# A Universal Framework For Pricing Financial and Insurance Risks

```A Universal Framework
For Pricing Financial
and Insurance Risks
Presentation at the ASTIN Colloquium
July 2001, Washington DC
Shaun Wang, 2001
Shaun Wang, FCAS, Ph.D.
SCOR Reinsurance Co.
Outline: A Puzzle Game




Present a new formula
to connect CAPM with
Black-Scholes
Piece together with
actuarial axioms
Empirical findings
Capital Allocations
CAPM
Price Data
?
Black-Scholes
Market Price of Risk

Asset return R has normal distribution

r --- the risk-free rate

={ E[R] r }/[R]
is “the market price of risk” or excess
return per unit of volatility.
Capital Asset Pricing Model
Let Ri and RM be the return for asset i
and market portfolio M.
i   ( Ri , RM )  M
The New Transform
F * ( x )   ( F ( x ))   
1
 is the standard normal cdf.

 extends the “market price of risk”
in CAPM to risks with non-normal
distributions
F * ( x)   ( F ( x))   
1

If FX is normal(), FX* is another
normal( )


E*[X] =  
If FX is lognormal( ), FX* is
another lognormal( )
Correlation Measure

Risks X and Y can be transformed to
normal variables:
1
X *   [ FX ( X )],
1
Y *   [ FY (Y )]
Define New Correlation
 * ( X , Y )   ( X *,Y *)
Why New Correlation ?

Let X ~ lognormal(0,1)
Let Y=X^b (deterministic)

(X,Y)  0 as b  +

For the new correlation:
*(X,Y)=1 for all b
Extending CAPM

The transform recovers CAPM for risks
with normal distributions

 extends the traditional meaning of
{ E[R] r }/[R]

New transform extends CAPM to risks with
non-normal distributions:
i   * ( Ri , RM )  M
Brownian Motion
dAi (t ) / Ai (t )  i dt   i dWi

Stock price Ai(T) ~ lognormal

To reproduce stock’s current value:
Ai(0) = E*[ Ai(T)] exp(rT)

Implies
i  T  (i  r ) /  i
Co-monotone Derivatives

For non-decreasing f, Y=f(X) is comonotone derivative of X.

e.g. Y=call option, X=underlying stock

Y and X have the same correlation *
with the market portfolio
 should be used for pricing the
underlying and its derivative
 Same
Commutable Pricing

Co-monotone derivative Y=f(X)

Equivalent methods:
a) Apply transform to FX to get FX*,
then derive FY* from FX*
b) Derive FY from FX, then apply
transform to FY to get FY*
Recover Black-Scholes



Apply transform with same i from
underlying stock to price options
Both i and the expected return i drop
out from the risk-adjusted stock price
distribution!!
We’ve just reproduced the B-S price!!
Option Pricing Example
A stock’s current price = \$1326.03.
Projection of 3-month price: 20 outcomes:
1218.71, 1309.51, 1287.08, 1352.47, 1518.84, 1239.06, 1415.00,
1387.64, 1602.70, 1189.37, 1364.62, 1505.44, 1358.41, 1419.09,
1550.21, 1355.32, 1429.04, 1359.02, 1377.62, 1363.84.
The 3-month risk-free rate = 1.5%.
How to price a 3-month European call
option with a strike price of \$1375 ?
Computation

Sample data: =4.08%, =8.07%
Use =(r)/ =0.320 as “starter”
 The transform yields a price =1328.14,
differing from current price=1326.03



Solve  to match current price. We get
=0.342
Use the true  to price options
Using New Transform (=0.342)
Sorted
Sample
x
1,189
1,219
1,239
Orig.
Prob.
f(x)
0.0500
0.0500
0.0500
1,519
1,550
1,603
0.0500
0.0500
0.0500
0.0318
0.0288
0.0235
1,380
1,360
1,346
1,326
Expected
Discounted
Trans.
Option
Prob.
Payoff
f*(x)
y(x)
0.0963
0.0774
0.0700
143.84
175.21
227.70
wtd
wtd
value
value
f(x) y(x) f*(x) y(x)
7.19
8.76
11.38
4.57
5.04
5.34
41.53
40.91
25.35
24.98
Loss vs Asset

Loss is negative asset: X= – A

New transform applicable to both assets
and losses, with opposite signs in 

Alternatively, …
Loss vs Asset
Use the same  without changing sign:
a) apply transform to FA for assets, but
b) apply transform to SX=1– FX for losses.

FA* ( x )    1 ( FA ( x ))   
S X* ( x )    1 ( S X ( x ))   
Actuarial World

Loss X with tail prob: SX(t) = Pr{ X&gt;t }.
E X  

S
X
(t )dt.
0

Layer X(a, a+h)=min[ max(Xa,0), h ]
E  X ( a , a  h ) 
a h
S
a
X
(t )dt.
Loss Distribution
Venter 1991 ASTIN Paper

Insurance prices by layer imply a
transformed distribution
– layer (t, t+dt) loss: SX(t) dt
– layer (t, t+dt) price: SX*(t) dt
– implied transform: SX(t)  SX*(t)
Graphic Intuition
Theoretical Choice
F * ( x )   ( F ( x ))   
1
• extends classic CAPM and
Black-Scholes,
• equilibrium price under more
relaxed distributional
assumptions than CAPM, and
• unified treatment of assets &amp;
losses
Reality Check
 Evidence
for 3-moment CAPM
which accounts for skewness
[Kozik/Larson paper]
 “Volatility smile” in option prices
 Empirical risk premiums for tail
events (CAT insurance and bond
default) are higher than implied by
the transform.
2-Factor Model

F * ( x)   b   ( F ( x))  
 1/b
1

is a multiple factor to the
normal volatility
 b&lt;1, depends on F(x), with smaller
 b adjusts for skewness &amp;
parameter uncertainty
Calibrate the b-function
F * ( x )  Q ( F ( x ))   
1
1)
Let Q be a symmetric distribution
with fatter tails than Normal(0,1):
 Normal-Lognormal Mixture
 Student-t
2)
Two calibrations lead to similar bfunctions at the tails
2-Factor Model: Normal-Lognormal
Calibration
gamma-parameter &amp; b-function
1.1
1.0
b-value
0.9
0.8
0.7
gamma=0.2
0.6
gamma=0.3
gamma=0.4
0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
F(x) value
0.7
0.8
0.9
Theoretical insights of bfunction




Relates closely to 3-moment CAPM.
Explains better investor behavior: distortion
by greed and fear
Explains “volatility smile” in option prices
Quantifies increased cost-of-capital for
gearing, non-liquidity markets, “stochastic
volatility”, information asymmetry, and
parameter uncertainty
Fit 2-factor model to 1999 transactions
Date Sources: Lane Financial LLC Publications
16.00%
12.00%
10.00%
8.00%
6.00%
4.00%
2.00%
Re
es
tic
C
on
R
ce
e
nt
ric
R
Ju e
no
R
es
R
id
e
en
Ke
tia
lvi
lR
n
e
1s
Ke
tE
lvi
ve
n
nt
2n
d
Ev
G
ol
en
d
t
Ea
gl
G
e
ol
A
d
Ea
gl
e
N
am
B
az
u
R
At
e
la
s
R
e
At
A
la
s
R
e
At
B
la
s
R
e
Se
ism C
ic
Lt
d
D
om
rd
2B
al
ya
ai
c
H
os
M
os
ai
c
2A
0.00%
M
14.00%
Transactions
lp
A
ha
in
d
W
in
d
20
00
20
FR
00
R
N
es
Pr
id
ef
en
Sh
tia
s
lR
e
20
00
M
ed
ite
N
rr
eH
an
M
i
ed
ea
ite
n
R
rr
e
an
A
e
an
P
ri
m
R
e
e
B
H
ur
ri
ca
P
ri
ne
m
e
W
E
Q
es
E
te
W
rn
G
C
ol
ap
d
ita
E
S
a
l
R
gl
e
W
20
in
d
01
S
C
R
la
W
ss
in
A
d
-1
C
la
ss
A
-2
W
lp
ha
A
Use 1999 parameters to price 2000 transactions
8.00%
7.00%
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
Transactions
2-factor model for corporate bonds: same
lambda but lower gamma than CAT-bond
1,400
1,200
1,000
800
600
400
200
0
AAA
AA
A
BBB
Bond Rating
BB
B
CCC
Universal Pricing
 Cross
Industry
Comparison
  and  by
industry: equity,
credit, CATbond, weather
and insurance
 Cross
Timehorizon
comparison
 Term-structure
of  and 
Capital Allocation
The pricing formula can