On the Applicability of the Wang Transform for Pricing Financial Risks

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##### On the Applicability of the Wang Transform for Pricing Financial Risks

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1. 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

2. 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 Wang Transform

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

4. 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”. Wang Transform (3)

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

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

7. Stochastic process Kolmogorov’s Backward Equation (KBE) Distribution function F(t,x;T,y) solves KBE with bound.condition F(T,x;T,y) = 1(x<y) Stochastic Calculus

8. Change in probability measure Girsanov’s Theorem Process Kt is Girsanov kernel Change in probability measure only affects dt-coefficient Stochastic Calculus (2)

9. Choose a traded asset with strictly positive price as numeraireNt. Express prices of all other traded assets in units of Nt. Stochastic process Xt in units of numeraire Euro-value of process: XtNt. Arbitrage-free pricing

10. Economy is arbitrage free & complete   unique (equivalent) martingale measure Application: use Girsanov’s Theorem to make Xt a martingale process: Unique choice: Market-price of risk Martingale measure Q* Arbitrage-free pricing (2)

11. All traded assets divided by numeraire are martingales under Q* In particular: Derivative with payoff f(XT) at time T Price ft / Nt must be martingale  t<T Wang-transform should yield same price Arbitrage-free pricing (3)

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

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

14. Wang-Tr is consistent with arb-free pricing iff KW = -(t,Xt)/(t,Xt) Substitute (t,x)KW = -(t,x) and simplify ODE in (t,T) Only valid solution if coefficients are functions of time only! Wang Transform (3)

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

16. Ornstein-Uhlenbeck process Expectation of process “seen from t=0” If x0=0 then E[x(t)]=0=x0 for all t>0 Not a martingale But, no “Wang-adjustment” needed Counter-example

17. 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) Conclusion