Zvi Wiener

1
Financial Risk Management

• Zvi Wiener
• Following
• P. Jorion, Financial Risk Manager Handbook

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Chapter 4Quantitative AnalysisMonte Carlo
Methods

• Following P. Jorion 2001
• Financial Risk Manager Handbook

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Monte Carlo
4
Monte Carlo Simulation
5
Simulating Markov Process

• The Wiener process

The Generalized Wiener process
The Ito process
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The Geometric Brownian Motion
Used for stock prices, exchange rates. ? is the
expected price appreciation ? ?total - q. S
follows a lognormal distribution.
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The Geometric Brownian Motion
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value
time
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Simulating Yields

• GBM processes are widely used for stock prices
  and currencies (not interest rates). A typical
  model of interest rates dynamics

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Simulating Yields

• ? 0 - Vasicek model, changes are normally
  distr.
• ? 1 - lognormal model, RiskMetrics.
• ? 0.5 - Cox, Ingersoll, Ross model (CIR).

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Other models

• Ho-Lee term-structure model
• HJM (Heath, Jarrow, Morton) is based on forward
  rates - no-arbitrage type.
• Hull-White model

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FRM-99, Question 18

• If S and Q follow a geometric Brownian Motion
  which of the following is true?
• A. Log(SQ) is normally distributed
• B. SQ is lognormally distributed
• C. SQ is normally distributed
• D. S Q is normally distributed

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FRM-99, Question 19

• Considering a one-factor CIR term structure model
  and the Vasicek model
• I. Drift coefficients are different
• II. Both include mean reversion
• III. Coefficients of the stochastic term, dz, are
  different.
• IV. CIR is a jump-diffusion model.
• A. All of the above is true
• B. I and III are true
• C. II, III, and IV are true
• D. II and III are true

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FRM-99, Question 19

• Considering a one-factor CIR term structure model
  and the Vasicek model
• I. Drift coefficients are different
• II. Both include mean reversion
• III. Coefficients of the stochastic term, dz, are
  different.
• IV. CIR is a jump-diffusion model.
• A. All of the above is true
• B. I and III are true
• C. II, III, and IV are true
• D. II and III are true

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FRM-99, Question 25

• The Vasicek modle defines a risk-neutral process
  for r which is dra(b-r)dt ?dz, where a, b, and
  ? are constants, and r represents the rate of
  interest. From the equation we conclude that the
  model is a
• A. Monte Carlo type model
• B. Single factor term structure model
• C. Two-factor term structure model
• D. Decision tree model

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FRM-99, Question 26

• The term a(b-r) in the equation
• dra(b-r)dt ?dz, represents which term?
• A. Gamma
• B. Stochastic
• C. Mean reversion
• D. Vega

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FRM-99, Question 30

• For which of the following currencies would it be
  most appropriate to choose a lognormal interest
  rate model over a normal model?
• A. USD
• B. JPY
• C. DEM
• D. GBP

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FRM-99, Question 30

• For which of the following currencies would it be
  most appropriate to choose a lognormal interest
  rate model over a normal model?
• A. USD
• B. JPY
• C. DEM
• D. GBP

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FRM-98, Question 23

• Which of the following interest rate term
  structure models tends to capture the mean
  reversion of interest rates?
• A. dra(b-r)dt ?dz
• B. dradt ?dz
• C. drardt ?dz
• D. dra(r-b)dt ?dz

Bad question
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FRM-98, Question 24

• Which of the following is a shortcoming of
  modeling a bond option by applying Black-Scholes
  formula to bond prices?
• A. It fails to capture convexity in a bond.
• B. It fails to capture the pull-to-par effect.
• C. It fails to maintain the put-call parity.
• D. It works for zero-coupon bond options only.

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FRM-00, Question 118

• Which group of term structure models do the
  Ho-Lee, Hull-White and Heath, Jarrow, Morton
  models belong to?
• A. No-arbitrage models.
• B. Two-factor models.
• C. Log normal models.
• D. Deterministic models.

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FRM-00, Question 118

• Which group of term structure models do the
  Ho-Lee, Hull-White and Heath, Jarrow, Morton
  models belong to?
• A. No-arbitrage models.
• B. Two-factor models.
• C. Log normal models.
• D. Deterministic models.

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FRM-00, Question 119

• A plausible stochastic process for the short-term
  rate is often considered to be one where the rate
  is pulled back to some long-run average level.
  Which one of the following term structure models
  does NOT include this?
• A. The Vasicek model.
• B. The Ho-Lee model.
• C. The Hull-White model.
• D. The Cox-Ingersoll-Ross model.

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FRM-00, Question 119

• A plausible stochastic process for the short-term
  rate is often considered to be one where the rate
  is pulled back to some long-run average level.
  Which one of the following term structure models
  does NOT include this?
• A. The Vasicek model.
• B. The Ho-Lee model.
• C. The Hull-White model.
• D. The Cox-Ingersoll-Ross model.

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Simulations for VaR

• Choose a stochastic process
• Generate a pseudo-sequence of variables
• Generate prices from these variables
• Calculate the value of the portfolio
• Repeat steps above many times
• Calculate VaR from the resulting distribution of
  values.

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Risk-neutral approach

• Standard approach assumes some risk aversion and
  utility function.
• Risk neutral approach - change probabilities in
  order to get

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Accuracy

• Sampling variability

Antithetic Variable Technique Control Variable
Technique Quasi-Random Sequences
Very difficult to use for American types.
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Monte Carlo
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Monte Carlo
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Monte Carlo
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Monte Carlo
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Speed of convergence
Whole circle
Upper triangle
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Smart Sampling
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Spectral Truncation
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Regular Grid

• An alternative to MC is using a regular grid to
  approximate the integral.
• Advantages
• The speed of convergence is error1/N.
• All areas are covered more uniformly.
• There is no need to generate random numbers.
• Disadvantages
• One cant improve it a little bit.
• It is more difficult to use it with a measure.

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FRM-99, Question 8

• VaR of a portfolio was estimated with 1,000
  independent log-normally distributed runs. The
  standard deviation of the results was 100,000.
  It was then decided to re-run the VaR calculation
  with 10,000 independent samples. The standard
  deviation of the result
• A. about 10,000 USD
• B. about 30,000 USD
• C. about 100,000 USD
• D. can not be determined from this information

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FRM-98, Question 34

• The value of an Asian option on the short rate.
  The Asian option gives the holder an amount equal
  to the average value of the short rate over the
  period to expiration less the strike rate. With a
  one-factor binomial model of interest rates what
  method you will recommend using?

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FRM-98, Question 34

• A. The backward induction method, since it is the
  fastest?
• B. The simulation method, using path averages,
  since the option is path dependent.
• C. The simulation method, using path averages,
  since the option is path independent.
• D. Either the backward induction or the
  simulation method since both methods give the
  same value.

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FRM-97, Question 17

• The measurement error in VaR, due to sampling
  variation should be greater with
• A. more observations and a high confidence level
  (e.g. 99).
• B. fewer observations and a high confidence
  level.
• C. more observations and a low confidence level.
  (e.g. 95).
• D. more observations and a low confidence level.

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Multiple Sources of Risk

• GBM model with j1,,N independent risk factors

correlated risk factors
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Multiple Sources of Risk

• Correlation matrix R
• Cholesky decomposition RA AT, where A is a lower
  triangular matrix with zeros in the upper left
  corner.
• Then ? A ?
• Example

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Cholesky Decomposition
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FRM-99, Question 29

• Covariance matrix

Let ?A AT, where A is lower triangular, be a
Cholesky decomposition. Then the four elements in
the upper left hand corner of A, a11, a12, a21,
a22, are respectively A. 3, 0, 4, 2 B. 3,
4, 0, 2 C. 3, 0, 2, 1 D. 2, 0, 3, 1
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FRM-99, Question 29

• Covariance matrix

Let ?A AT, where A is lower triangular, be a
Cholesky decomposition. Then the four elements in
the upper left hand corner of A, a11, a12, a21,
a22, are respectively A. 3, 0, 4, 2 B. 3,
4, 0, 2 C. 3, 0, 2, 1 D. 2, 0, 3, 1
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FRM-99, Question 29
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FRM-99, Question 29
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FRM-99, Question 29

• Given the following covariance matrix

Given the following covariance matrix A.
Log(SQ) is normally distributed B. SQ is
lognormally distributed C. SQ is normally
distributed D. S Q is normally distributed