Implementation of Stochastic Volatility Models
Project description
StochVolModels
Implement pricing analytics and Monte Carlo simulations for stochastic volatility models including log-normal SV model and Heston SV model The analytics for the lognormal is based on the paper
Log-normal Stochastic Volatility Model with Quadratic Drift: Applications to Assets with Positive Return-Volatility Correlation and to Inverse Martingale Measures by Artur Sepp and Parviz Rakhmonov
Table of contents
Running model calibration to sample Bitcoin options data
Model Interface
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
Implementation of Lognormal SV model is based on paper https://github.com/ArturSepp/StochVolModels/blob/main/docs/lognormal_stoch_vol_paper.pdf
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
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)
Comparision 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
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