Download presentation

Presentation is loading. Please wait.

Published byMegan Purcell Modified over 4 years ago

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
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

3
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

4
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.

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

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

7
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<y)

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

9
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.

10
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)

11
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<T Wang-transform should yield same price

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

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

14
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!

15
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

16
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

17
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)

Similar presentations

OK

CAS Spring Meeting, Puerto Rico, May 8th 2006 Pragmatic Insurance Option Pricing by Jon Holtan If P&C Insurance Company Norway/Sweden/Denmark/Finland.

CAS Spring Meeting, Puerto Rico, May 8th 2006 Pragmatic Insurance Option Pricing by Jon Holtan If P&C Insurance Company Norway/Sweden/Denmark/Finland.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on complex numbers class 11th sample Ppt on climate of europe Ppt on water cycle for class 4 Ppt on single phase transformer Ppt on vehicle security system Ppt on ms excel 2010 Ppt on our planet earth Ppt on the art of war by sun Download ppt on turbo generator for cars Ppt on foreign exchange market in india