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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) For the traditional correlation: (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>t }. E X S X (t )dt. 0 Layer X(a, a+h)=min[ max(Xa,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 & 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<1, depends on F(x), with smaller values at tails (higher adjustment) b adjusts for skewness & 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 & 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 Yield Spread for Insurance-Linked Securities 16.00% Model-Spread Empirical-Spread 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 Yield Spread 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 Yield Spread Use 1999 parameters to price 2000 transactions Fitted versus Empirical Spread 8.00% 7.00% 6.00% 5.00% 4.00% 3.00% Model-Spread 2.00% 1.00% Empirical-Spread 0.00% Transactions 2-factor model for corporate bonds: same lambda but lower gamma than CAT-bond Bond Rating and Yield Spread 1,400 Model Fitted Spread 1,200 Spread (basis points) Actual Spread 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 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