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Bayesian Change Point Detection

Framework for Exotic Derivatives

Section

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

./diagrams/call.png

Put Chart

./diagrams/put.png

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 early redeem 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

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
    • pricing and risk management
    • barrier analysis
    • payments and other lifecycle events

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

cliquetDemo_v2
  ( 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