1. Suppose that the price of an asset at close of trading yesterday was $300 and its volatility was estimated as 1.3% per day. The price at the close of trading today is $298. Update the volatility estimate using:

(a) The EWMA model with λ = 0.94.

Answer: Using the EWMA model, the variance is updated to

0.94 × 0.0132 + 0.06 × (−2/300)2  = 0.00016153

so that the new daily volatility is ^0.00016153 = 0.01271 or 1.271%.

(b) The GARCH(1,1) model with ω = 0.000002,α = 0.04, and β = 0.94.

Answer: Using GARCH(1,1), the variance is updated to

0.000002 + 0.94 × 0.0132 + 0.04 × (−2/300)2  = 0.00016264

so that the new daily volatility is ^0.00016264 = 0.01275 or 1.275%.

2. Suppose that the parameters in a GARCH(1,1) model are α = 0.03,β = 0.95, and ω = 0.000002.

(a) What is the long-run average volatility?

Answer: The long-run average variance, VL , is

ω            0.000002

1 − α − β          0.02

The long-run average volatility is ^0.0001 = 0.01 or 1% per day.

(b) If the current volatility is 1.5% per day, what is your estimate of the volatility in 20, 40, and 60 days?

Answer: The expected variance in 20 days is

0.0001 + 0.9820 (0.0152 − 0.0001) = 0.000183.

The expected daily volatility is therefore ^0.000183 = 0.0135 or 1.35%.  Similarly, the expected volatilities in 40 and 60 days are 1.25% and 1.17% per day, respectively.

(c) What volatility should be used to price 20-, 40, and 60-day options?

Answer: We have a = ln ( ) = 0.0202. The variance used to price 20-day options is 252 × [0.0001 + 1  (0.0152 − 0.0001)] = 0.051

so that the annual volatility is 22.61%. Similarly, the volatilities that should be used for 40- and 60-day options are 21.63% and 20.85% per annum, respectively.

(d) Suppose that there is an event that increases the current volatility from 1.5% per day to 2% per day. Estimate the effect on the volatility in 20, 40, and 60 days.

Answer: The expected variance in 20 days is now

0.0001 + 0.9820 (0.022 − 0.0001) =  0.0003.

The expected daily volatility in 20 days is therefore ^0.0003 = 0.0173 or 1.73%.  Similarly, the expected daily volatilities in 40 and 60 days are 1.53% and 1.38%, respectively.

(e) Estimate by how much the event increases the volatilities used to price 20-, 40, and 60-day options. Answer: When today’s volatility increases from 1.5% per day to 2% per day, the volatility used to price a 20-day option changes by

1 − e −0 .0202×20          0.015 

or 6.88% on an annual basis, bringing the volatility up to 29.49%. Similarly, the 40- and 60-day volatilities increase to 27.63% and 26.10%, respectively.