analytics, this