A Universal A Universal Framework Framework For Pricing Financial For Pricing Financial

and Insurance Risksand Insurance Risks

Presentation at the ASTIN Colloquium Presentation at the ASTIN Colloquium July 2001, Washington DCJuly 2001, Washington DC

Shaun Wang, FCAS, Ph.D.Shaun Wang, FCAS, Ph.D.

SCOR Reinsurance Co.SCOR Reinsurance Co.Shaun Wang, 2001Shaun Wang, 2001

Outline: A Puzzle Outline: A Puzzle GameGame

Present Present a new a new formulaformula to to connectconnect CAPM with Black-CAPM with Black-ScholesScholes

Piece together with Piece together with actuarial axiomsactuarial axioms

Empirical findingsEmpirical findings Capital AllocationsCapital Allocations

CAPMCAPM

Black-ScholesBlack-Scholes

Price DataPrice Data

??

Market Price of Market Price of RiskRisk

Asset return Asset return RR has normal distribution has normal distribution

rr --- the risk-free rate --- the risk-free rate

={ E[R] ={ E[R] r }/r }/[R][R]

isis “ “the market price of riskthe market price of risk” or excess ” or excess return per unit of volatility.return per unit of volatility.

Capital Asset Pricing Capital Asset Pricing ModelModel

Let Let RRii and and RRMM be the return for asset be the return for asset ii

and market portfolio and market portfolio MM. .

MMii RR ),(

The New TransformThe New Transform

extends theextends the “ “market price of riskmarket price of risk” ” in CAPM to risks with non-normal in CAPM to risks with non-normal distributionsdistributions

))(()(* 1 xFxF

is the standard normal cdf.is the standard normal cdf.

If If FFXX is normal( is normal(), ), FFXX** is another is another

normalnormal(( ))

E*[E*[XX] = ] =

If If FFXX is lognormal( is lognormal( ), ), FFXX** is is

another lognormalanother lognormal(( ) )

))(()(* 1 xFxF

Correlation MeasureCorrelation Measure

Risks X and Y can be transformed to Risks X and Y can be transformed to normal variables:normal variables:

)]([*

)],([*1

1

YFY

XFX

Y

X

Define New Correlation

*)*,(),(* YXYX

Why New Correlation ?Why New Correlation ?

Let X ~ lognormal(0,1)Let X ~ lognormal(0,1) Let Y=X^bLet Y=X^b ( (deterministicdeterministic)) For the traditional correlation:For the traditional correlation:

(X,Y) (X,Y) 0 as b 0 as b + +

For the new correlation:For the new correlation:

**((X,Y)=1 for all bX,Y)=1 for all b

Extending CAPMExtending CAPM The transformThe transform recoversrecovers CAPM for risksCAPM for risks

with normal distributionswith normal distributions extends extends the traditional meaning ofthe traditional meaning of { E[R] { E[R] r }/r }/[R][R]

New transformNew transform extendsextends CAPM to risks with CAPM to risks with non-normal distributions:non-normal distributions:

MMii RR ),(*

To reproduce stock’s current value:To reproduce stock’s current value:

iiiii dWdttAtdA )(/)(

iii rT /)(

AAii(0) = E*[ A(0) = E*[ Aii(T)] exp((T)] exp(rT)rT)

Brownian MotionBrownian Motion

Stock price AStock price Aii(T) ~ lognormal(T) ~ lognormal

ImpliesImplies

Co-monotone Co-monotone DerivativesDerivatives

For non-decreasing f, Y=f(X) is co-For non-decreasing f, Y=f(X) is co-monotone derivative of X.monotone derivative of X.

e.g. Y=call option, X=underlying stocke.g. Y=call option, X=underlying stock Y and X have the sameY and X have the same correlation correlation **

with the market portfoliowith the market portfolio Same Same should be used for pricingshould be used for pricing thethe

underlyingunderlying and itsand its derivativederivative

Commutable PricingCommutable Pricing

Co-monotone derivative Y=f(X) Co-monotone derivative Y=f(X)

Equivalent methods:Equivalent methods:

a)a) ApplyApply transformtransform to to FFXX to get to get FFXX**, ,

then derive then derive FFYY** from from FFXX**

b)b) Derive Derive FFYY from from FFXX, then apply, then apply

transform transform to to FFYY to get to get FFYY**

ApplyApply transformtransform with same with same ii from from

underlying stock to price optionsunderlying stock to price options

Both Both ii and the expected return and the expected return ii drop drop

out from the risk-adjusted stock price out from the risk-adjusted stock price

distribution!! distribution!!

We’ve just reproduced the B-S price!!We’ve just reproduced the B-S price!!

Recover Black-Recover Black-ScholesScholes

Option Pricing ExampleOption Pricing Example

A stock’s current price = $1326.03. A stock’s current price = $1326.03. Projection of 3-month price: 20 outcomes: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%.The 3-month risk-free rate = 1.5%.How to price a 3-month European call How to price a 3-month European call option with a strike price of option with a strike price of $1375$1375 ? ?

ComputationComputation

Sample data: Sample data: =4.08%, =4.08%, =8.07%=8.07% UseUse =(=(r)/r)/ =0.320 as “starter” =0.320 as “starter” The transform yields a price =1328.14, The transform yields a price =1328.14,

differing from current price=1326.03differing from current price=1326.03 SolveSolve to match current price. We getto match current price. We get

=0.342=0.342 Use the trueUse the true to price optionsto price options

Using New TransformUsing New Transform ((=0.342)=0.342)

Sorted Orig. Trans. Option wtd wtdSample Prob. Prob. Payoff value value

x f(x) f*(x) y(x) f(x) y(x) f*(x) y(x)1,189 0.0500 0.0963 - - - 1,219 0.0500 0.0774 - - - 1,239 0.0500 0.0700 - - -

1,519 0.0500 0.0318 143.84 7.19 4.57 1,550 0.0500 0.0288 175.21 8.76 5.04 1,603 0.0500 0.0235 227.70 11.38 5.34

Expected 1,380 1,346 41.53 25.35 Discounted 1,360 1,326 40.91 24.98

Loss is negative asset: X= – ALoss is negative asset: X= – A

New transform applicable to both assets New transform applicable to both assets

and losses, with opposite signs in and losses, with opposite signs in

Alternatively, …Alternatively, …

Loss vs AssetLoss vs Asset

Loss vs AssetLoss vs Asset

Use the same Use the same without changing sign without changing sign::

a)a) apply apply transformtransform to F to FAA for assets, but for assets, but

b)b) apply apply transformtransform to S to SXX=1– F=1– FXX for losses. for losses.

))(()(

))(()(1*

1*

xSxS

xFxF

XX

AA

Loss X with Loss X with tail probtail prob: S: SXX(t) = Pr{ X>t }.(t) = Pr{ X>t }.

.)(),( dttShaaXEha

a

X

0

.)( dttSXE X

LayerLayer X(a, a+h)=min[ max(X X(a, a+h)=min[ max(Xa,0), h ]a,0), h ]

Actuarial WorldActuarial World

Loss Loss DistributionDistribution

Insurance prices by layer imply a

transformed distribution– layer (t, t+dt) loss: Slayer (t, t+dt) loss: SXX(t) dt (t) dt

– layer (t, t+dt) price: Slayer (t, t+dt) price: SXX*(t) dt*(t) dt

– implied transform: Simplied transform: SXX(t) (t) S SXX*(t)*(t)

Venter 1991 ASTIN PaperVenter 1991 ASTIN Paper

Graphic IntuitionGraphic Intuition

Theoretical ChoiceTheoretical Choice

• extends classic CAPM and Black-Scholes,

• equilibrium price under more relaxed distributional assumptions than CAPM, and

• unified treatment of assets & losses

))(()(* 1 xFxF

Reality CheckReality Check

Evidence for 3-moment CAPM Evidence for 3-moment CAPM which accounts for skewness which accounts for skewness [[Kozik/Larson paperKozik/Larson paper]]

““Volatility smile” in option pricesVolatility smile” in option prices Empirical risk premiums for tail Empirical risk premiums for tail

events (events (CAT insurance and bond CAT insurance and bond defaultdefault) are higher than implied by ) are higher than implied by the transform.the transform.

2-Factor Model2-Factor Model

11/b /b is a multiple factor to the is a multiple factor to the normal volatilitynormal volatility

b<b<11,, depends on depends on FF((xx)), , with smaller with smaller values at tails (higher adjustment)values at tails (higher adjustment)

bb adjusts for skewness & adjusts for skewness & parameter uncertaintyparameter uncertainty

))(()(* 1 xFbxF

Calibrate the Calibrate the bb-function-function

1)1) Let Let QQ be a symmetric distribution be a symmetric distribution with fatter tails than Normal(0,1):with fatter tails than Normal(0,1):

Normal-Lognormal MixtureNormal-Lognormal Mixture Student-tStudent-t

2)2) Two calibrations lead to similar Two calibrations lead to similar bb--functions at the tailsfunctions at the tails

))(()(* 1 xFQxF

2-Factor Model: Normal-2-Factor Model: Normal-Lognormal CalibrationLognormal Calibration

gamma-parameter & b-function

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

F(x) value

b-v

alu

e

gamma=0.2

gamma=0.3

gamma=0.4

Theoretical insights of Theoretical insights of bb--functionfunction

Relates closely to 3-moment CAPM.Relates closely to 3-moment CAPM. Explains better investor behavior: Explains better investor behavior:

distortion by distortion by greedgreed and and fearfear Explains “volatility smile” in option Explains “volatility smile” in option

pricesprices Quantifies increased cost-of-capital for Quantifies increased cost-of-capital for

gearing, non-liquidity markets, gearing, non-liquidity markets, “stochastic volatility”, information “stochastic volatility”, information asymmetry, and parameter uncertaintyasymmetry, and parameter uncertainty

Fit 2-factor model to 1999 Fit 2-factor model to 1999 transactionstransactions

Date Sources: Lane Financial LLC PublicationsDate Sources: Lane Financial LLC Publications

Yield Spread for Insurance-Linked Securities

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

Transactions

Yie

ld S

prea

d

Model-Spread

Empirical-Spread

Use 1999 parameters to price 2000 Use 1999 parameters to price 2000 transactionstransactions

Fitted versus Empirical Spread

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

Transactions

Yie

ld S

pre

ad

Model-Spread

Empirical-Spread

2-factor model for corporate bonds: 2-factor model for corporate bonds: same same lambdalambda but lower but lower gammagamma than than

CAT-bondCAT-bond

Bond Rating and Yield Spread

0

200

400

600

800

1,000

1,200

1,400

AAA AA A BBB BB B CCC

Bond Rating

Sp

rea

d (

ba

sis

po

ints

)

Model Fitted Spread

Actual Spread

Universal PricingUniversal Pricing

Cross Industry Cross Industry ComparisonComparison

and and by by industry: equity, industry: equity, credit, CAT-credit, CAT-bond, weather bond, weather and insuranceand insurance

Cross Time-Cross Time-horizon horizon comparisoncomparison

Term-structure Term-structure of of and and

Capital AllocationCapital Allocation

The pricing formula can The pricing formula can serve as a bridge linking serve as a bridge linking riskrisk, , capitalcapital and and returnreturn..

Pricing parameters are Pricing parameters are readily comparable to readily comparable to other industries.other industries.

A more robust method A more robust method than many current ERM than many current ERM practicespractices