Market Risk Modelling
By A.V. Vedpuriswar
July 31, 2009
Volatility
1
Basics of volatility
Volatility is a huge issue in risk management.
Volatility is a key parameter in modelling market risk
The science of volatility measurement has advanced a lot in
recent years.
Here we look at some basic concepts and tools.
2
Estimating Volatility

Calculate daily return u1 = ln Si / Si-1

1 m
2

(
u

u
)
n 1

m  1 i 1

Variance rate per day

We can simplify this formula by making the following
simplifications.
ui = (Si – Si-1) / Si-1
ū=0
m-1 = m
m
1
n 2   u 2 n 1
m i 1
If we want to weight
n 2    i u 2 n 1
(  i  1)
3
Estimating Volatility

Exponentially weighted moving average model means
weights decrease exponentially as we go back in time.
n 2 = 2n-1+ (1 - ) u2n-1
=  [n-22 + (1- )un-22] + (1- )un-12
= (1- )[un-12 + un-22] + 2n-22
= (1-) [un-12 + u2n-2 + 2un-32 ] + 3 2n-3

If we apply GARCH model,
n2 = Y VL + un-12 + 2n-1
VL = Long run average variance rate
Y +  +  = 1.

If Y = 0,  = 1-, = , it becomes
exponentially weighted model.
GARCH incorporates the property of mean reversion.
4
Problem
 The current estimate of daily volatility is 1.5%. The closing
price of an asset yesterday was $30. The closing price of
the asset today is $30.50. Using the EWMA model, with λ =
0.94, calculate the updated estimate of volatility .
5
Solution
ht
=
λ σ2t-1 + ( 1 – λ) rt-12
 λ
=
.94
 rt-1 =
 ht
ln[(30.50 )/ 30]
=
.0165
=
(.94) (.015)2 + (1-.94) (.0165)2
Volatility = .01509 = 1.509 %
6
Greeks
7
Introduction
Greeks help us to measure the risk associated with derivative
positions.
Greeks also come in handy when we do local valuation of
instruments.
This is useful when we calculate value at risk.
8
Delta

Delta is the rate of change in option price with respect to
the price of the underlying asset.

It is the slope of the curve that relates the option
price to the underlying asset price.

A position with delta of zero is called delta neutral.

Delta keeps changing.

So the investor’s position may remain delta neutral for
only a relatively short period of time.

The hedge has to be adjusted periodically.

This is known as rebalancing.
9
Gamma

The gamma is the rate of change of the portfolio’s
delta with respect to the price of the underlying asset.

It is the second partial derivative of the portfolio price
with respect to the asset price.

If gamma is small, it means delta is changing slowly.

So adjustments to keep a portfolio delta neutral
made only relatively infrequently.

However, if gamma is large, it means the delta is
highly sensitive to the price of the underlying asst.

It is then quite risky to leave a delta neutral portfolio
unchanged for any length of time.
can be
10
Theta
Theta of a portfolio is the rate of change of value of the
portfolio with respect to change of time.
Theta is also called the time decay of the portfolio.
Theta is usually negative for an option.
As time to maturity decreases with all else remaining the
same, the option loses value.
11
Vega

The Vega of a portfolio of derivatives is the rate of
change of the value of the portfolio with respect to
the
volatility of the underlying asset.

High Vega means high sensitivity to small changes in
volatility.

A position in the underlying asset has zero Vega.

The Vega can be changed by adding options.

If V is Vega of the portfolio and VT is the Vega of the
traded option, a position of –V/ VT in the traded option
makes the portfolio Vega neutral.

If a hedger requires the portfolio to be both gamma
and Vega neutral, at least two traded derivatives
dependent on the underlying asset must usually be
used.
12
Rho
 Rho of a portfolio of options is the rate of change of value of
the portfolio with respect to the interest rate.
13
Problem
Suppose an existing short option position is delta neutral
and has a gamma of －6000. Here, gamma is negative
because we have sold options. Assume
there exists
a traded option with a delta of 0.6 and gamma of 1.25.
Create a gamma neutral position.
14
Solution
To gamma hedge, we must buy 6000/1.25 = 4800 options.
Then we must sell (4800) (.6) = 2880 shares to maintain
a gamma neutral and original delta neutral position.
15
Problem
A delta neutral position has a gamma of －3200. There is
an option trading with a delta of 0.5 and gamma of 1.5.
How can we generate a gamma neutral position for the
existing portfolio while maintaining a delta neutral
hedge?
16
Solution

Buy 3200/1.5
=
2133 options

Sell (2133) (.5)
=
1067 shares
17
Problem
 Suppose a portfolio is delta neutral, with gamma
= - 5000 and vega = - 8000. A traded option has
gamma =
.5, vega = 2.0 and delta = 0.6. How do we achieve vega
neutrality?

18
To achieve Vega neutrality we can add 4000 options.
 Delta increases by (.6) (4000) = 2400
So we sell 2400 units of asset to maintain delta neutrality.
As the same time, Gamma changes from – 5000 to
((.5) (4000) – 5000 = - 3000.
19
 Suppose there is a second traded option with gamma
0.8, vega = 1.2 and delta = 0.5.
=
 if w1 and w2 are the weights in the portfolio,

w1 = 400
- 5000 + .5w1 + .8w2 = 0
- 8000 + 2.0w1 + 1.2w2 = 0
w2 = 6000.
This makes the portfolio gamma and vega neutral.
Now let us examine delta neutrality.
Delta = (400) (.6) + (6000) (.5) = 3240
3240 units of the underlying asset will have to be sold to
maintain delta neutrality.

20
Value at Risk
21
Introduction
Value at Risk (VAR) is probably the most important tool for
measuring market risk.
VAR tells us the maximum loss a portfolio may suffer at a
given confidence interval for a specified time horizon.
If we can be 95% sure that the portfolio will not suffer more
than $ 10 million in a day, we say the 95% VAR is $ 10
million.
22
Illustration
Average revenue = $5.1 million per day
Total no. of observations = 254.
Std dev = $9.2 million
Confidence level = 95%
No. of observations < - $10 million = 11
No. of observations < - $ 9 million = 15
23
Find the point such that the no. of observations to the left
= (254) (.05) = 12.7
(12.7 – 11) /( 15 – 11 )
=
1.7 / 4
So required point = - (10 - .4)
=
- $9.6 million
VAR = E (W) – (-9.6)
= 5.1 – (-9.6) = $14.7 million
If we assume a normal distribution,
Z at 95% ( one tailed) confidence interval = 1.645
VAR = (1.645) (9.2) =
$ 15.2 million
≈ .4
Problem
 The VAR on a portfolio using a one day horizon is USD 100
million. What is the VAR using a 10 day horizon ?
25
Solution
 Variance scales in proportion to time.
 So variance gets multiplied by 10

And std deviation by √10
 VAR = 100 √10 = (100) (3.16)
=
316
 (σN2 = σ12 + σ22 ….. = Nσ2)
26
Problem
 If the daily VAR is $12,500, calculate the weekly,
monthly, semi annual and annual VAR. Assume 250
days and 50 weeks per year.
27
Solution
Weekly VAR
＝
(12,500) (√5)
＝ 27,951
Monthly VAR
＝
( 12,500) (√20)
＝ 55,902
Semi annual VAR
Annual VAR
＝ (12,500) (√125)
＝ 139,754
＝ (12,500) (√250)
＝ 197,642
28
Variance Covariance Method
29
Problem
 Suppose we have a portfolio of $10 million in shares of
Microsoft. We want to calculate VAR at 99% confidence
interval over a 10 day horizon. The volatility of Microsoft is
2% per day. Calculate VAR.
30
Solution

σ = 2%
= (.02) (10,000,000)
= $200,000

Z (P = .01)
= Z (P =.99)
= 2.33
 Daily VAR
= (2.33) (200,000)
= $ 466,000
 10 day VAR
= 466,000 √10
= $ 1,473,621
Ref : Options, futures and other derivatives, By John Hull
31
Problem
 Consider a portfolio of $5 million in AT&T shares with a
daily volatility of 1%. Calculate the 99% VAR for 10 day
horizon.
32
Solution

σ =


1%
=
(.01) (5,000,000)
= $ 50,000
Daily VAR
=
(2.33) (50,000)
= $ 116,500
10 day VAR
=
$ 111,6500 √10
= $ 368,405
33
Problem
 Now consider a combined portfolio of AT&T and Microsoft
shares. Assume the returns on the two shares have a
bivariate normal distribution with the correlation of 0.3.
What is the portfolio VAR.?
34
Solution

σ2 =
w12 σ12 + w22 σ22 + 2 ῤPw1 W2 σ1 σ2

=
(200,000)2 + (50,000)2 + (2) (.3) (200,000) (50,000)
=
220,277

σ

Daily VAR = (2.33) (220,277)
=
513,129

10 day VAR = (513,129) √10
=
$1,622,657
 Effect of diversification = (1,473,621 + 368,406) – (1,622,657)
= 219,369
35
Monte Carlo Simulation
36
What is Monte Carlo VAR?
The Monte Carlo approach involves generating many price
scenarios (usually thousands) to value the assets in a
portfolio over a range of possible market conditions.
The portfolio is then revalued using all of these price
scenarios.
 Finally, the portfolio revaluations are ranked to select the
required level of confidence for the VAR calculation.
37
Step 1: Generate Scenarios
The first step is to generate all the price and rate scenarios
necessary for valuing the assets in the relevant portfolio, as
well as the required correlations between these assets.
There are a number of factors that need to be considered
when generating the expected prices/rates of the assets:
– Opportunity cost of capital
– Stochastic element
– Probability distribution
38
Opportunity Cost of Capital
A rational investor will seek a return at least equivalent to the
risk-free rate of interest.
Therefore, asset prices generated by a Monte Carlo simulation
must incorporate the opportunity cost of capital.
39
Stochastic Element

A stochastic process is one that evolves randomly over time.
Stock market and exchange rate fluctuations are examples of
stochastic processes.
The randomness of share prices is related to their volatility.
The greater the volatility, the more we would expect a share
price to deviate from its mean.
40
Probability Distribution
Monte Carlo simulations are based on random draws from a
variable with the required probability distribution, usually the
normal distribution.
The normal distribution is useful when modeling market risk in
many cases.
But it is the returns on asset prices that are normally
distributed, not the asset prices themselves.
So we must be careful while specifying the distribution.
41
Step 2: Calculate the Value of the Portfolio

Once we have all the relevant market price/rate scenarios, the
next step is to calculate the portfolio value for each scenario.
For an options portfolio, depending on the size of the portfolio, it
may be more efficient to use the delta approximation rather than
a full option pricing model (such as Black-Scholes) for ease of
calculation.
Δoption = Δ(ΔS)
Thus the change in the value of an option is the product of the
delta of the option and the change in the price of the underlying.
42
Other approximations
There are also other approximations that use delta, gamma (Γ)
and theta (Θ) in valuing the portfolio.
By using summary statistics, such as delta and gamma, the
computational difficulties associated with a full valuation can be
reduced.
 Approximations should be periodically tested against a full
revaluation for the purpose of validation.
When deciding between full or partial valuation, there is a tradeoff between the computational time and cost versus the
accuracy of the result.
 The Black-Scholes valuation is the most precise, but tends to
be slower and more costly than the approximating methods.
43
Step 3: Reorder the Results
After generating a large enough number of scenarios and
calculating the portfolio value for each scenario:
– the results are reordered by the magnitude of the change in the
value of the portfolio (Δportfolio) for each scenario
– the relevant VAR is then selected from the reordered list
according to the required confidence level
 If 10,000 iterations are run and the VAR at the 95% confidence
level is needed, then we would expect the actual loss to exceed
the VAR in 5% of cases (500).
So the 501st worst value on the reordered list is the required
VAR.
Similarly, if 1,000 iterations are run, then the VAR at the 95%
confidence level is the 51st highest loss on the reordered list.
44
Formula used typically in Monte Carlo for stock price
modelling
45
Advantages of Monte Carlo
 This method can cope with the risks associated with nonlinear positions.
 We can choose data sets individually for each variable.
 This method is flexible enough to allow for missing data
periods to be excluded from the VAR calculation.
 We can incorporate factors for which there is no actual
historical experience.
 We can estimate volatilities and correlations using different
statistical techniques.
46
Problems with Monte Carlo
Cost of computing resources can be quite high.
Speed can be slow.
Random Numbers may not be all that random.
 Pseudo random numbers are only a substitute for true
random numbers and tend to show clustering effects.
Quasi-Monte Carlo techniques have been developed to
produce quasi-random numbers that are more uniformly
spaced.
47
Monte Carlo is based on random draws from a variable with the
required probability distribution, often normal distribution.
As with the variance-covariance approach, the normal
distribution assumption can be problematic .
Monte Carlo can however, be performed with alternative
distributions.
Model risk is the risk of loss arising from the failure of a model to
sufficiently match reality, or to otherwise deliver the required
results.
For Monte Carlo simulations, the results (value at risk estimate)
depend critically on the models used to value (often complex)
financial instruments.
48
Historical Simulation
49
Introduction
Historical simulation is one of the three most common
approaches used to calculate value at risk.
 Unlike the Monte Carlo approach, it uses the actual historical
distribution of returns to simulate the VAR of a portfolio.
Use of real data, coupled with ease of implementation, has
made historical simulation a very popular approach to
estimating VAR.
50
Few assumptions
Historical simulation avoids the assumption that returns on
the assets in a portfolio are normally distributed.
Instead, it uses actual historical returns on the portfolio assets
to construct a distribution of potential future portfolio losses.
From this distribution, the VAR can be read.
This approach requires minimal analytics.
All we need is a sample of the historic returns on the portfolio
whose VAR we wish to calculate.
51
Steps
Collect data
Generate scenarios
Calculate portfolio returns
Arrange in order.
52
Problem
% Returns
Frequency
Cumulative Frequency
- 16
1
1
- 14
1
2
- 10
1
3
-7
2
5
-5
1
6
-4
3
9
-3
1
10
-1
2
12
0
3
15
1
1
16
2
2
18
4
1
19
6
1
20
7
1
21
8
1
22
9
1
23
11
1
24
12
1
26
14
2
27
18
1
28
21
1
29
23
1
30
What is VAR (90%) ?
53
Solution
 10% of the observations, i.e, (.10) (30)
= 3 lie below -7
So VAR = -7
54
Advantages
Simple
No normality assumption
Non parametric
55
Disadvantages
Reliance on the past
Length of estimation period
Weighting of data
Data issues
56
Comparison of different VAR modeling techniques
57
Simulation vs Variance Covariance
Simulation approaches are preferred by global banks due to:
– flexibility in dealing with the ever-increasing range of complex instruments
in financial markets
– the advent of more efficient computational techniques in recent years
– the falling costs in information technology
However, the variance-covariance approach might be the most
appropriate method for many smaller firms, particularly when:
– they do not have significant options positions
– they prefer to outsource the data requirement component of their risk
calculations to a company such as RiskMetrics
– significant savings can often be made by using outsourced volatility and
correlation data, compared to internally storing the daily price histories
required for simulation techniques
58
Model Validation
59
Basel Committee Standards
Banks that prefer to use internal models must meet, on a daily
basis, a capital requirement that is the higher of either:
– the previous day's value at risk
– the average of the daily value at risk of the preceding 60 business days
multiplied by a minimum factor of three
VAR must be computed on a daily basis.
A one-tailed confidence interval of 99% must be used.
The minimum holding period should be 10 trading days .
The minimum historical observation period should be one
year.
60
Banks should update their data sets at least once every three
months.
Banks can recognize correlations within broad risk categories.
Provided the relevant supervisory authority is satisfied with the
bank's system for measuring correlations , they may also
recognize correlations across broad risk factor categories.
Banks' internal models are required to accurately capture the
unique risks associated with options and option-like instruments.
The Basel Committee has also specified qualitative factors that
banks must meet before they are permitted to use internal models.
61
The Basel Committee prescribes an increase in capital
requirements if, based on a sample of 250 observations (a
one-year observation period), the VAR model underpredicts
the number of exceptions (losses exceeding the 99%
confidence level).
 For such purposes, three 'zones' have been distinguished by
the Committee.
Green Zone :
0-4 exceptions
Yellow zone :
5-9 exceptions
Red zone
:
10 or more exceptions
62
Stress Testing
63
Introduction
Stress testing involves analysing the effects of exceptional
events in the market on a portfolio's value.
These events may be exceptional, but they are also plausible.
And their impact can be severe.
Historical scenarios or hypothetical scenarios can be used.
64
Two approaches to Stress testing
Single-factor stress testing (sensitivity testing) involves
applying a shift in a specific risk factor to a portfolio in order to
assess the sensitivity of the portfolio to changes in that risk
factor.
Multiple-factor stress testing (scenario analysis) involves
applying simultaneous moves in multiple risk factors to a
portfolio to reflect a risk scenario or event that looks plausible
in the near future.
65
Conducting Stress Tests
From a computational viewpoint, stress testing can be thought
of as a variant of simulation methods.
It merely uses a different technique to generate scenarios.
Once scenarios have been developed, the next step is to
analyze the effect of each scenario on portfolio value.
 This can sometimes be done in the same way as a simulation
to calculate VAR.
Stress tests can typically be run by inputting the stressed
values of the risk factors into existing models and
recalculating the portfolio value using the new data.
66
Extreme Value Theory
EVT is a branch of statistics dealing with the extreme
deviations from the mean of statistical distributions.
The key aspect of EVT is the extreme value theorem.
According to EVT, given certain conditions, the distribution of
extreme returns in large samples converges to a particular
known form, regardless of the initial or parent distribution of
the returns.
67
EVT Parameters
This distribution is characterized by three parameters –
location, scale and shape (tail).
The tail parameter is the most important as it gives an
indication of the heaviness (or fatness) of the tails of the
distribution.
The EVT approach is very useful because the distributions
from which return observations are drawn are very often
unknown.
EVT does not make strong assumptions about the shape of
this unknown distribution.
68