Problem 1:

The implied volatilities of a traded asset (from its traded option prices) and the corresponding vega exposure that a trader has in those buckets (for a book of vanilla options) by:

Days	Implied		Days	Vega Exposure
	Vol			USD
0 - 30	16.00%		0 - 30	$ 1,000,000
0 - 60	15.00%		0 - 60	$ 850,000
0 - 90	15.85%		0 - 90	$ (1,750,000)
0 - 180	14.25%		0 - 180	$ (2,350,000)
0 - 360	13.00%		0 - 360	$ 900,000

(a) Generate the forward volatility curve from the spot volatilities given above (hint:
      decompose the straight (spot) volatility exposure into buckets of 0 - 30, 30 - 60,
      60 - 90 and so on);
(b) The risk manager, noting the unevenness of the buckets, has decided to construct
      narrower slices of the vega exposure at the beginning and wider ones at the end
      such that: 50% of the vega exposure of the one year option will be mapped onto
      the 180 - 360 bucket, 25% of the vega exposure of the one year option in the 90 -
     180 bucket and 8.33% of its vega exposure in each of the 0 - 30, 30 - 60 and 60 -
      90 buckets. Under this mapping technique what will be the multifactor (modified)
      vega exposure of the option book above.

Problem 2

(a)   The implied volatility of a call is 15% and the underlying spot is at 100. What is
        the alpha of the option? If the volatility of an option increase by 30% and the spot
        goes down by 15% how will the alpha of the option change?
(b)   If the probability of an early exercise for a one year American style binary call
        option on a stock is negligible then what is the omega of this option? Now
        suppose, the rise in 100 basis points in the dividend payout ratio increases the
        price of the binary call by 73 basis points. What is the expected life of the binary?
(c)   The delta of a 30 day put option with 15% volatility is 0.64. When the option is
        re-priced with 29 days the delta is 0.59. What is the delta bleed?
(d)   Explain the "risk neutrality" argument using Put-Call parity.

Problem 3

If is a portfolio of vanilla options on an underlying asset then the change in the portfolio's value is expressed as:

where is the portfolio's delta and is the gamma of the portfolio. and represent a small change in the portfolio and the asset respectively. If the delta of the portfolio (in

Dollar terms) is $5 million and the gamma of the portfolio is $275,000 then what is the worst case scenario (worst case loss) for the portfolio.

Problem 4

(a) What is the Excel? function to calculate the value of a random normal number (a
      random number drawn from a normal distribution with mean of zero and standard
      deviation of one)?
(b) How would you write the expression for a random normal number in Excel? using a Sobol
      sequence?
(c) When you run a VBA macro you get a message "run time error". What does this mean?
(d) What is the difference between a subroutine and a function in VBA. Explain with an example.

Problem 5

Explain intuitively the significance of Cholesky Matrix in multi-asset risk decomposition. Why is a Cholesky matrix lower triangular? If M is a Cholesky matrix and is the variance-covariance matrix then what is the mathematical relationship between the two. If in a real life situation the Cholesky factorization breaks down then how do we clean the correlation matrix (hint: used quite often in Value at Risk problems and refer to technical document on VaR).

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