Financial Risk Management

Quantitative Analysis Monte Carlo Methods Following P. Jorion, Financial Risk Management Chapter 4

Daniel HERLEMONT

Monte Carlo

1

0.5

-1

-0.5

0.5

1

-0.5

-1

Daniel HERLEMONT

Page 1 1

Monte Carlo Simulation 15 10 5

10

20

30

40

-5 -10 -15

Daniel HERLEMONT

Simulating Markov Process The Wiener process

∆z = N (0, ∆t ) The Generalized Wiener process

∆x = a∆t + b∆z The Ito process

∆x = a ( x, t ) ∆t + b( x, t ) ∆z Daniel HERLEMONT

Page 2 2

The Geometric Brownian Motion

∆S = µS∆t + σS∆z Used for stock prices, exchange rates. µ is the expected price appreciation: µ = µtotal - q. S follows a lognormal distribution.

Daniel HERLEMONT

The Geometric Brownian Motion

 σ2  τ + σ τ ε N ( 0,1) ln (ST ) = ln (St ) +  µ − 2   ∆S ~ N µ ∆t , σ 2 ∆t S

(

(

)

S t +1 = S t + S t µ∆t + σε N ( 0,1) ∆t

)

Daniel HERLEMONT

Page 3 3

value

time

Daniel HERLEMONT

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

∆rt = k (b − rt )∆t + σrtγ ∆zt

Speed of mean reversion Long term mean Daniel HERLEMONT

Page 4 4

Simulating Yields

∆rt = k (b − rt )∆t + σrtγ ∆zt γ = 0 - Vasicek model, changes are normally distr. γ = 1 - lognormal model, RiskMetrics. γ = 0.5 - Cox, Ingersoll, Ross model (CIR).

Daniel HERLEMONT

Other models

∆rt = θ (t )∆t + σ∆zt Ho-Lee term-structure model HJM (Heath, Jarrow, Morton) is based on forward rates no-arbitrage type. Hull-White model:

∆rt = (θ (t ) − art )∆t + σ∆zt Daniel HERLEMONT

Page 5 5

FRM-99, Question 18 If S and Q follow a geometric Brownian Motion which of the following is true? A. Log(S+Q) is normally distributed B. S*Q is lognormally distributed C. S*Q is normally distributed D. S + Q is normally distributed

Daniel HERLEMONT

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

Daniel HERLEMONT

Page 6 6

FRM-99, Question 25 The Vasicek modle defines a risk-neutral process for r which is dr=a(b-r)dt +σdz, where a, b, and σ 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

Daniel HERLEMONT

FRM-99, Question 26 The term a(b-r) in the equation dr=a(b-r)dt +σdz, represents which term? A. Gamma B. Stochastic C. Mean reversion D. Vega

Daniel HERLEMONT

Page 7 7

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

Daniel HERLEMONT

FRM-98, Question 23 Which of the following interest rate term structure models tends to capture the mean reversion of interest rates? A. dr=a*(b-r)*dt +σ*dz B. dr=a*dt +σ*dz C. dr=a*r*dt +σ*dz D. dr=a*(r-b)*dt +σ*dz

Daniel HERLEMONT

Page 8 8

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.

Daniel HERLEMONT

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.

Daniel HERLEMONT

Page 9 9

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.

Daniel HERLEMONT

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.

Daniel HERLEMONT

Page 10 10

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

price = e − rT E [ payoffT ]

Daniel HERLEMONT

Accuracy Sampling variability

σ estimate =

σ var iable N

Antithetic Variable Technique Control Variable Technique Quasi-Random Sequences Very difficult to use for American types. Daniel HERLEMONT

Page 11 11

Monte Carlo

Daniel HERLEMONT

Monte Carlo 1

0.5

-1

-0.5

0.5

1

-0.5

-1 Daniel HERLEMONT

Page 12 12

Monte Carlo 1

0.5

-1

-0.5

0.5

1

-0.5

-1

Daniel HERLEMONT

Monte Carlo 1

0.5

-1

-0.5

0.5

1

-0.5

-1 Daniel HERLEMONT

Page 13 13

Speed of convergence

0.08 0.06

Whole circle

0.04 0.02 20

40

60

80

100

-0.02 -0.04

Upper triangle

Daniel HERLEMONT

Smart Sampling

Daniel HERLEMONT

Page 14 14

Spectral Truncation

Daniel HERLEMONT

Regular Grid An alternative to MC is using a regular grid to approximate the integral. Advantages: The speed of convergence is error~1/N. All areas are covered more uniformly. There is no need to generate random numbers. Disadvantages: One can’t improve it a little bit. It is more difficult to use it with a measure.

Daniel HERLEMONT

Page 15 15

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

Daniel HERLEMONT

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?

Daniel HERLEMONT

Page 16 16

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.

Daniel HERLEMONT

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.

Daniel HERLEMONT

Page 17 17

Multiple Sources of Risk GBM model with j=1,…,N independent risk factors

∆S j ,t = S j ,t −1µ j ∆t + S j ,tσ j ε j ,t ∆t correlated risk factors

ε1 = λ1 ε 2 = ρλ1 + 1 − ρ 2 λ2 Daniel HERLEMONT

Multiple Sources of Risk Correlation matrix R Cholesky decomposition R=A AT, where A is a lower triangular matrix with zeros in the upper left corner. Then ε = A λ Example:

 4 0 0   1 2 0  1 2 3  

Daniel HERLEMONT

Page 18 18

Cholesky Decomposition

a11a21   1 ρ   a11 0  a11 a21   a112    =    =  2 2   ρ 1   a21 a22  0 a22   a11a21 a21 + a22 

a112 =1 a11a21 = ρ 2 2 a21 + a22 =1

Daniel HERLEMONT

FRM-99, Question 29 Covariance matrix:

 0.09% 0.06% 0.03%    Σ =  0.06% 0.05% 0.04%   0.03% 0.04% 0.06%   

Let Σ=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% Daniel HERLEMONT

Page 19 19