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A Universal Framework for Pricing Financial and Insurance Risks

This presentation showcases a new formula that connects the Capital Asset Pricing Model (CAPM) with the Black-Scholes options pricing framework, extending it to risks with non-normal distributions. It introduces a novel correlation measure and a transformation method to recover CAPM principles for assets and losses. The empirical findings highlight practical application in pricing options and insurance, accommodating skewness and tail events, and discusses adjustments for parameter uncertainty. This unified approach offers insights into investor behaviors and market dynamics.

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A Universal Framework for Pricing Financial and Insurance Risks

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

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

  3. 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.

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

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

  6. If FX is normal(), FX* is another normal( )  E*[X] =   • If FX is lognormal( ), FX* is another lognormal( )

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

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

  9. Extending CAPM • The transformrecoversCAPM for riskswith normal distributions •  extends the traditional meaning of { E[R] r }/[R] • New transformextendsCAPM to risks with non-normal distributions:

  10. Brownian Motion • To reproduce stock’s current value: • Stock price Ai(T) ~ lognormal Ai(0) = E*[ Ai(T)] exp(rT) • Implies

  11. Co-monotone Derivatives • For non-decreasing f, Y=f(X) is co-monotone derivative of X. • e.g. Y=call option, X=underlying stock • Y and X have the samecorrelation *with the market portfolio • Same should be used for pricingtheunderlyingand itsderivative

  12. Commutable Pricing • Co-monotone derivative Y=f(X) • Equivalent methods: • Applytransformto FX to get FX*, then derive FY* from FX* • Derive FY from FX, then applytransform to FY to get FY*

  13. Recover Black-Scholes • Applytransformwith 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!!

  14. 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 ?

  15. 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

  16. Using New Transform (=0.342)

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

  18. Use the same  without changing sign: apply transform to FA for assets, but apply transform to SX=1– FX for losses. Loss vs Asset

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

  20. Loss Distribution

  21. 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)

  22. Graphic Intuition

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

  24. 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.

  25. 2-Factor Model • 1/b is a multiple factor to the normal volatility • b<1, depends on F(x), with smaller values at tails (higher adjustment) • b adjusts for skewness & parameter uncertainty

  26. Calibrate the b-function • Let Q be a symmetric distribution with fatter tails than Normal(0,1): • Normal-Lognormal Mixture • Student-t • Two calibrations lead to similar b-functions at the tails

  27. 2-Factor Model: Normal-Lognormal Calibration

  28. Theoretical insights of b-function • 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

  29. Fit 2-factor model to 1999 transactionsDate Sources: Lane Financial LLC Publications

  30. Use 1999 parameters to price 2000 transactions

  31. 2-factor model for corporate bonds: same lambda but lower gamma than CAT-bond

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

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

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