Market Risk ModellingBy A.V. Vedpuriswar

July 31, 2009

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Volatility

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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.

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Estimating Volatility Calculate daily return u1 = ln Si / Si-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

If we want to weight

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1

)(1

1

uum n

m

i

12

1

2 1

n

m

i

um

n

122

ni un )1( i

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Estimating Volatility Exponentially weighted moving average model

means weights decrease exponentially as we go back in time.

n 2 = 2n-1+ (1 - ) u2

n-1

= [n-22 + (1- )un-2

2] + (1- )un-12

= (1- )[un-12 + un-2

2] + 2n-22

= (1-) [un-12 + u2

n-2 + 2un-32 ] + 3 2

n-3

If we apply GARCH model,n

2 = Y VL + un-12 + 2

n-1

VL = Long run average variance rateY + + = 1. If Y = 0, = 1-, = , it

becomes exponentially weighted model.

GARCH incorporates the property of mean reversion.

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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 .

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Solution ht = λ σ2

t-1 + ( 1 – λ) rt-12

λ = .94 rt-1 = ln[(30.50 )/ 30]

= .0165 ht = (.94) (.015)2 + (1-.94) (.0165)2

Volatility = .01509 = 1.509 %

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Greeks

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.

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1010

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.

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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 can be 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.

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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.

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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.

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Rho Rho of a portfolio of options is the rate of change of

value of the portfolio with respect to the interest rate.

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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.

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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.

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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?

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Solution Buy 3200/1.5 = 2133 options Sell (2133) (.5) = 1067 shares

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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?

Problem

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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.

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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, - 5000 + .5w1 + .8w2 = 0

- 8000 + 2.0w1 + 1.2w2 = 0 w1 = 400 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.

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Value at Risk

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.

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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

Illustration

Find the point such that the no. of observations to the left

= (254) (.05) = 12.7 (12.7 – 11) /( 15 – 11 ) = 1.7 / 4 ≈ .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

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Problem The VAR on a portfolio using a one day horizon is

USD 100 million. What is the VAR using a 10 day horizon ?

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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 (σN

2 = σ12 + σ2

2 ….. = Nσ2)

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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.

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Solution

Weekly VAR ＝ (12,500) (√5) ＝ 27,951

Monthly VAR ＝ ( 12,500) (√20) ＝ 55,902

Semi annual VAR ＝ (12,500) (√125) ＝ 139,754

Annual VAR ＝ (12,500) (√250) ＝ 197,642

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Variance Covariance Method

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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.

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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

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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.

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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

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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.?

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Solution

σ2 = w12 σ1

2 + w22 σ2

2 + 2 ῤPw1 W2 σ1 σ2 = (200,000)2 + (50,000)2 + (2) (.3) (200,000)

(50,000) σ = 220,277 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

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Monte Carlo Simulation

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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.

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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

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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.

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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.

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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.

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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.

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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 trade-off 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.

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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.

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Formula used typically in Monte Carlo for stock price modelling

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Advantages of Monte Carlo

This method can cope with the risks associated with non-linear 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.

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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.

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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.

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Historical Simulation

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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.

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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.

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Steps

Collect data Generate scenarios Calculate portfolio returns Arrange in order.

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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 2 120 3 151 1 162 2 184 1 196 1 207 1 218 1 229 1 23

11 1 2412 1 2614 2 2718 1 2821 1 2923 1 30

What is VAR (90%) ?

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10% of the observations, i.e, (.10) (30) = 3 lie below -7

So VAR = -7

Solution

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Advantages

Simple No normality assumption Non parametric

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Disadvantages

Reliance on the past Length of estimation period Weighting of data Data issues

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Comparison of different VAR modeling techniques

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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

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Model Validation

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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.

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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.

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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

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Stress Testing

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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.

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Two approaches to Stress testingSingle-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.

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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.

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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.

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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.