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On the Applicability of the Wang Transform for Pricing Financial Risks Antoon Pelsser ING - Corp. Insurance Risk Mgt. Erasmus University Rotterdam http://www.few.eur.nl/few/people/pelsser

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2 Wang Transform General framework for pricing risks Inspired by Black-Scholes pricing for options –Adjust mean of probability distribution –Easy for (log)normal distribution –Generalisation for general distributions

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3 Wang Transform (2) Given probability distribution F(t,x;T,y) as seen from time t Adjust pricing distribution F W with distortion operator – is cumulative normal distribution function

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4 Wang Transform (3) Wang (2000) and (2001) shows that this distortion operator yields correct answer for –CAPM (normal distribution) –Black-Scholes economy (lognormal distr.) Wang then proposes this distortion operator as A Universal Framework for Pricing Financial and Insurance Risks.

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5 Pricing Financial Risk Well-established theory: arbitrage-free pricing –Harrison-Kreps (1979), Harrison-Pliska (1981) Economy is arbitrage-free martingale probability measure

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6 Pricing Financial Risk (2) Calculate price via Wang-transform Calculate price via arbitrage-free pricing Investigate conditions for both approaches to be equivalent

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7 Stochastic Calculus Stochastic process Kolmogorovs Backward Equation (KBE) Distribution function F(t,x;T,y) solves KBE –with bound.condition F(T,x;T,y) = 1 (x

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8 Stochastic Calculus (2) Change in probability measure –Girsanovs Theorem –Process K t is Girsanov kernel Change in probability measure only affects dt-coefficient

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9 Arbitrage-free pricing Choose a traded asset with strictly positive price as numeraire N t. Express prices of all other traded assets in units of N t. Stochastic process X t in units of numeraire –Euro-value of process: X t N t.

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10 Economy is arbitrage free & complete unique (equivalent) martingale measure Application: use Girsanovs Theorem to make X t a martingale process: Unique choice: –Market-price of risk –Martingale measure Q * Arbitrage-free pricing (2)

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11 Arbitrage-free pricing (3) All traded assets divided by numeraire are martingales under Q* In particular: –Derivative with payoff f(X T ) at time T Price f t / N t must be martingale t

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12 Wang Transform Probability distribution F W : Solve (t,T) from Adjust mean to equal forward price at time t Weaker condition than martingale!

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13 Wang Transform (2) Find Girsanov kernel K W implied by Wang Transform from KBE: Solving for K W gives:

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14 Wang Transform (3) Wang-Tr is consistent with arb-free pricing iff K W = - (t,X t )/ (t,X t ) Substitute (t,x) K W = - (t,x) and simplify ODE in (t,T) –Only valid solution if coefficients are functions of time only!

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15 Wang Transform (4) Wang-Tr is consistent with arb-free pricing iff Very restrictive conditions –E.g.: (t,X t )/ (t,X t ) function of time only

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16 Counter-example Ornstein-Uhlenbeck process Expectation of process seen from t=0 If x 0 =0 then E[x(t)]=0=x 0 for all t>0 –Not a martingale –But, no Wang-adjustment needed

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17 Conclusion Wang-Transform cannot be a universal pricing framework for financial and insurance risks More promising approach: incomplete markets –Distinguish hedgeable & unhedgeable risks –Musiela & Zariphopoulou (Fin&Stoch, 2004ab)

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