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Options, Futures, and Other Derivatives

John C. Hull

Chapter 23

Estimating volatilities and correlations - all with Video Answers

Educators

Chapter Questions

Problem 1

Explain the exponentially weighted moving average (EWMA) model for estimating volatility from historical data.

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01:00

Problem 2

What is the difference between the exponentially weighted moving average model and the GARCH $(1,1)$ model for updating volatilities?

Nick Johnson
Nick Johnson
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Problem 3

The most recent estimate of the daily volatility of an asset is $$1.5 \%$$ and the price of the asset at the close of trading yesterday was $$\$ 30.00$$. The parameter $\lambda$ in the EWMA model is 0.94. Suppose that the price of the asset at the close of trading today is $$\$ 30.50$$. How will this cause the volatility to be updated by the EWMA model?

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Problem 4

A company uses an EWMA model for forecasting volatility. It decides to change the parameter $\lambda$ from 0.95 to 0.85 . Explain the likely impact on the forecasts.

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02:12

Problem 5

The volatility of a certain market variable is $30 \%$ per annum. Calculate a $99 \%$ confidence interval for the size of the percentage daily change in the variable.

Adriano Chikande
Adriano Chikande
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Problem 6

A company uses the $\operatorname{GARCH}(1,1)$ model for updating volatility. The three parameters are $\omega, \alpha$, and $\beta$. Describe the impact of making a small increase in each of the parameters while keeping the others fixed.

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Problem 7

The most recent estimate of the daily volatility of the U.S. dollar/sterling exchange rate is $0.6 \%$ and the exchange rate at $4 \mathrm{p} . \mathrm{m}$. yesterday was 1.5000 . The parameter $\lambda$ in the EWMA model is 0.9. Suppose that the exchange rate at 4 p.m. today proves to be 1.4950 . How would the estimate of the daily volatility be updated?

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Problem 8

Assume that S\&P 500 at close of trading yesterday was 1,040 and the daily volatility of the index was estimated as $1 \%$ per day at that time. The parameters in a GARCH$(1,1)$ model are $\omega=0.000002, \alpha=0.06$, and $\beta=0.92$. If the level of the index at close of trading today is 1,060 , what is the new volatility estimate?

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Problem 9

Suppose that the daily volatilities of asset A and asset B, calculated at the close of trading yesterday, are $1.6 \%$ and $2.5 \%$, respectively. The prices of the assets at close of trading yesterday were $\$ 20$ and $\$ 40$ and the estimate of the coeflicient of correlation between the returns on the two assets was 0.25 . The parameter $\lambda$ used in the EWMA model is 0.95 .
(a) Calculate the current estimate of the covariance between the assets.
(b) On the assumption that the prices of the assets at close of trading today are $$\$ 20.5$$ and $$\$ 40.5$$, update the correlation estimate.

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Problem 10

The parameters of a GARCH $(1,1)$ model are estimated as $\omega=0.000004, \alpha=0.05$, and $\beta=0.92$. What is the long-run average volatility and what is the equation describing the way that the variance rate reverts to its long-run average? If the current volatility is $20 \%$ per year, what is the expected volatility in 20 days?

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Problem 11

Suppose that the current daily volatilities of asset $\mathrm{X}$ and asset $\mathrm{Y}$ are $1.0 \%$ and $1.2 \%$, respectively. The prices of the assets at close of trading yesterday were $$\$ 30$$ and $$\$ 50$$ and the estimate of the coefficient of correlation between the returns on the two assets made at this time was 0.50 . Correlations and volatilities are updated using a $\mathrm{GARCH}(1,1)$ model. The estimates of the model's parameters are $\alpha=0.04$ and $\beta=0.94$. For the correlation $\omega=0.000001$, and for the volatilities $\omega=0.000003$. If the prices of the two assets at close of trading today are $$\$ 31$$ and $$\$ 51$$, how is the correlation estimate updated?

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Problem 12

Suppose that the daily volatility of the FTSE 100 stock index (measured in pounds sterling) is $1.8 \%$ and the daily volatility of the dollar/sterling exchange rate is $0.9 \%$. Suppose further that the correlation between the FTSE 100 and the dollar/sterling exchange rate is 0.4 . What is the volatility of the FTSE 100 when it is translated to U.S. dollars? Assume that the dollar/sterling exchange rate is expressed as the number of U.S. dollars per pound sterling. (Hint: When $Z=X Y$, the percentage daily change in $Z$ is approximately equal to the percentage daily change in $X$ plus the percentage daily change in $Y_{\text {) }}$

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01:45

Problem 13

Suppose that in Problem 23.12 the correlation between the S\&P 500 Index (measured in dollars) and the FTSE 100 Index (measured in sterling) is 0.7 , the correlation between the S\&P 500 Index (measured in dollars) and the dollar/sterling exchange rate is 0.3 , and the daily volatility of the S\&P 500 index is $1.6 \%$. What is the correlation between the S\&P 500 index (measured in dollars) and the FTSE 100 index when it is translated to dollars? (Hint: For three variables $X, Y$, and $Z$, the covariance between $X+Y$ and $Z$ equals the covariance between $X$ and $Z$ plus the covariance between $Y$ and $Z$.)

Michelle Z.
Michelle Z.
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27:31

Problem 14

Show that the GARCH $(1,1)$ model $\sigma_n^2=\omega+\alpha u_{n-1}^2+\beta \sigma_{n-1}^2$ in equation (23.9) is equivalent to the stochastic volatility model $d V=a\left(V_L-V\right) d t+\xi V d z$, where time is measured in days, $V$ is the square of the volatility of the asset price, and
$$
a=1-\alpha-\beta, \quad V_L=\frac{\omega}{1-\alpha-\beta}, \quad \xi=\alpha \sqrt{2}
$$
What is the stochastic volatility model when time is measured in years? (Hint: The variable $u_{n-1}$ is the return on the asset price in time $\Delta t$. It can be assumed to be normally distributed with mean zero and standard deviation $\sigma_{n-1}$. It follows from the moments of the normal distribution that the mean and variance of $u_{n-1}^2$ are $\sigma_{n-1}^2$ and $2 \sigma_{n-1}^4$, respectively.)

Jeremiah Mbaria
Jeremiah Mbaria
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Problem 15

At the end of Section 23.8 , the VaR and ES for the four-index example were calculated using the model-building approach. How do the VaR and ES estimates change if the investment is \$$2.5 million in each index? Carry out calculations when (a) volatilities and correlations are estimated using the equally weighted model and (b) when they are estimated using the EWMA model with $\lambda=0,94$$. Use the spreadsheets on the author's website.

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01:19

Problem 16

What is the effect of changing $\lambda$ from 0.94 to 0.97 in the EWMA calculations in the fourindex example at the end of Section 23.8. Use the spreadsheets on the author's website.

Ajay Singhal
Ajay Singhal
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Problem 17

Suppose that the price of gold at close of trading yesterday was $$\$ 600$$ and its volatility was estimated as $1.3 \%$ per day. The price at the close of trading today is $$\$ 596$$. Update the volatility estimate using
(a) The EWMA model with $\lambda=0.94$
(b) The GARCH $(1,1)$ model with $\omega=0.000002, \alpha=0.04$, and $\beta=0.94$,

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Problem 18

Suppose that in Problem 23.17 the price of silver at the close of trading yesterday was $$\$ 16$$, its volatility was estimated as $1.5 \%$ per day, and its correlation with gold was estimated as 0.8 . The price of silver at the close of trading today is unchanged at $$\$ 16$$. Update the volatility of silver and the correlation between silver and gold using the two models in Problem 23.17. In practice, is the $\omega$ parameter likely to be the same for gold and silver?

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Problem 19

An Excel spreadsheet containing over 900 days of daily data on a number of different exchange rates and stock indices can be downloaded from the author's website:
www-2.rotman.utoronto.ca/ hull/data.
Choose one exchange rate and one stock index. Estimate the value of $\lambda$ in the EWMA model that minimizes the value of $\sum_i\left(v_i-\beta_i\right)^2$, where $v_i$ is the variance forecast made at the end of day $i-1$ and $\beta_i$ is the variance calculated from data between day $i$ and day $i+25$. Use the Solver tool in Excel. Set the variance forecast at the end of the first day equal to the square of the return on that day to start the EWMA calculations.

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Problem 20

Suppose that the parameters in a GARCH $(1,1)$ model are $\alpha=0.03, \beta=0.95$, and $\omega=0.000002$.
(a) What is the long-run average volatility?
(b) If the current volatility is $1.5 \%$ per day, what is your estimate of the volatility in 20 , 40 , and 60 days?
(c) What volatility should be used to price 20 -, 40 -, and 60 -day options?
(d) Suppose that there is an event that increases the current volatility by $0.5 \%$ to $2 \%$ per day. Estimate the effect on the volatility in 20,40 , and 60 days.
(c) Estimate by how much the event increases the volatilities used to price $20-, 40-$, and 60-day options.

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Problem 21

The calculations for the four-index example at the end of Section 23.8 assume that the investments in the DJIA, FTSE 100, CAC 40, and Nikkei 225 are $$\$4 million$$, $$\$3 million$$, $$\$ 1$$ million, and $$\$ 2$$ million, respectively. How do the VaR and ES estimates change if the investments are $$\$ 3$$ million, $$\$3 million$$, $$\$1 million$$, and $$\$3 million$$, respectively? Carry out calculations when (a) volatilities and correlations are estimated using the equally weighted model and (b) when they are estimated using the EWMA model. What is the effect of changing $\lambda$ from 0.94 to 0.90 in the EWMA calculations? Use the spreadsheets on the author's website.

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03:08

Problem 22

Estimate parameters for EWMA and $\operatorname{GARCH}(1,1)$ from data on the euro-USD exchange rate between July 27,2005 , and July 27, 2010. This data can be found on the author's website:
www-2.rotman.utoronto.ca/ hull/data.

Alejandro Ruiz
Alejandro Ruiz
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