The Argo Hedge fund has the following portfolio of options of the XYZ stock

Type	Position	Delta of Option	Gamma of Option
Call	−1,000	0.50	0.2
Call	−500	0.80	0.6
Put	−2,000	−0.40	1.3
Call	−500	0.70	1

A new option becomes available with a delta of 0.6, a gamma of 0.5

What position in the new option and in XYZ stock would make the portfolio both gamma neutral and delta neutral?

Question 2

Suppose that the price of Trisco Corp at close of trading yesterday was $275 and its volatility was estimated as 2% per day. The price of Trisco at the close of trading today is $278. Suppose further that the price of Baldi Corp. at the close of trading yesterday was $100, its volatility was estimated as 1.5% per day, and its correlation with Trisco was estimated as 0.8. The price of Baldi at the close of trading today is $98. Update the volatility of Trisco and Baldi and the correlation between Trisco and Baldi using

(a) The EWMA model with l= 0.94

(b) The GARCH(1,1) model with w = 0.0000018, a= 0.03, and b= 0.96.

Question 3

You have a portfolio of options:

1. 200 long European calls on ABC corporation shares maturing in 9 months. The current price of ABC shares is $45 and its annual volatility is 30%. These options are “”at the money”.
2. 300 short European puts on VWX corporation shares maturing in 9 months. The current price of VWX shares is $50 and the strike price for your options is $51. The volatility of VWX is 20% per year.

The correlation coefficient between returns on ABC and VWX is 0.25

Both corporations don’t pay dividends. The risk free rate is expected to be at 2% per year. Use 10-day time horizon and 99% confidence level.

1. Calculate VaR of your portfolio using the delta linear model.
2. Calculate VaR of your portfolio using the delta-gamma quadratic model. Assume that each option’s price depends on only one risk factor (i.e. cross-gammas are 0)

Question 4

Consider the following data on three ETF funds A, B and C:

		s
A	0.1	0.1
B	0.08	0.09
C	0.15	0.2

The correlation matrix is given by:

	A	B	C
A	1
B	0.2	1
C	0.4	-0.5	1

1. What is the variance and the expected return on the equally weighted portfolio of A, B, and C.
2. Assuming that returns on A, B, and C are jointly normally distributed, calculate the 95% VaR for this portfolio
3. What are the benefits of diversification from the VaR perspective?
4. Calculated the 95% expected shortfall for this portfolio

Question 5

You have a portfolio of two assets with independently distributed returns. You invested $10M in Asset X and $20M in Asset Y. The distributions of returns on both assets are given below:

Probability	Return on Asset X	Probability	Returns on Asset Y
0.01	-25%	0.015	-20%
0.05	-5%	0.045	-5%
0.94	8%	0.95	10%

1. What is the 95% VaR of this portfolio?
2. What is the 95% expected shortfall of this portfolio?
3. Suppose that the lowest possible expected return on Asset X was reassessed to be instead of with the same 1% probability. Recalculate VaR and ES of this portfolio, taking this change into account.
4. Based on your calculations, what is your opinion about shortcomings of VaR?

Question 6

The value of the XYZ’s equity is $10 million and the volatility of its equity is 40%. The debt that will have to be repaid in three years is $10 million. The risk-free interest rate is 3% per annum.

1. Use Merton’s model to estimate the probability of default
2. Find the credit spread implied by Merton’s model

-25% -25% -25%