Probabilistic Programming in Quantitative Finance by Thomas Wiecki, Lead Data Scientist at Quantopian

Probabilistic Programming in Quantitative FinanceProbabilistic Programming in Quantitative Finance
Thomas WieckiThomas Wiecki
@twiecki@twiecki
bayesian_risk_perf_v3 slides http://twiecki.github.io/bayesian_risk_perf_v3.slides.html?print-pdf#/
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About meAbout me
Lead Data Scientist at : Building a
crowd sourced hedge fund.
PhD from Brown University -- research on computational neuroscience and machine
learning using Bayesian modeling.
Quantopian Inc (https://www.quantopian.com)
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The problem we're gonna solveThe problem we're gonna solve
Two real-money strategies:
In [76]: plot_strats()
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Types of risk
Systematic and Unsystematic Risk
Volatility
Tailrisk
Beta
D
raw
dow
n
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Sharpe RatioSharpe Ratio
Sharpe =
mean returns
volatility
In [24]: print "Sharpe ratio strategy etrade =", data_0.mean() / data_0.std() * np.sqrt(252)
print "Sharpe ratio strategy IB =", data_1.mean() / data_1.std() * np.sqrt(252)
Sharpe ratio strategy etrade = 0.627893606355
Sharpe ratio strategy IB = 1.43720181575
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Types of risk
Systematic and Unsystematic Risk
Model misspeciﬁcation
Estimation Uncertainty
Programming errors
Data issuesModelRisk
Volatility
Tailrisk
Beta
D
raw
dow
n
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Short primer on random variablesShort primer on random variables
Represents our beliefs about an unknown state.
Probability distribution assigns a probability to each possible state.
Not a single number (e.g. most likely state).
"When I bet on horses, I never lose. Why? I bet on all the horses." Tom Haverford
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You already know what a variable is...You already know what a variable is...
In [8]: coin = 0 # 0 for tails
coin = 1 # 1 for heads
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A random variable assigns all possible values a certainA random variable assigns all possible values a certain
probabilityprobability
In [ ]: coin = {0: 50%,
1: 50%}
Alternatively:Alternatively:
coin ~ Bernoulli(p=0.5)
coin is a random variable
Bernoulli is a probability distribution
~ reads as "is distributed as"
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This was discrete (binary), what about the continuousThis was discrete (binary), what about the continuous
case?case?
returns ~ Normal( , )μ σ2
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In [77]: from scipy import stats
sns.distplot(data_0, kde=False, fit=stats.norm)
plt.xlabel('returns')
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How to estimateHow to estimate andand ??
Naive: point estimate
Set mu = mean(data) and sigma = std(data)
Maximum Likelihood Estimate
Correct answer as
μ σ
n → ∞
Bayesian analysisBayesian analysis
Most of the time ...
Uncertainty about and
Turn and into random variables
How to estimate?
n ≠ ∞
μ σ
μ σ
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Bayes Formula!Bayes Formula!
BayesPrior
Data
Posterior
Use prior knowledge and data to update our beliefs.
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In [78]: interactive(gen_plot, n=(0, 600), bayes=True)
Out[78]:
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Probabilistic ProgrammingProbabilistic Programming
Model unknown causes (e.g. ) of a phenomenon as random variables.
Write a programmatic story of how unknown causes result in observable data.
Use Bayes formula to invert generative model to infer unknown causes.
μ
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Approximating the posterior with MCMC samplingApproximating the posterior with MCMC sampling
In [81]: plot_want_get()
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PyMC3PyMC3
Probabilistic Programming framework written in Python.
Allows for construction of probabilistic models using intuitive syntax.
Features advanced MCMC samplers.
Fast: Just-in-time compiled by Theano.
Extensible: easily incorporates custom MCMC algorithms and unusual probability
distributions.
Authors: John Salvatier, Chris Fonnesbeck, Thomas Wiecki
Upcoming beta release!
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Model returns distribution: Specifying ourModel returns distribution: Specifying our
priorspriors
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In [82]: x = np.linspace(-.3, .3, 500)
plt.plot(x, T.exp(pm.Normal.dist(mu=0, sd=.1).logp(x)).eval())
plt.title(u'Prior: mu ~ Normal(0, $.1^2$)'); plt.xlabel('mu'); plt.ylabel('Probabili
ty Density'); plt.xlim((-.3, .3));
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In [83]: x = np.linspace(-.1, .5, 500)
plt.plot(x, T.exp(pm.HalfNormal.dist(sd=.1).logp(x)).eval())
plt.title(u'Prior: sigma ~ HalfNormal($.1^2$)'); plt.xlabel('sigma'); plt.ylabel('Pr
obability Density');
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Bayesian Sharpe ratioBayesian Sharpe ratio
μ ∼ Normal(0, ).12
← Prior
σ ∼ HalfNormal( ).12
← Prior
returns ∼ Normal(μ, )σ2
← Observed!
Sharpe =
μ
σ
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Graphical model of returnsGraphical model of returns
Bayes
PosteriorsPriors
Data
µ ~
σ ~
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This is what the data looks likeThis is what the data looks like
In [9]: print data_0.head()
2013-12-31 21:00:00 0.002143
2014-01-02 21:00:00 -0.028532
2014-01-03 21:00:00 -0.001577
2014-01-06 21:00:00 -0.000531
2014-01-07 21:00:00 0.011310
Name: 0, dtype: float64
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In [14]: import pymc as pm
with pm.Model() as model:
# Priors on Random Variables
mean_return = pm.Normal('mean return', mu=0, sd=.1)
volatility = pm.HalfNormal('volatility', sd=.1)
# Model returns as Normal
obs = pm.Normal('returns',
mu=mean_return,
sd=volatility,
observed=data_0)
sharpe = pm.Deterministic('sharpe ratio',
mean_return / volatility * np.sqrt(252))
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In [15]: with model:
# Instantiate MCMC sampler
step = pm.NUTS()
# Draw 500 samples from the posterior
trace = pm.sample(500, step)
[-----------------100%-----------------] 500 of 500 complete in 0.4 sec
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Analyzing the posteriorAnalyzing the posterior
In [84]: sns.distplot(results_normal[0][0]['mean returns'], hist=False, label='etrade')
sns.distplot(results_normal[1][0]['mean returns'], hist=False, label='IB')
plt.title('Posterior of the mean'); plt.xlabel('mean returns')
Out[84]: <matplotlib.text.Text at 0x7fde80cb5850>
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In [85]: sns.distplot(results_normal[0][0]['volatility'], hist=False, label='etrade')
sns.distplot(results_normal[1][0]['volatility'], hist=False, label='IB')
plt.title('Posterior of the volatility')
plt.xlabel('volatility')
Out[85]: <matplotlib.text.Text at 0x7fde80e58310>
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In [86]: sns.distplot(results_normal[0][0]['sharpe'], hist=False, label='etrade')
sns.distplot(results_normal[1][0]['sharpe'], hist=False, label='IB')
plt.title('Bayesian Sharpe ratio'); plt.xlabel('Sharpe ratio');
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In [28]: print 'P(Sharpe ratio IB > 0) = %.2f%%' %
(np.mean(results_normal[1][0]['sharpe'] > 0) * 100)
P(Sharpe ratio IB > 0) = 96.48%
In [29]: print 'P(Sharpe ratio IB > Sharpe ratio etrade) = %.2f%%' %
(np.mean(results_normal[1][0]['sharpe'] > results_normal[0][0]['sharpe']) * 100)
P(Sharpe ratio IB > Sharpe ratio etrade) = 80.06%
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Value at Risk with uncertaintyValue at Risk with uncertainty
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In [88]: ppc_etrade = post_pred(var_cov_var_normal, results_normal[0][0], 1e6, .05, samples=8
00)
ppc_ib = post_pred(var_cov_var_normal, results_normal[1][0], 1e6, .05, samples=800)
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Interim summaryInterim summary
Bayesian stats allows us to reformulate common risk metrics, use priors and
quantify uncertainty.
IB strategy seems better in almost every regard. Is it though?
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So far, only added confidenceSo far, only added confidence
In [89]: sns.distplot(results_normal[0][0]['sharpe'], hist=False, label='etrade')
sns.distplot(results_normal[1][0]['sharpe'], hist=False, label='IB')
plt.title('Bayesian Sharpe ratio'); plt.xlabel('Sharpe ratio');
plt.axvline(data_0.mean() / data_0.std() * np.sqrt(252), color='b');
plt.axvline(data_1.mean() / data_1.std() * np.sqrt(252), color='g');
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Is this a good model?Is this a good model?
In [93]: sns.distplot(data_1, label='data IB', kde=False, norm_hist=True, color='.5')
for p in ppc_dist_normal:
plt.plot(x, p, c='r', alpha=.1)
plt.plot(x, p, c='r', alpha=.5, label='Normal model')
plt.xlabel('Daily returns')
plt.legend();
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Can it be improved? Yes!Can it be improved? Yes!
Identical model as before, but instead, use a heavy-tailed T distribution:
returns ∼ T(ν, μ, )σ2
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In [94]: sns.distplot(data_1, label='data IB', kde=False, norm_hist=True, color='.5')
for p in ppc_dist_t:
plt.plot(x, p, c='y', alpha=.1)
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Lets compare posteriors of the normal and TLets compare posteriors of the normal and T
modelmodel
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Mean returnsMean returns
In [96]: sns.distplot(results_normal[1][0]['mean returns'], hist=False, color='r', label='nor
mal model')
sns.distplot(results_t[1][0]['mean returns'], hist=False, color='y', label='T model'
)
plt.xlabel('Posterior of the mean returns'); plt.ylabel('Probability Density');
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Bayesian T-Sharpe ratioBayesian T-Sharpe ratio
In [97]: sns.distplot(results_normal[1][0]['sharpe'], hist=False, color='r', label='normal mo
del')
sns.distplot(results_t[1][0]['sharpe'], hist=False, color='y', label='T model')
plt.xlabel('Bayesian Sharpe ratio'); plt.ylabel('Probability Density');
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But why? T distribution is more robust!But why? T distribution is more robust!
In [98]: sim_data = list(np.random.randn(75)*.01)
sim_data.append(-.2)
sns.distplot(sim_data, label='data', kde=False, norm_hist=True, color='.5'); sns.dis
tplot(sim_data, label='Normal', fit=stats.norm, kde=False, hist=False, fit_kws={'col
or': 'r', 'label': 'Normal'}); sns.distplot(sim_data, fit=stats.t, kde=False, hist=F
alse, fit_kws={'color': 'y', 'label': 'T'})
plt.xlabel('Daily returns'); plt.legend();
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Estimating tail risk using VaREstimating tail risk using VaR
In [99]: ppc_normal = post_pred(var_cov_var_normal, trace_normal, 1e6, .05, samples=800)
ppc_t = post_pred(var_cov_var_t, trace_t, 1e6, .05, samples=800)
sns.distplot(ppc_normal, label='Normal', norm_hist=True, hist=False, color='r')
sns.distplot(ppc_t, label='T', norm_hist=True, hist=False, color='y')
plt.legend(loc=0); plt.xlabel('5% daily Value at Risk (VaR) with $1MM capital (in
$)'); plt.ylabel('Probability density'); plt.xticks(rotation=15);
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Comparing the Bayesian T-Sharpe ratiosComparing the Bayesian T-Sharpe ratios
In [101]: sns.distplot(results_t[0][0]['sharpe'], hist=False, label='etrade')
sns.distplot(results_t[1][0]['sharpe'], hist=False, label='IB')
plt.xlabel('Bayesian Sharpe ratio'); plt.ylabel('Probability Density');
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In [42]: print 'P(Sharpe ratio IB > Sharpe ratio etrade) = %.2f%%' %
(np.mean(results_t[1][0]['sharpe'] > results_t[0][0]['sharpe']) * 100)
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ConclusionsConclusions
Bayesian statistics allows us to quantify uncertainty -- measure orthogonal sources
of risk.
Rich statistical framework to compare different models against each other.
Blackbox inference algorithms allow estimation of complex models.
PyMC3 puts advanced samplers at your fingertips.
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Further readingFurther reading
-- Develop trading algorithms like this in
your browser.
-- IPython
Notebook book on Bayesian stats using PyMC2.
-- Great book by Kruschke.
Twitter:
Quantopian (https://www.quantopian.com)
My blog for Bayesian linear regression (financial alpha and beta)
(https://twiecki.github.io)
Probilistic Programming for Hackers (http://camdavidsonpilon.github.io
/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/)
Doing Bayesian Data Analysis (http://www.indiana.edu/~kruschke
/DoingBayesianDataAnalysis/)
PyMC3 repository (https://github.com/pymc-devs/pymc3)
@twiecki (https://twitter.com/twiecki)
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Probabilistic Programming in Quantitative Finance by Thomas Wiecki, Lead Data Scientist at Quantopian

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Probabilistic Programming in Quantitative Finance by Thomas Wiecki, Lead Data Scientist at Quantopian