Discussion of Ruhm Arbitrage Paper

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Discussion of Ruhm Arbitrage Paper
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Arbitrage-Free Pricing(No chance of profits
unless losses possible )

• General idea is that means under transformed
  probabilities give arbitrage-free prices
• But there are details
• Probabilities transformed are of events
• States of nature
• Probability zero events same before and after
• Transforming probabilities of outcomes of deals
  does not does not always lead to arbitrage-free
  pricing Ruhms example shows this for stock
  options
• Transforming probabilities of stock prices does
  work

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States of Nature

• For stock option prices these are the stock
  prices
• For insurance they are frequency and severity
  distributions
• These are probabilities that have to be
  transformed
• Transforming aggregate loss probabilities will
  not necessarily give arbitrage-free prices
• Seen already with Wangs 1998 transform paper

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Requirement on Transform

• Equivalent martingale transform
• Equivalent requires same impossible states
• Martingale requires that mean at any future point
  is the current value
• For insurance prices such a transform would
  weight adverse scenarios more heavily in order to
  compensate for profit loadings (so no transformed
  expected profit on present value basis)
• Then all layers priced as transformed mean

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Why Arbitrage-Free?

• In complete markets arbitrage opportunities are
  quickly competed away
• In incomplete markets it may be impossible to
  realize theoretical arbitrage opportunities
• But competitive pressures could still penalize
  pricing that violates arbitrage principles
• Pricing that would create arbitrage profits would
  be too good to last for long
• In complete markets principle of no-arbitrage
  actually determines unique prices

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Arbitrage-free Pricing in Incomplete Markets

• Not impossible in fact problem is a
  proliferation of choices
• Requires a probability transform which makes
  prices the expected losses under transformed
  probabilities
• In complete markets the transform can be uniquely
  determined and has a no-risk hedging strategy
• Incomplete markets have many possible transforms
  but all hedges are imperfect

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Transforms for Compound Poisson Process

• Møller (2003 ASTIN Colloquium) shows how to
  create such transforms
• Co-ordinated transforms of frequency and severity
• Starts with f(y) function that is gt 1
• Frequency parameter l is transformed to
  l1Ef(Y). Severity g(y) transformed to
  g(y)1f(y)/1 Ef(Y).
• Scaled so that transformed mean total loss is
  price of ground-up coverage

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Two Popular Transforms in Finance

• Minimum martingale transform
• Corresponds to hedge with minimum variance
• Minimum entropy martingale transform
• Minimizes a more abstract information distance
• Recognizes that markets are sensitive to risk
  beyond quadratic

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Application to Insurance Surplus Process

• Process is premium flow less loss flow
• Transformed probabilities make this a martingale
• Makes expected transformed losses premium
• Møller paper 2003 ASTIN demonstrated what the
  minimum martingale and minimum entropy martingale
  transforms would be for this process
• Frequency and severity get linked transforms

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MMM and MEM for Surplus Process

• Start with actual expected claim count l and size
  g(y)
• Minimum martingale measure with 0ltslt1
• l l/(1 s)
• g(y) 1 s sy/EYg(y)
• Claim sizes above the mean get increased
  probability and below the mean get decreased
• No claim size probability decreases more than the
  frequency increases
• Thus no layers have prices below expected losses
• s selected to give desired ground-up profit load

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Minimum Entropy Measure

• Has parameter c
• l lEecY only works if moment exists
• g(y) g(y)ecy/EecY
• Severity probability increases iff y gt
  ln(EecY)1/c
• For small claims g(y) gt g(y) gt g(y)/EecY so
  probability never decreases more than frequency
  probability increases
• Avoids potential problem Mack noted for many
  transforms of negative loading of lower layers

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Hypothetical Example

• l2500, g(y) 0.00012/(1y/10,000)2.2, policy
  limit 10M
• To get a load of 20, take the MMM s 0.45
• l l/(1 s) 2511
• g(y) 1ssy/EYg(y) (.9955y/187,215)g(y)
• Probability at 10M goes to 0.055 from 0.025
• 4M x 1M gets load of 62.3, 5M x 5M gets 112.8
• For MEM these are 50.8 and 209.1, as more
  weight is in the far tail
• 89 of the risk load is above 1M for MEM 73
  for MMM

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Testing with Pricing Data

• Had prices and cat model losses for a group of
  reinsurance treaties
• Fit MMM, MEM and a mixture of them to this data
  with transforms based on industry loss
  distribution distribution of sum across
  companies
• Had separate treaties and modeled losses for
  three perils H, E, and FE
• Mixture always fit best, but not usually much
  better than MEM alone, which was better than MMM
• Fit by minimizing expected squared relative errors

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Fits

• H Error E Error FE
  Error
• MMM s .017 .381 .021 .470 .036 .160
• MEM ln c -28.2 .308 -26.3 .311 -26.6
  .082
• Mixed .011 .298 0 .220 .116 .064
  -27.0
  -25.5 -26.7
• Quadratic effects not enough to predict prices
• Especially problematic in high layers

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Graphs
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Beyond No Arbitrage

• Principle of no good deals
• Good deal defined as risk everyone would want to
  buy, no one would want to sell
• Involves a cutoff point
• Expanding literature on how to define cutoff
• Restricts prices more than does no arbitrage
• Some similarities to risk transfer testing