- 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.
- [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.
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;
}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);
}
}
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)
- Stan uses “burn-in” to initialise.
- PyMC3 uses 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
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
- ADVI probably breaks because the posterior is not continuous
- Idea: use sigmoid (soft step) rather than (hard) step function.
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)
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;
}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);
}
- 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
| 125 | 447 |
| 167 | 1051 |
| 226 | 490 |
| 230 | 88 |
| 233 | 190 |
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.
call k x = max (x - k) 0
put k x = max (k - x) 0
- 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
- 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
- \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
- Trade type is Haskell script
- Trade parameters e.g. start date, strike, expiration date, barrier levels, etc
- Fixings e.g. prices on Asian in
- Pricing via MC or PDE
- \LaTeX
- Payments
- Barriers
- Mathematica
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) assetscliquet
( 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\begin{center} \small $$ \mathrm{pay}\Bigg(t{PD},min\Bigg({GC},max\Bigg({GF}, ∑i=1\mathrm{len(t^D)}min\Bigg({LC},\frac{STOP(t_i^D)}{STOP(ti-1^D)} - 1\Bigg) \Bigg) \Bigg)\Bigg) $$ \end{center}
\begin{center}
\small
\begin{tabular}{ l l l }
\bf{Variable} & \bf{Description} & \bf{Type}
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