Discussion of Ruhm Arbitrage
Paper
Guy Carpenter Instrat®
One Madison Avenue
New York New York 10010
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 –
Ruhm’s example shows this for stock options
Transforming probabilities of stock prices does work
Guy Carpenter
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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 Wang’s 1998 transform paper
Guy Carpenter
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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
Guy Carpenter
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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
Guy Carpenter
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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 > –1
Frequency parameter l is transformed to l[1+Ef(Y)].
Severity g(y) transformed to g(y)[1+f(y)]/[1+ Ef(Y)].
Scaled so that transformed mean total loss is price
of ground-up coverage
Guy Carpenter
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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
Guy Carpenter
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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 0<s<1
l* = l/(1 – s)
g*(y) = [1 – s + sy/EY]g(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
Guy Carpenter
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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 > ln[(EecY)1/c]
For small claims g(y) > g*(y) > 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
Guy Carpenter
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Hypothetical Example
l=2500, g(y) = 0.00012/(1+y/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) = [1–s+sy/EY]g(y) = (.9955+y/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
Guy Carpenter
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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
Guy Carpenter
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Fits
H
MMM s
.017
Error
.381
MEM ln c: -28.2 .308
Mixed
…
.011 .298
-27.0
E
.021
Error
FE Error
.470
.036 .160
-26.3 .311
-26.6 .082
0 .220
-25.5
.116 .064
-26.7
Quadratic effects not enough to predict prices
Especially problematic in high layers
Guy Carpenter
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Graphs
2.3
Loading Factors for Martingale Pricing of Hurricane
2.1
Loading
1.9
MMMM Loading
1.7
MEMM Loading
Mixed Loading
Premium Loading
1.5
1.3
1.1
0
Guy Carpenter
0.07
0.14
Expected Loss on Line
0.21
0.28
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5
Loading Factors for Martingale Pricing of Earthquake
4.25
Loading
3.5
MMM Loading
MEM Loading
2.75
Mixed Loading
Premium Loading
2
1.25
0
Guy Carpenter
0.01
0.03
0.02
Expected Loss on Line
0.04
0.05
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7.4
Loading Factors for Martingale Pricing of FE
6.4
Loading
5.4
MMM Loading
MEM Loading
Mixed Loading
4.4
Premium Loading
3.4
2.4
0
Guy Carpenter
0.005
0.01
Expected Loss on Line
0.015
0.02
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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
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