Market Risk Modelling
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Basics of Market risk modelling

Basics of Market risk modelling

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Market Risk Modelling Market Risk Modelling Presentation Transcript

  • Market Risk Modelling By A.V. Vedpuriswar July 31, 2009
  • Volatility
  • 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.
  • Estimating Volatility
    • Calculate daily return u 1 = ln S i / S i-1
    • Variance rate per day
    • We can simplify this formula by making the following simplifications.
    • u i = (S i – S i-1) / S i-1
    • ū = 0 m-1 = m
    • If we want to weight
  • Estimating Volatility
      • Exponentially weighted moving average model means weights decrease exponentially as we go back in time.
      •  n 2 =   2 n-1 + (1 -  ) u 2 n-1
      • =  [  n-2 2 + (1-  )u n-2 2 ] + (1-  )u n-1 2
      • = (1-  )[u n-1 2 +  u n-2 2 ] +  2  n-2 2
      • = (1-  ) [u n-1 2 +  u 2 n-2 +  2 u n-3 2 ] +  3  2 n-3
      • If we apply GARCH model,
      •  n 2 = Y V L +  u n-1 2 +  2 n-1
      • V L = Long run average variance rate
      • Y +  +  = 1. If Y = 0,  = 1-  ,  =  , it becomes exponentially weighted model.
      • GARCH incorporates the property of mean reversion.
  • 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 .
  • Solution
    • h t = λ σ 2 t-1 + ( 1 – λ ) r t-1 2
    • λ = .94
    • r t-1 = ln[(30.50 )/ 30]
    • = .0165
    • h t = (.94) (.015) 2 + (1-.94) (.0165) 2
    • Volatility = .01509 = 1.509 %
  • 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.
  • 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.
  • 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.
  • 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.
    • T heta is usually negative for an option.
    • As time to maturity decreases with all else remaining the same, the option loses value.
  • 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 V T is the Vega of the traded option, a position of –V/ V T 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.
  • Rho
    • Rho of a portfolio of options is the rate of change of value of the portfolio with respect to the interest rate.
  • 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.
  • 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.
  • 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?
  • Solution
    • Buy 3200/1.5 = 2133 options
    • Sell (2133) (.5) = 1067 shares
    • 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
    • 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.
    • Suppose there is a second traded option with gamma = 0.8, vega = 1.2 and delta = 0.5.
    • if w 1 and w 2 are the weights in the portfolio,
    • - 5000 + .5w 1 + .8w 2 = 0 - 8000 + 2.0w 1 + 1.2w 2 = 0
    • w 1 = 400 w 2 = 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.
  • 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.
    • 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
  • Problem
    • The VAR on a portfolio using a one day horizon is USD 100 million. What is the VAR using a 10 day horizon ?
  • 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
    • ( σ N 2 = σ 1 2 + σ 2 2 ….. = N σ 2 )
  • 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.
  • 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
  • Variance Covariance Method
  • 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.
  • 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
  • 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.
  • 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
  • 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.?
  • Solution
    • σ 2 = w 1 2 σ 1 2 + w 2 2 σ 2 2 + 2 ῤ Pw 1 W 2 σ 1 σ 2
    • = (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
  • Monte Carlo Simulation
  • 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.
  • 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
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • Formula used typically in Monte Carlo for stock price modelling
  • 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.
  • Problems with Monte Carlo
    • Cost of computing resources c an 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.
    • 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.
  • Historical Simulation
  • 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.
  • 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.
  • Steps
    • Collect data
    • Generate scenarios
    • Calculate portfolio returns
    • Arrange in order.
  • Problem What is VAR (90%) ? % 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
    • 10% of the observations, i.e, (.10) (30)
    • = 3 lie below -7
    • So VAR = -7
    Solution
  • Advantages
    • Simple
    • No normality assumption
    • Non parametric
  • Disadvantages
    • Reliance on the past
    • Length of estimation period
    • Weighting of data
    • Data issues
  • Comparison of different VAR modeling techniques
  • 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
  • Model Validation
  • 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.
    • 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.
    • 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
  • Stress Testing
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.