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## PowerPoint Slideshow about ' Price of Risk' - dane-dodson

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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

Other Positions within ABN AMRO

- Development teams for trading models
- Asset Management and Asset Allocation
- Department of Economics
- Developers of rating models

ABN AMRO Career

Career Development Programs.

- Introduction Course (+/- 6 weeks)
- Industry conferences / courses

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

ABN AMRO Wereldwijd KoopkrachtGarantie 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

ABN AMRO Wereldwijd KoopkrachtGarantie Note 2005-2015

- The purchasing power of your investment

HICP - Harmonised Index of Consumer

Prices excluding Tobacco

20%

Option

Option

payoff

Gain/loss

Option

100%

Stock

Stock

value

Stock

value

100

100

-100%

Three types of financial instruments:

- Bank account (virtually risk-less)
- Share (moderate risk)
- Option (very risky)

Risk neutral valuationOne step binomial model

S(1)=110

V(1)= 5

S(0)=100

V(0)= ?

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/e0.05×1 = 21.4
- Option value today V(0) = 0.25·100 – 21.4 = 3.6

Risk neutral valuationRisk 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.

Mathematical methods

- Trees (binomial, trinomial)

S(1)=121

V(1)=16

- Monte Carlo

S(1/2)=110

V(1/2)= 9.8

S(1)=99

V(1)= 0

S(0)=100

V(0)= 6.0

S(1/2)=90

V(1/2)= 0

- Partial differential equations

S(1)=81

V(1)= 0

- Analytical solutions (Black-Sholes equation, for example)

References

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

ABN AMROQuantitative 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

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