1. 0 1 1## Price of Risk Ton Vorst Global Head of Quantitative Risk Analytics October 7, 2005 ABN AMRO

2. 0 2 2## Quantitative Risk Analytics  Staff: 56(roughly 50% Ph-D’s)  Most in Amsterdam (roughly 25% foreign). Some in London and New York  Activities: Validation of Trading Models, Credit Portfolio and Counterparty Risk Models, Quantitative consultancy  After a while people move to other positions within ABN AMRO

3. 0 3 3## Other Positions within ABN AMRO  Development teams for trading models  Asset Management and Asset Allocation  Department of Economics  Developers of rating models

4. 0 4 4## ABN AMRO Career Career Development Programs.  Introduction Course (+/- 6 weeks)  Industry conferences / courses

5. 0 5 5## Financial instruments Product Analysis within QRA Equity (stock: KPN, Shell, IBM,…; indices: AEX, DJ, Nikkei)  Currency (Foreign Exchange, FX)  Interest rates (Bonds, LIBOR)  Commodities  Derivatives: Futures, Options

6. 0 6 6## ABN AMRO Wereldwijd Koopkracht Garantie Note 2005-2015 EUR 100,000,000 Capital Protected Securities Linked to the Performance of an Inflation Index and Basket of Indices, due 2015  125% of your investment  Equal purchasing power of your investment  75% of market rise + your investment back Best of

7. 0 7 7## ABN AMRO Wereldwijd Koopkracht Garantie Note 2005-2015  The purchasing power of your investment HICP - Harmonised Index of Consumer Prices excluding Tobacco 20%

8. 0 8 8## ABN AMRO Wereldwijd Koopkracht Garantie Note 2005-2015 Basket of IndicesWeight S&P 50055% Dow Jones EURO STOXX 5030% Nikkei 22510% Hang Seng China5%  75% of Market Rise Basket value

9. 0 9 9## Option Stock value 100 Option payoff Gain/loss -100% 100% 100 Option Stock value Three types of financial instruments:  Bank account (virtually risk-less)  Share (moderate risk)  Option (very risky)

10. 0 10 10## Risk neutral valuation One step binomial model S(0)=100 V(0)= ? S(1)=110 V(1)= 5 S(1)=90 V(1)= 0  We create a portfolio:  A number of shares, Δ  Sell one option with strike 105 and unknown value V(0)  The value of the portfolio P(t) = Δ·S(t) – V(t)  Find Δ such that value of portfolio, P(1), is independent of the stock value  Stock goes UP: P(1) = Δ·110 – 5  Stock goes DOWN: P(1) = Δ·90 – 0  UP = DOWN follows Δ·110 – 5 = Δ·90 and Δ = 0.25  P(1) = 22.5  Risk-less portfolio must earn the risk-free interest rate, say 5% per year  Portfolio value today is P(0) = 22.5/e 0.05×1 = 21.4  Option value today V(0) = 0.25·100 – 21.4 = 3.6

11. 0 11 11## Risk neutral valuation Risk neutral world  Risk neutral valuation can be generalized:  We can assume that all assets grow with the risk-free interest rate, if we can hedge all risks  Mathematically, this corresponds to using a “risk free measure”  Put it in a mathematical form where r is the risk free interest rate. The risk free interest rate is used to calculate a future value and to discount them.

12. 0 12 12## Mathematical methods  Monte Carlo  Trees (binomial, trinomial)  Partial differential equations  Analytical solutions (Black-Sholes equation, for example) S(0)=100 V(0)= 6.0 S(1/2)=110 V(1/2)= 9.8 S(1/2)=90 V(1/2)= 0 S(1)=121 V(1)=16 S(1)=99 V(1)= 0 S(1)=81 V(1)= 0

13. 0 13 13## References  Probability theory  Stochastic calculus  Measure theory  C++, MATLAB,…  John Hull, Options, Futures, and Other Derivatives  http://www.wilmott.com (Forums) http://www.wilmott.com

14. 0 14 14## ABN AMRO Quantitative Risk Analytics Group Market Risk Management Quantitative Risk Analytics Ton Vorst Market Risk Modelling & Product Analysis Credit Risk Modelling & Product Analysis Quantitative Consultancy & Operations Research