Transcript WangTransform

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
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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Wang Transform (2)
• 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
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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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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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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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Stochastic Calculus
• Stochastic process
• Kolmogorov’s Backward Equation (KBE)
• Distribution function F(t,x;T,y) solves KBE

t
F 

x
F  
1
2
2

2
x
2
F 0
– with bound.condition F(T,x;T,y) = 1(x<y)
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Stochastic Calculus (2)
• Change in probability measure
– Girsanov’s Theorem
– Process Kt is Girsanov kernel
• Change in probability measure only affects
dt-coefficient
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Arbitrage-free pricing
• Choose a traded asset with strictly positive
price as numeraire Nt.
• Express prices of all other traded assets in
units of Nt.
• Stochastic process Xt in units of numeraire
– Euro-value of process: XtNt.
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Arbitrage-free pricing (2)
• 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*
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Arbitrage-free pricing (3)
• 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
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Wang Transform
• Probability distribution FW:
• Solve (t,T) from
• Adjust mean to equal forward price at time t
• Weaker condition than martingale!
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Wang Transform (2)
• Find Girsanov kernel KW implied by Wang
Transform from KBE:
• Solving for KW gives:
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Wang Transform (3)
• 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!
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Wang Transform (4)
• Wang-Tr is consistent with arb-free pricing iff
• Very restrictive conditions
– E.g.: (t,Xt)/(t,Xt) function of time only
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Counter-example
• Ornstein-Uhlenbeck process
dx   axdt   dW
• Expectation of process “seen from t=0”
E [ x ( t )]  x 0 e
 at
• If x0=0 then E[x(t)]=0=x0 for all t>0
– Not a martingale
– But, no “Wang-adjustment” needed
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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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