2. 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

3. 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 ?

4. 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.

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

6. 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.

7. 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 (   )

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

9. 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

10. 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:

11. 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

12. 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

13. 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 *

14. 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

15. 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 ?

16. Computation n Sample data:  =4.08%,  =8.07% n Use =(  r)/  =0.320 as “starter” n The transform yields a price =1328.14, differing from current price=1326.03 n Solve to match current price. We get =0.342 n Use the true to price options

17. Using New Transform ( =0.342)

18. 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

19. 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.

20. 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

21. Loss Distribution

22. 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

23. Graphic Intuition

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

25. 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.

26. 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

27. 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

28. 2-Factor Model: Normal-Lognormal Calibration

29. 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

30. Fit 2-factor model to 1999 transactions Date Sources: Lane Financial LLC Publications

31. Use 1999 parameters to price 2000 transactions

32. 2-factor model for corporate bonds: same lambda but lower gamma than CAT-bond

33. 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 

34. 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