Question

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 $605. Moreover, 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 sliver at the close of trading today is unchanged at $16. a. Update the volatilities for gold and silver using the EWMA model with λ = 0.94. b. Update the correlation between gold and sliver using the EWMA model with λ = 0.94.

Answer 1

Ans a)

EWMA model with λ = 0.94.

Estimated Volatility S

Variance = λ * S² + ( 1 - λ) * p

Here p = Proportional Price Change

Ans a)

for gold :

Variance = λ * S² + ( 1 - λ) * p²   

S = 1.3%

p = Proportional Price Change = Change in Price / Previous Price =( 605 - 600) / 600 = 5 / 600 = 0.00833

Variance = 0.94 * (1.3% )² + ( 1- 0.94) * ( 0.00833 )²  

= 0.00015886 +  0.000004167

= 0.00016302667

volatilities for gold = Square root of Variance = SQRT (0.00016302667)

= 0.01276819

= 1.2768%

for silver :

Variance = λ * S² + ( 1 - λ) * p²   

Now Proportional change in price p = 0

So

Variance = λ * S²

S = 1.5%

Variance = 0.94 * (1.5% )²   = 0.0002115

Volatilities for Silver = Square root of Variance = SQRT (0.0002115 )

= 0.014543

= 1.4543%

Ans:

volatilities for gold = 1.2768%

Volatilities for Silver =  1.4543%

Ans b)

Initial Covariance = Correlation * Volatility of Gold * Volatility of Silver = 0.8 * 1.3% * 1.5% = 0.000156

New Covariance using EWMA

COV =  λ * (Initial Covariance) + ( 1 - λ) * p

Now price change p for Silver = 0

So,

COV =  λ * (Initial Covariance) = 0.94 * ( 0.000156) =  0.00014664

Corelation = COV / (volatilities for gold)*  (volatilities for Silver)

= 0.7934

Updated correlation between gold and sliver using the EWMA = 0.7934