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Bayesian Change Point Detection

Introduction

Two Presentations for the Price of One

A Framework for Exotic Financial Derivatives

• At the end of 2006 Barclays managed more than 35 templates, representing a total population of several thousand (exotic) trades.
• The various downstream systems need to be able to process the new trade type.
• Templates took months to release.

Change Point Detection

• [2016-12-16 Fri], UK Chancellor tweets

Arrived in South Korea, a growing trade opportunity for the UK where exports have doubled over the last year. Now worth nearly £11bn.

• But these days, UK Office of National Statistics makes this easy to verify.

Change Point Detection

Office for National Statistics (ONS)

Quarterly Trade Data

Annual Trade Data

Monthly Trade Data

Naive Model

Naive Stan I

data {
  int<lower=1> N;
  vector[N] x;
  vector[N] y;
}

parameters {
  real tau;
  real mu1;
  real mu2;
  real gamma1;
  real gamma2;

  real<lower=0> sigma1;
  real<lower=0> sigma2;
}

Naive Stan II

model {
  real mu;
  real gamma;
  real sigma;

  mu1 ~ normal(0, 10);
  mu2 ~ normal(0, 10);
  gamma1 ~ normal(0, 10);
  gamma2 ~ normal(0, 10);
  sigma1 ~ normal(0, 10);
  sigma2 ~ normal(0, 10);
  tau ~ uniform(0,N+1);

  for (i in 1:N) {
    mu = i < tau ? mu1 : mu2;
    gamma = i < tau ? gamma1 : gamma2;
    sigma = i < tau ? sigma1 : sigma2;
    y[i] ~ normal(mu * x[i] + gamma, sigma);
  }
}

Some Test Data

Inferred Change Point

PyMC3

The Same in Python

chg_model = Model()

with chg_model:
  
  alpha1 = Normal('alpha1', mu=0, sd=10)
  alpha2 = Normal('alpha2', mu=0, sd=10)
  beta1  = Normal( 'beta1', mu=0, sd=10)
  beta2  = Normal( 'beta2', mu=0, sd=10)
  sigma1 = HalfNormal('sigma1', sd=10)
  sigma2 = HalfNormal('sigma2', sd=10)
  tau = Uniform('tau', lower=0, upper=len(w) + 1)
  
  alpha = switch(tau >= v, alpha1, alpha2)
  beta  = switch(tau >= v,  beta1,  beta2)
  sigma = switch(tau >= v, sigma1, sigma2)
  
  mu = alpha + beta * v
  
  Y_obs = Normal('Y_obs', mu=mu, sd=sigma, observed=w)

The Results

What Went Wrong?

• Stan uses “burn-in” to initialise.
• PyMC3 uses ADVI.

Variational inference (VI) is a scalable technique for approximate Bayesian inference. Automatic differentiation variational inference (ADVI) is a way of automating VI so that all that is needed is the model and the data.

Diagnostics for Stan

Python Diagnostics ADVI

pd.DataFrame(list(start_advi.items()))

        0          1
0   beta1   1.307549
1  alpha2   2.251540
2  alpha1   3.327910
3  sigma2   8.354255
4   beta2   1.512960
5     tau  73.073964
6  sigma1   6.992683

Python Diagnostics MAP

pd.DataFrame(list(start_map.items()))

                0                   1
0           beta1  0.9680002398415648
1          alpha2  -37.86057522849939
2          alpha1  10.729059293077137
3  tau_interval__                 0.0
4           beta2  1.9750114311467755
5    sigma2_log__  0.7086888732834176
6    sigma1_log__  0.8616867832117518

Not Continuous

• ADVI probably breaks because the posterior is not continuous
• Idea: use sigmoid (soft step) rather than (hard) step function.

PyMC3 Second Attempt

with chg_cont_model:
    
    alpha1 = Normal('alpha1', mu=0, sd=10)
    alpha2 = Normal('alpha2', mu=0, sd=10)
    beta1  = Normal( 'beta1', mu=0, sd=10)
    beta2  = Normal( 'beta2', mu=0, sd=10)
    sigma1 = HalfNormal('sigma1', sd=10)
    sigma2 = HalfNormal('sigma2', sd=10)
    
    tau = Uniform('tau', lower=0, upper=len(w) + 1)
    
    weight = tt.nnet.sigmoid(2 * (v - tau))
    
    alpha = weight * alpha2 + (1 - weight) * alpha1
    beta = weight * beta2 + (1 - weight) * beta1
    sigma = weight * sigma2 + (1 - weight) * sigma1
    
    mu = alpha + beta * v
    
    Y_obs = Normal('Y_obs', mu=mu, sd=sigma, observed=w)

The Results

Stan Again

Stan Second Attempt I

functions {
  vector interp(vector w, vector wc, vector a) {
    return w * a[1] + wc * a[2];
  }
}

data {
  int<lower=1> N;
  vector[N] x;
  vector[N] y;
}

transformed data {
  vector[N] steps;
  for (n in 1:N) steps[n] = 2 * n;
}

parameters {
  real<lower=0, upper=N+1> tau;
  vector[2] mu;
  vector[2] gamma;
  vector<lower=0>[2] sigma;
}

Stan Second Attempt II

model {
  vector[N] w = inv_logit(steps - 2 * tau);
  vector[N] wc = 1 - w;

  y ~ normal(x .* interp(w, wc, mu) + interp(w, wc, gamma),
             interp(w, wc, sigma));

  mu ~ normal(0, 10);
  gamma ~ normal(0, 10);
  sigma ~ normal(0, 10);
}

Inferred Change Point

The Promise

Four Variants

• I promised four variants
• But here are two more for free
• Stan manual implies making time discrete and the marginalising it out
• PyMC3 documentation also implies making time discrete but using Metropolis-Hastings for this variable rather than NUTS

Back to the Real World

Finally, Did UK / South Korea Change?

Where Precisely?

125	447
167	1051
226	490
230	88
233	190

Fitting the Model

Framework for Exotic Derivatives

Background

What is an Option

An option or derivative is a contract giving the owner the right, but not the obligation, to buy (call) or sell (put) an underlying asset at a specified price (aka the strike), on or before a specified date.

Mathematically

$$ c = (x - k)^+ $$

$$ p = (k - x)^+ $$

In Haskell

call k x = max (x - k) 0
put k x = max (k - x) 0

Call Chart

Put Chart

Exotic

• Baskets
  • An option on a portfolio of underlyings
• Compound options
  • Options on other options, e.g. a call on a call
• Path dependent options
  • Barrier options — payout locked-in when underlying hits trigger
  • Lookback options — payout based on highest or lowest price during the lookback period
  • Asian options–payout derived from average value of underlying over a specified window
  • Autocallables — will redeem early if a particular barrier condition is met
  • Knock-in put

Trade Lifecycle

• Sales interact with the customers
• Structurers create new products, often on customer request
• Quants provide mathematical models and formal description of trades (payout functions)
• Risk management validate and sign-off the payout functions
• Traders derive the final price, manage the trade over its lifetime and analyse upcoming events
• Payments systems handle payment events throughout the lifetime of the trade

The Framework

Functional Payout Framework

• \cite{Jones_2000} \citeauthor{Jones_2000} \citetitle{Jones_2000}
• Barclays 2006
• A standardized representation for describing payoffs
• A common suite of tools for trades which use this representation
  • Pricing via C / Monte Carlo
  • Mathematical / \LaTeX representation / Mathematica for risk management
  • Barrier analysis
  • Payments and other lifecycle events
  • Pricing via C / PDE

Functional Payout Framework

Specifying a Trade

• Trade type is Haskell script
• Trade parameters e.g. start date, strike, expiration date, barrier levels, etc
• Fixings e.g. prices on Asian in

Backends

• Pricing via MC or PDE
• \LaTeX
• Payments
• Barriers
• Mathematica

Some Examples

perf :: Date -> Date -> Asset -> Double
perf t1 t2 asset =
  observe asset t2 / observe asset t1 - 1

bestOf :: (List Asset, Date, Date) -> Double
bestOf (assets', startDate', endDate') =
  foldl1 max perfs where
    assets = name "Assets" assets'
    startDate = name "Starting date" startDate'
    endDate = name "End date" endDate'
    perfs = map (perf startDate endDate) assets

Some Examples

cliquet
  ( name "Asset" -> asset
  , name "Global floor" -> gf
  , name "Global cap" -> gc
  , name "Local floor" -> lf
  , name "Local cap" -> lc
  , name "Initial date" -> inDate
  , name "Dates" -> dates
  , name "Payment date" -> payDate
  )
  = max gf $ min gc $ sum perfs
  where
    cliquet d d' = (d', max lf $ min lc $ perf d d' asset)
    (_, perfs) = mapAccumL cliquet inDate dates

The \LaTeX

\begin{center} \small $$ \mathrm{pay}\Bigg(t^{PD},min\Bigg({GC},max\Bigg({GF}, ∑_i=1^\mathrm{len(t^D)}min\Bigg({LC},\frac{S^TOP(t_i^D)}{S^TOP(t_i-1^D)} - 1\Bigg) \Bigg) \Bigg)\Bigg) $$ \end{center}

\begin{center} \small \begin{tabular}{ l l l } \bf{Variable} & \bf{Description} & \bf{Type}
$TOP$ & Top-level input & Tuple of $(S^TOP, GF, GC, LF, LC, t^ID, t^D, t^PD)$ \ \quad $S^TOP$ & Asset & Asset \ \quad $GC$ & Global floor & Double \ \quad $GF$ & Global cap & Double \ \quad $LC$ & Local floor & Double \ \quad $LF$ & Local cap & Double \ \quad $t^ID$ & Initial date & Date \ \quad $t^D$ & Dates & List of Date \ \quad $t^PD$ & Payment date & Date \end{tabular} \end{center}

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