Administrative details

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MFE 230HSection 4
Bradyn Breon-Drish breon_at_haas.berkeley.edu Sept.
11, 2007
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Assignment 5

• Building a Risk Management System
• Delta-normal VAR CVAR
• Historical simulation VAR CVAR
• Stress testing
• Monte Carlo simulation VAR

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Example

• As of Sept. 2, 2004, we have a portfolio that is
• Short 100 units of SP 500 basket at 1118.31
• Long 5000 shares of Apple Computer at 35.66,
  implied vol 40
• Write calls on 1000 shares of Apple
• K35, T1 month

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Q1 Delta-normal approach

• We want the net delta on each risk factor (in )
• Risk factors are the index and the stock
• First compute hedge ratio (delta) of option
• Then use Delta Delta of instrument Dollar
  value of notional

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Q1 Delta-normal Component VAR

• Component VAR (see p.172-174) is an additive
  decomposition of VAR that includes
  diversification effects
• Caution The beta and correlation here must be
  relative to the portfolio, not the market

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Q1 Delta-normal Component VAR (cont.)

• Vector of betas relative to portfolio can be
  computed by
• Individual correlations are then

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Q2 Historical-simulation Component VAR

• Using historical simulation, there are two ways
  to compute CVAR
• Find date of the 1st percentile return. Compute
  the actual returns to the two positions on that
  day. Can also adjust for means.
• By construction, we then have

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Q2 Historical-simulation Component VAR (cont.)

• Alternately, we can use
• where the VAR is the historical simulation
  version.
• Note there is still a normality assumption
  hiding in this method.

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Q2 Expected tail loss

• Expected tail loss is the expected loss,
  conditional on exceeding VAR
• Take arithmetic average of returns less than VAR
  and multiply by portfolio value
• In Excel, can use SUMIF(.) and COUNTIF(.) commands

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Q3 Stress test

• SP drops by 15
• First scenario, assume AAPL does not change.
  Simply move SP down by 15.
• Second scenario, use conditional scenario method
  (p.366-367) to account for AAPL move

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Q3 Stress test (cont.)

• For conditional scenario method first regress
  other factor on SP, producing a beta
• For a given return of x in the SP, compute the
  conditional expected return on the other factor
  as x beta.
• Using these returns, calculate the new price for
  SP and the stock, then re-price the option using
  B-S.
• Finally, with all of the new prices, calculate
  the portfolio value and compare to the starting
  value

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Q4 Monte Carlo

• Considering only the second subportfolio, compute
  the 99 VAR using lognormal distribution for
  prices
• There are two steps involved
• Simulate ending stock price
• Use B-S to price option using ending stock price

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Q4 Monte Carlo Step 1

• Adjust mean and volatility for the time horizon
  (mean t mu, vol sqrt(t) sigma), generate
  random variables
• Generate normal random variables
• In Excel can use
• NORMINV(RAND(),mean,vol)
• Compute ending stock price as
• S(0) Exp(x), where x random variable from above

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Q4 Monte Carlo Step 2

• With simulated stock price, re-price the option
• Be sure to account for the fact that the
  time-to-maturity of the option is now shortened

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Q4 Monte Carlo Calculate VAR

• Using ending stock and option prices calculated
  above, find the value of the subportfolio
• Compute VAR as shown in eq.12.3 on p.315
• Why might this be larger than VAR in Q1 2?