1. The returns of two stocks were x_n=0.02 and y_n=0.07, their estimated daily volatilities were _x,n=0.01 and _y,n=0.05, and their estimated correlation was _n=0.5. Assuming =0.94, what is the updated correlation forecast _n+1, given the above information, under the exponentially weighted moving average (EWMA) model?

a.

0.45

b.

0.38

c.

0.64

d.

0.57

2. Consider a long forward contract to buy a certain commodity for $55 in 2 years. Assume the current spot price is S₀=50, with expected return 5%, risk-free rate r=8%, volatility =40%. What is the level of loss that will not be exceeded 97.5% of the time in ten days?

a.

39.5

b.

11.4

c.

4.3

d.

5.5

3. Suppose that two random variables V₁ and V₂ have uniform distributions where all values between 0 and 1 are equally likely. Use a Gaussian copula to define the correlation structure between V₁ and V₂. Let (x,y;) denote the standard bivariate normal cumulative distribution function with correlation . What is the probability Prob(V₁<0.3, V₂<0.4) if =0.4?

a.

(-0.52,-0.25;0.4) > 0.12

b.

(0.3,0.4;0.4) < 0.12

c.

(-0.52,-0.25;0.4) < 0.12

d.

(0.3,0.4;0.4) > 0.12

4. Suppose that a GARCH (1,1) model is estimated from daily data as

(_n)² = 0.000002 + 0.14(u_n-1)² + 0.83(_n-1)²

What volatility should be used to price 30-day options? Assume current volatility is 4% per day.

a.

0.19

b.

0.27

c.

0.52

d.

0.44