Implements Path Shadowing Monte Carlo [1], which can be used for volatility prediction and option pricing.
This methods averages future quantities over generated price paths (grey) whose past history matches, or `shadows', the actual observed history (blue).
For a quick start, see tutorial.ipynb
Path Shadowing: finding close paths in a generated dataset.
From a venv with python>=3.10 run the commands below to install the required packages.
pip install git+https://github.com/RudyMorel/scattering_spectra@v2.0.2
pip install git+https://github.com/RudyMorel/shadowingOn a given date, path shadowing involves scanning through generated price paths whose past history matches, or shadows, the actual observed past history (blue). These generated price paths can be produced using the Scattering Spectra model [2] with the script snp_generation.py.
fromscatspectraimportSPDaily, TimeSeriesDatasetfromshadowingimport (
PathShadowing, Identity, RelativeMSE, PredictionContext, plot_shadow
)
# current past history of SnPsnp=SPDaily()
dlnx_current=snp.dlnx[0,0,-20:]
# generated dataset # can be generated using the script `generate_dataset.py`dpath=Path("/path/to/your/generated/dataset")
dataset=TimeSeriesDataset(dpath, R=32768) # dataset of 32768 long trajectories# shadowingembedding=Identity(dlnx_current.shape[-1]) # naive embeddingdistance=RelativeMSE() # distance in the embedding space obj=PathShadowing(
embedding=embedding,
distance=distance,
dataset=dataset,
context=PredictionContext(horizon=20),
)
distances, close_paths, _=obj.shadow(
dlnx_current,
k=1024, # get the 1024 closest paths from the generated datasetcuda=True
)
plot_shadow(dlnx_current, distances[0,:], close_paths[0,:,:,:], eta=0.09, date=snp.dts[-1], color='blue');
Shadow of a path.
Once the closest paths have been collected, you can aggregate their future volatility with a weighted average, to obtain a volatility prediction. The weighted average gives more importance to paths that are close to the current one.
fromshadowingimportplot_volatilityTs=np.array([5,10,20]) # the time scales at which we want to predict the volatility# the function to apply to each path to_predict=lambdax: realized_variance(x, Ts=Ts, vol=False)[:,:,0,:]
# now aggreate predictions on the 1024 closest paths with a weighted averagevars, _=obj.predict_from_paths(
distances,
close_paths,
to_predict=to_predict,
proba_name='softmax',
eta=0.1,
)
# helper function for plotting vol predictionsplot_volatility(dlnx_current, (vars[0,:]**0.5), Ts, distances[0,:], close_paths[0,...], eta=0.09, date=snp.dts[-1], color='blue')
Volatility prediction obtained with Path Shadowing Monte Carlo.
In the same way, one can price options on the collected close paths through Path Shadowing (hedged) Monte Carlo [3]
fromscatspectraimportSoftmax, PriceData, compute_smile# averaging operator in Hedged HMCave=Softmax(distances[-1,:], eta=0.9)
# the shadowing paths extending the currently observed pastx=PriceData(dlnx=close_paths[-1,:,0,20:], x_init=100.0).x# HMC on such shadowing pathsMs=np.linspace(-2,2,9)
smile_conditional=compute_smile(x, Ts, Ms, ave=ave)
# plotting_, ax=plt.subplots(1, 1, figsize=(4,3))
smile_conditional.plot(ax=ax, color='black', rescale=True, legend=True)
plt.xticks(fontsize=20); plt.yticks(fontsize=20)
plt.xlabel(r"$\mathcal{M}$", fontsize=20);
Option smile obtained through Path Shadowing (hedged) Monte Carlo.
See tutorial.ipynb for more advanced examples.
[1] "Path Shadowing Monte Carlo"
Rudy Morel et al. - https://arxiv.org/abs/2308.01486
[2] "Scale Dependencies and Self-Similar Models with Wavelet Scattering Spectra"
Rudy Morel et al. - https://arxiv.org/abs/2204.10177
[3] "Hedged Monte-Carlo: low variance derivative pricing with objective probabilities"
M. Potters, J.-P. Bouchaud, D. Sestovic - https://arxiv.org/abs/cond-mat/0008147