A Universal Framework For Pricing Financial and Insurance Risks Presentation at the ASTIN Colloquium July 2001, Washington DC Shaun Wang, FCAS, Ph.D. SCOR Reinsurance Co.  Shaun Wang, 2001

Outline: A Puzzle Game  Present a new formula to connect CAPM with Black-Scholes  Piece together with actuarial axioms  Empirical findings  Capital Allocations CAPM Black-Scholes Price Data ?

Market Price of Risk n Asset return R has normal distribution n r --- the risk-free rate n ={ E[R]  r }/  [R] is “the market price of risk” or excess return per unit of volatility. is “the market price of risk” or excess return per unit of volatility.

Capital Asset Pricing Model Let R i and R M be the return for asset i and market portfolio M.

The New Transform n extends the “market price of risk” in CAPM to risks with non-normal distributions is the standard normal cdf.  is the standard normal cdf.

n If F X is normal(  ), F X * is another normal (   )  E*[X] =    E*[X] =   n If F X is lognormal(   ), F X * is another lognormal (   )

Correlation Measure n Risks X and Y can be transformed to normal variables: Define New Correlation

Why New Correlation ? n Let X ~ lognormal(0,1) n Let Y=X^b (deterministic) n For the traditional correlation:  (X,Y)  0 as b  +   (X,Y)  0 as b  +  n For the new correlation:  *(X,Y)=1 for all b  *(X,Y)=1 for all b

Extending CAPM n The transform recovers CAPM for risks with normal distributions n extends the traditional meaning of { E[R]  r }/  [R] { E[R]  r }/  [R] n New transform extends CAPM to risks with non-normal distributions:

n To reproduce stock’s current value: A i (0) = E*[ A i (T)] exp(  rT) Brownian Motion n Stock price A i (T) ~ lognormal n Implies

Co-monotone Derivatives n For non-decreasing f, Y=f(X) is co- monotone derivative of X. e.g. Y=call option, X=underlying stock e.g. Y=call option, X=underlying stock n Y and X have the same correlation  * with the market portfolio n Same should be used for pricing the underlying and its derivative

Commutable Pricing n Co-monotone derivative Y=f(X) n Equivalent methods: a)Apply transform to F X to get F X *, then derive F Y * from F X * b)Derive F Y from F X, then apply transform to F Y to get F Y *

n Apply transform with same i from underlying stock to price options n Both i and the expected return  i drop out from the risk-adjusted stock price distribution!! n We’ve just reproduced the B-S price!! Recover Black-Scholes

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

Computation n Sample data:  =4.08%,  =8.07% n Use =(  r)/  =0.320 as “starter” n The transform yields a price = , differing from current price= n Solve to match current price. We get =0.342 n Use the true to price options

Using New Transform ( =0.342)

n Loss is negative asset: X= – A n New transform applicable to both assets and losses, with opposite signs in n New transform applicable to both assets and losses, with opposite signs in n Alternatively, … Loss vs Asset

n Use the same without changing sign: a)apply transform to F A for assets, but b)apply transform to S X =1– F X for losses.

n Loss X with tail prob: S X (t) = Pr{ X>t }. n Layer X(a, a+h)=min[ max(X  a,0), h ] Actuarial World

Loss Distribution

n Insurance prices by layer imply a transformed distribution –layer (t, t+dt) loss: S X (t) dt –layer (t, t+dt) price: S X *(t) dt –implied transform: S X (t)  S X *(t) Venter 1991 ASTIN Paper

Graphic Intuition

Theoretical Choice extends classic CAPM and Black-Scholes, equilibrium price under more relaxed distributional assumptions than CAPM, and unified treatment of assets & losses

Reality Check n Evidence for 3-moment CAPM which accounts for skewness [Kozik/Larson paper] n “Volatility smile” in option prices n Empirical risk premiums for tail events (CAT insurance and bond default) are higher than implied by the transform.

2-Factor Model n 1/b is a multiple factor to the normal volatility n b<1, depends on F(x), with smaller values at tails (higher adjustment) n b adjusts for skewness & parameter uncertainty

Calibrate the b-function 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 b- functions at the tails

2-Factor Model: Normal-Lognormal Calibration

Theoretical insights of b- function n Relates closely to 3-moment CAPM. n Explains better investor behavior: distortion by greed and fear n Explains “volatility smile” in option prices n 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

Use 1999 parameters to price 2000 transactions

2-factor model for corporate bonds: same lambda but lower gamma than CAT-bond

Universal Pricing n Cross Industry Comparison n and  by industry: equity, credit, CAT- bond, weather and insurance n Cross Time- horizon comparison n Term-structure of and 

Capital Allocation n The pricing formula can serve as a bridge linking risk, capital and return. n Pricing parameters are readily comparable to other industries. n A more robust method than many current ERM practices