Probabilisticprogramminginquantitativefinancebythomaswieckileaddatascientistatquantopian...

Probabilistic Programming in Quantitative FinanceProbabilistic Programming in Quantitative FinanceThomas 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 meLead Data Scientist at : Building acrowd sourced hedge fund.PhD from Brown University -- research on computational neuroscience and machinelearning using Bayesian modeling.

Quantopian Inc (https://www.quantopian.com)


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The problem we're gonna solveThe problem we're gonna solveTwo real-money strategies:

In [76]: plot_strats()


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Types of risk

Systematic and Unsystematic RiskVola

tilityTail

riskBet

aDra

wdown


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Sharpe RatioSharpe RatioSharpe = mean returnsvolatility

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 = 0.627893606355Sharpe ratio strategy IB = 1.43720181575


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Types of risk

Systematic and Unsystematic Risk

Model misspecication

Estimation Uncertainty

Programming errors

Data issuesMo

del R

isk

Volatility

Tail risk

Beta

Drawdo

wn


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Short primer on random variablesShort primer on random variablesRepresents 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 certainprobabilityprobabilityIn [ ]: coin = {0: 50%,

1: 50%}

Alternatively:Alternatively:coin ~ Bernoulli(p=0.5)

coin is a random variableBernoulli 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 continuouscase?case?returns ~ Normal( , ) 2


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In [77]: from scipy import statssns.distplot(data_0, kde=False, fit=stats.norm)plt.xlabel('returns')


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How to estimate How to estimate and and ??Naive: point estimateSet mu = mean(data) and sigma = std(data)Maximum Likelihood EstimateCorrect answer as

n

Bayesian analysisBayesian analysisMost of the time ...Uncertainty about and Turn and into random variablesHow to estimate?

n


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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 ProgrammingModel 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.


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Approximating the posterior with MCMC samplingApproximating the posterior with MCMC samplingIn [81]: plot_want_get()


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PyMC3PyMC3Probabilistic 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 probabilitydistributions.Authors: John Salvatier, Chris Fonnesbeck, Thomas WieckiUpcoming beta release!


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Model returns distribution: Specifying ourModel returns distribution: Specifying ourpriorspriors


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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('Probability 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('Probability Density');


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Bayesian Sharpe ratioBayesian Sharpe ratio Normal(0, ).12 Prior HalfNormal( ).12 Priorreturns Normal(, )2 Observed!Sharpe =


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Graphical model of returnsGraphical model of returns

Bayes

PosteriorsPriors

Data

~ ~


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This is what the data looks likeThis is what the data looks likeIn [9]: print data_0.head()

2013-12-31 21:00:00 0.0021432014-01-02 21:00:00 -0.0285322014-01-03 21:00:00 -0.0015772014-01-06 21:00:00 -0.0005312014-01-07 21:00:00 0.011310Name: 0, dtype: float64


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In [14]: import pymc as pmwith 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 posteriorIn [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]:


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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]:


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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=800)ppc_ib = post_pred(var_cov_var_normal, results_normal[1][0], 1e6, .05, samples=800)


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Interim summaryInterim summaryBayesian stats allows us to reformulate common risk metrics, use priors andquantify uncertainty.IB strategy seems better in almost every regard. Is it though?


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So far, only added confidenceSo far, only added confidenceIn [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(,, )2


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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 Tmodelmodel


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Mean returnsMean returnsIn [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 ratioIn [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.distplot(sim_data, label='Normal', fit=stats.norm, kde=False, hist=False, fit_kws={'color': 'r', 'label': 'Normal'}); sns.distplot(sim_data, fit=stats.t, kde=False, hist=False, fit_kws={'color': 'y', 'label': 'T'})plt.xlabel('Daily returns'); plt.legend();


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Estimating tail risk using VaREstimating tail risk using VaRIn [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 ratiosIn [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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ConclusionsConclusionsBayesian statistics allows us to quantify uncertainty -- measure orthogonal sourcesof 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.

-- IPythonNotebook 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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