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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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

Install using

pip install stochvolmodels

Upgrade using

pip install --upgrade stochvolmodels

Close using

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

• 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)

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

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

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

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

Basic features are implemented in

examples/run_lognormal_sv_pricer.py

Imports:

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

# 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)

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)

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) 

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

examples/plots_for_paper

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()

As illustrations of different analytics, this packadge includes module my_papers with codes for computations and visualisations featured in several papers for

1. "Log-normal Stochastic Volatility Model with Quadratic Drift" by Artur Sepp and Parviz Rakhmonov: https://www.worldscientific.com/doi/10.1142/S0219024924500031

stochvolmodels/my_papers/logsv_model_wtih_quadratic_drift

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

stochvolmodels/my_papers/volatility_models

3. "Valuation and Hedging of Cryptocurrency Inverse Options" by Artur Sepp and Vladimir Lucic, SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4606748

stochvolmodels/my_papers/inverse_options

4. "Unified Approach for Hedging Impermanent Loss of Liquidity Provision" by Artur Sepp, Alexander Lipton and Vladimir Lucic, SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4887298

stochvolmodels/my_papers/il_hedging

5. "Stochastic Volatility for Factor Heath-Jarrow-Morton Framework" by Artur Sepp and Parviz Rakhmonov, SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4646925

stochvolmodels/my_papers/sv_for_factor_hjm

StochVolModels/ ├── stochvolmodels/ │ ├── pricers/ │ │ ├── model_pricer.py # Generic model interface │ │ ├── logsv_pricer.py # Log-normal SV implementation │ │ └── heston_pricer.py # Heston SV implementation │ ├── data/ │ │ └── option_chain.py # Option chain data structures │ └── my_papers/ # Research paper implementations │ ├── logsv_model_with_quadratic_drift/ │ ├── volatility_models/ │ ├── inverse_options/ │ ├── il_hedging/ │ └── sv_for_factor_hjm/ ├── examples/ │ ├── run_lognormal_sv_pricer.py │ ├── run_heston_sv_pricer.py │ ├── run_heston.py │ └── plots_for_paper/ └── README.md 

Contributions are welcome! Please feel free to submit a Pull Request. For major changes, please open an issue first to discuss what you would like to change.

This project is licensed under the MIT License - see the LICENSE file for details.

If you use this package in your research, please cite the relevant papers:

@misc{sepp2024stochvolmodels, title={StochVolModels: Python Implementation of Stochastic Volatility Models}, author={Sepp, Artur}, year={2024}, howpublished={\url{https://github.com/ArturSepp/StochVolModels}}, note={Python package for pricing analytics and Monte Carlo simulations} } @article{sepprakhmonov2023, title={Log-normal stochastic volatility model with quadratic drift}, author={Sepp, Artur and Rakhmonov, Parviz}, journal={International Journal of Theoretical and Applied Finance}, volume={26}, number={8}, year={2023}, url={https://www.worldscientific.com/doi/epdf/10.1142/S0219024924500031} } @article{sepprakhmonov2023b, title={What is a robust stochastic volatility model}, author={Sepp, Artur and Rakhmonov, Parviz}, year={2023}, note={Working paper}, url={http://ssrn.com/abstract=4647027} } @article{lucicsepp2024, title={Valuation and hedging of cryptocurrency inverse options}, author={Lucic, Vladimir and Sepp, Artur}, journal={Quantitative Finance}, volume={24}, number={7}, pages={851--869}, year={2024}, url={https://www.tandfonline.com/doi/full/10.1080/14697688.2024.2364804} } @article{lucicsepp2024, title={Valuation and hedging of cryptocurrency inverse options}, author={Lucic, Vladimir and Sepp, Artur}, journal={Quantitative Finance}, volume={24}, number={7}, pages={851--869}, year={2024}, url={https://www.tandfonline.com/doi/full/10.1080/14697688.2024.2364804} } @article{sepprakhmonov2024, title={Stochastic volatility for factor Heath-Jarrow-Morton framework}, author={Sepp, Artur and Rakhmonov, Parviz}, year={2025}, journal={Review of Derivatives Research}, note={Accepted}, url={http://ssrn.com/abstract=4646925} }

Special thanks to co-authors and collaborators:

• Parviz Rakhmonov
• Vladimir Lucic
• Alexander Lipton

For additional research and advanced analytics, see the companion modules and papers included in this package.

If you use StochVolModels in your research, please cite it as:

@software{stochvolmodels2024, author={Sepp, Artur}, title={StochVolModels: Python implementation of pricing analytics and Monte Carlo simulations for stochastic volatility models}, year={2024}, url={https://github.com/ArturSepp/StochVolModels}, }