Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.1
Value at Risk
Chapter 14
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.2
The Question Being Asked in
Value at Risk (VaR)
,What loss level is such that we are X%
confident it will not be exceeded in N
business days?”
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.3
Meaning is Probability
2( 0,)
P r ( )
( ) *
YN
Yp
pN
(1-?) %
%
Z?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.4
VaR and Regulatory Capital
Regulators require banks to keep capital for
market risk equal to the average of VaR
estimates for past 60 trading days using
X=99 and N=10,times a multiplication factor.
(Usually the multiplication factor equals 3)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.5
Advantages of VaR
It captures an important aspect of risk in a
single number
It is easy to understand
It asks the simple question:,How bad can
things get?”
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.6
Daily Volatilities
In option pricing we express volatility as
volatility per year
In VaR calculations we express volatility as
volatility per day
day
y e a r?
252
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.7
Daily Volatility (continued)
Strictly speaking we should define?day as the
standard deviation of the continuously
compounded return in one day
In practice we assume that it is the standard
deviation of the proportional change in one day
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.8
IBM Example (p,343)
We have a position worth $10 million in IBM
shares
The volatility of IBM is 2% per day (about 32%
per year)
We use N=10 and X=99
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.9
IBM Example (continued)
The standard deviation of the change in the
portfolio in 1 day is $200,000
The standard deviation of the change in 10
days is
200 000 10 456,$632,?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.10
IBM Example (continued)
We assume that the expected change in the
value of the portfolio is zero (This is OK for
short time periods)
We assume that the change in the value of the
portfolio is normally distributed
Since N(0.01)=-2.33,the VaR is
2 33 632 456 473 621.,$1,,
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.11
AT&T Example
Consider a position of $5 million in AT&T
The daily volatility of AT&T is 1% (approx 16%
per year)
The STD per 10 days is
The VaR is
50 000 10 144,$158,?
158 114 2 33 405,,$368,
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.12
Portfolio (p,344)
Now consider a portfolio consisting of both IBM
and AT&T
Suppose that the correlation between the
returns is 0.7
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.13
STD of Portfolio
A standard result in statistics states that
In this case?x = 632,456 and?Y=158,114 and
r = 0.7,The standard deviation of the change in
the portfolio value is therefore 751,665
rX Y X Y X Y2 2 2
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.14
VaR for Portfolio
The VaR for the portfolio is
The benefits of diversification are
(1,473,621+368,405)-1,751,379=$90,647
What is the incremental effect of the AT&T
holding on VaR?
751 665 2 33 751 379,,$1,,
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.15
The Linear Model
We assume
The change in the value of a portfolio is linearly
related to the change in the value of market
variables
The changes in the values of the market
variables are normally distributed
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.16
The General Linear Model
continued (Equation 14.5)
P x
i
i i
i
n
P i j i j ij
j
n
i
n
P i i
i j
i j i j ij
i
n
i
P
r
r
1
2
11
2 2 2
1
2
w h e r e is th e v o latil ity o f v a r iab le
and is th e p o r tf o li o ' s sta n d a r d d e v iatio n
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.17
Handling Interest Rates
We do not want to define every interest rate as
a different market variable
An approach is to use the duration relationship
P=-DP?y so that?P=DPy?y,where?y is the
volatility of yield changes and?P is as before
the standard deviation of the change in the
portfolio value
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.18
Alternative,Cash Flow
Mapping (p,347)
We choose as market variables zero-coupon
bond prices with standard maturities (1mm,
3mm,6mm,1yr,2yr,5yr,7yr,10yr,30yr)
Suppose that the 5yr rate is 6% and the 7yr rate
is 7% and we will receive a cash flow of
$10,000 in 6.5 years.
The volatilities per day of the 5yr and 7yr bonds
are 0.50% and 0.58% respectively
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.19
Cash Flow Mapping (continued)
We interpolate between the 5yr rate of 6% and
the 7yr rate of 7% to get a 6.5yr rate of 6.75%
The PV of the $10,000 cash flow is
5 40,6
0 67 5.1
0 00,10
5.6?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.20
Cash Flow Mapping (continued)
We interpolate between the 0.5% volatility for
the 5yr bond price and the 0.58% volatility for
the 7yr bond price to get 0.56% as the volatility
for the 6.5yr bond
We allocate? of the PV to the 5yr bond and (1-
) of the PV to the 7yr bond
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.21
Cash Flow Mapping (continued)
Suppose that the correlation between
movement in the 5yr and 7yr bond prices is 0.6
To match variances
This gives?=0.074
2 2 2 2 20,5 6 0,5 0,5 8 ( 1 ) 2 0,6 0,5 0,5 8 ( 1 )
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.22
Cash Flow Mapping (continued)
The cash flow of 10,000 in 6.5 years is replaced by
in 5 years and by
in 7 years.
This cash flow mapping preserves value and
variance
648$074.006.1540,6 5
725,9$926.007.1540,6 7
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.23
When Linear Model Can be
Used
Portfolio of stocks
Portfolio of bonds
Forward contract on foreign currency
Interest-rate swap
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.24
The Linear Model and Options
(p,350)
Consider a portfolio of options dependent on a
single stock price,S,Define
and
S
P
S
Sx
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.25
Linear Model and Options
(continued)
As an approximation
Similar when there are many underlying
market variables
where?i is the delta of the portfolio with
respect to the ith asset
xSSP
i
iii xSP?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.26
Example
Consider an investment in options on IBM
and AT&T,Suppose the stock prices are 120
and 30 respectively and the deltas of the
portfolio with respect to the two stock prices
are 1,000 and 20,000 respectively
As an approximation
where?x1 and?x2 are the proportional
changes in the two stock prices
21 000,2030000,1120 xxP
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.27
Skewness
The linear model fails to capture
skewness in the probability distribution
of the portfolio value,
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.28
Quadratic Model (p,352)
For a portfolio dependent on a single stock price
(by Taylor expansion)
this becomes
Where the proportional change
21
2P S S
22
2
1 xSxSP
2/ (0,),x S S N
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.29
Moments of?P (p,354)
E P S
E P S S
E P S S
( )
( )
( )
1
2
3
4
9
2
15
8
2 2
2 2 2 2 4 2 4
3 4 2 4 6 3 6
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.30
Quadratic Model (continued)
With many market variables and each
instrument dependent on only one
where?i and?i are the delta and gamma
of the portfolio with respect to the ith
variable
n
i
n
i
iiiiii xSxSP
1 1
22
2
1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.31
Quadratic Model (continued)
When the change in the portfolio value
has the form
we can calculate the moments of?P
analytically if the?xi are assumed to be
normal
P x xi i
i
n
i i
i
n
1
2
1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.32
Quadratic Model (continued)
Once we have done this we can use the
Cornish Fisher expansion to calculate
fractiles of the distribution of?P
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.33
Monte Carlo Simulation
(p,355)
The stages are as follows
Value portfolio today
Sample once from the multivariate distributions
of the?xi
Use the?xi to determine market variables at
end of one day
Revalue the portfolio at the end of day
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.34
Monte Carlo Simulation
Calculate?P
Repeat many times to build up a probability
distribution for?P
VaR is the appropriate fractile of the distribution
times square root of N
For example,with 1,000 trial the 1 percentile is
the 10th worst case.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.35
Speeding Up Monte Carlo
Use the quadratic approximation to calculate
P
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.36
Historical Simulation
(p,356)
Create a database of the daily movements in all
market variables.
The first simulation trial assumes that the percentage
changes in all market variables are as on the first day
The second simulation trial assumes that the
percentage changes in all market variables are as on
the second day
and so on
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.37
Stress Testing
(p,357)
This involves testing how well a portfolio
performs under some of the most
extreme market moves seen in the last
10 to 20 years
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.38
Back-Testing
(p,357)
Tests how well VaR estimates would
have performed in the past
We could ask the question,How often
was the loss greater than the 99%/10
day VaR?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.39
Principal Components Analysis
(p,357)
Suppose that a portfolio depends on a
number of related variables (eg interest rates)
We define factors as a scenarios where there
is a certain movement in each market
variable (The movements are known as factor
loadings)
The observations on the variables can often
be largely explained by two or three factors
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.40
Results for Interest Rates
(Table 14.4)
The first factor is a roughly parallel shift
(83.1% of variation explained)
The second factor is a twist (10% of
variation explained)
The third factor is a bowing (2.8% of
variation explained)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.41
Assignments
14.1 – 14.5,14.10 – 14.14
Assignment Questions
Tang Yincai,? 2003,Shanghai Normal University
14.1
Value at Risk
Chapter 14
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.2
The Question Being Asked in
Value at Risk (VaR)
,What loss level is such that we are X%
confident it will not be exceeded in N
business days?”
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.3
Meaning is Probability
2( 0,)
P r ( )
( ) *
YN
Yp
pN
(1-?) %
%
Z?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.4
VaR and Regulatory Capital
Regulators require banks to keep capital for
market risk equal to the average of VaR
estimates for past 60 trading days using
X=99 and N=10,times a multiplication factor.
(Usually the multiplication factor equals 3)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.5
Advantages of VaR
It captures an important aspect of risk in a
single number
It is easy to understand
It asks the simple question:,How bad can
things get?”
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.6
Daily Volatilities
In option pricing we express volatility as
volatility per year
In VaR calculations we express volatility as
volatility per day
day
y e a r?
252
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.7
Daily Volatility (continued)
Strictly speaking we should define?day as the
standard deviation of the continuously
compounded return in one day
In practice we assume that it is the standard
deviation of the proportional change in one day
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.8
IBM Example (p,343)
We have a position worth $10 million in IBM
shares
The volatility of IBM is 2% per day (about 32%
per year)
We use N=10 and X=99
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.9
IBM Example (continued)
The standard deviation of the change in the
portfolio in 1 day is $200,000
The standard deviation of the change in 10
days is
200 000 10 456,$632,?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.10
IBM Example (continued)
We assume that the expected change in the
value of the portfolio is zero (This is OK for
short time periods)
We assume that the change in the value of the
portfolio is normally distributed
Since N(0.01)=-2.33,the VaR is
2 33 632 456 473 621.,$1,,
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.11
AT&T Example
Consider a position of $5 million in AT&T
The daily volatility of AT&T is 1% (approx 16%
per year)
The STD per 10 days is
The VaR is
50 000 10 144,$158,?
158 114 2 33 405,,$368,
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.12
Portfolio (p,344)
Now consider a portfolio consisting of both IBM
and AT&T
Suppose that the correlation between the
returns is 0.7
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.13
STD of Portfolio
A standard result in statistics states that
In this case?x = 632,456 and?Y=158,114 and
r = 0.7,The standard deviation of the change in
the portfolio value is therefore 751,665
rX Y X Y X Y2 2 2
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.14
VaR for Portfolio
The VaR for the portfolio is
The benefits of diversification are
(1,473,621+368,405)-1,751,379=$90,647
What is the incremental effect of the AT&T
holding on VaR?
751 665 2 33 751 379,,$1,,
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.15
The Linear Model
We assume
The change in the value of a portfolio is linearly
related to the change in the value of market
variables
The changes in the values of the market
variables are normally distributed
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.16
The General Linear Model
continued (Equation 14.5)
P x
i
i i
i
n
P i j i j ij
j
n
i
n
P i i
i j
i j i j ij
i
n
i
P
r
r
1
2
11
2 2 2
1
2
w h e r e is th e v o latil ity o f v a r iab le
and is th e p o r tf o li o ' s sta n d a r d d e v iatio n
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.17
Handling Interest Rates
We do not want to define every interest rate as
a different market variable
An approach is to use the duration relationship
P=-DP?y so that?P=DPy?y,where?y is the
volatility of yield changes and?P is as before
the standard deviation of the change in the
portfolio value
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.18
Alternative,Cash Flow
Mapping (p,347)
We choose as market variables zero-coupon
bond prices with standard maturities (1mm,
3mm,6mm,1yr,2yr,5yr,7yr,10yr,30yr)
Suppose that the 5yr rate is 6% and the 7yr rate
is 7% and we will receive a cash flow of
$10,000 in 6.5 years.
The volatilities per day of the 5yr and 7yr bonds
are 0.50% and 0.58% respectively
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.19
Cash Flow Mapping (continued)
We interpolate between the 5yr rate of 6% and
the 7yr rate of 7% to get a 6.5yr rate of 6.75%
The PV of the $10,000 cash flow is
5 40,6
0 67 5.1
0 00,10
5.6?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.20
Cash Flow Mapping (continued)
We interpolate between the 0.5% volatility for
the 5yr bond price and the 0.58% volatility for
the 7yr bond price to get 0.56% as the volatility
for the 6.5yr bond
We allocate? of the PV to the 5yr bond and (1-
) of the PV to the 7yr bond
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.21
Cash Flow Mapping (continued)
Suppose that the correlation between
movement in the 5yr and 7yr bond prices is 0.6
To match variances
This gives?=0.074
2 2 2 2 20,5 6 0,5 0,5 8 ( 1 ) 2 0,6 0,5 0,5 8 ( 1 )
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.22
Cash Flow Mapping (continued)
The cash flow of 10,000 in 6.5 years is replaced by
in 5 years and by
in 7 years.
This cash flow mapping preserves value and
variance
648$074.006.1540,6 5
725,9$926.007.1540,6 7
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.23
When Linear Model Can be
Used
Portfolio of stocks
Portfolio of bonds
Forward contract on foreign currency
Interest-rate swap
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.24
The Linear Model and Options
(p,350)
Consider a portfolio of options dependent on a
single stock price,S,Define
and
S
P
S
Sx
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.25
Linear Model and Options
(continued)
As an approximation
Similar when there are many underlying
market variables
where?i is the delta of the portfolio with
respect to the ith asset
xSSP
i
iii xSP?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.26
Example
Consider an investment in options on IBM
and AT&T,Suppose the stock prices are 120
and 30 respectively and the deltas of the
portfolio with respect to the two stock prices
are 1,000 and 20,000 respectively
As an approximation
where?x1 and?x2 are the proportional
changes in the two stock prices
21 000,2030000,1120 xxP
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.27
Skewness
The linear model fails to capture
skewness in the probability distribution
of the portfolio value,
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.28
Quadratic Model (p,352)
For a portfolio dependent on a single stock price
(by Taylor expansion)
this becomes
Where the proportional change
21
2P S S
22
2
1 xSxSP
2/ (0,),x S S N
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.29
Moments of?P (p,354)
E P S
E P S S
E P S S
( )
( )
( )
1
2
3
4
9
2
15
8
2 2
2 2 2 2 4 2 4
3 4 2 4 6 3 6
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.30
Quadratic Model (continued)
With many market variables and each
instrument dependent on only one
where?i and?i are the delta and gamma
of the portfolio with respect to the ith
variable
n
i
n
i
iiiiii xSxSP
1 1
22
2
1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.31
Quadratic Model (continued)
When the change in the portfolio value
has the form
we can calculate the moments of?P
analytically if the?xi are assumed to be
normal
P x xi i
i
n
i i
i
n
1
2
1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.32
Quadratic Model (continued)
Once we have done this we can use the
Cornish Fisher expansion to calculate
fractiles of the distribution of?P
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.33
Monte Carlo Simulation
(p,355)
The stages are as follows
Value portfolio today
Sample once from the multivariate distributions
of the?xi
Use the?xi to determine market variables at
end of one day
Revalue the portfolio at the end of day
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.34
Monte Carlo Simulation
Calculate?P
Repeat many times to build up a probability
distribution for?P
VaR is the appropriate fractile of the distribution
times square root of N
For example,with 1,000 trial the 1 percentile is
the 10th worst case.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.35
Speeding Up Monte Carlo
Use the quadratic approximation to calculate
P
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.36
Historical Simulation
(p,356)
Create a database of the daily movements in all
market variables.
The first simulation trial assumes that the percentage
changes in all market variables are as on the first day
The second simulation trial assumes that the
percentage changes in all market variables are as on
the second day
and so on
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.37
Stress Testing
(p,357)
This involves testing how well a portfolio
performs under some of the most
extreme market moves seen in the last
10 to 20 years
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.38
Back-Testing
(p,357)
Tests how well VaR estimates would
have performed in the past
We could ask the question,How often
was the loss greater than the 99%/10
day VaR?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.39
Principal Components Analysis
(p,357)
Suppose that a portfolio depends on a
number of related variables (eg interest rates)
We define factors as a scenarios where there
is a certain movement in each market
variable (The movements are known as factor
loadings)
The observations on the variables can often
be largely explained by two or three factors
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.40
Results for Interest Rates
(Table 14.4)
The first factor is a roughly parallel shift
(83.1% of variation explained)
The second factor is a twist (10% of
variation explained)
The third factor is a bowing (2.8% of
variation explained)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
14.41
Assignments
14.1 – 14.5,14.10 – 14.14
Assignment Questions
