Question

Suppose that the current daily volatilities of asset A and asset B are 1.65% and 2.32%,

respectively. The prices of the assets at close of trading yesterday were $35 and $52 and

the estimate of the coefficient of correlation between the returns on the two assets made

at that time was 0.26. The parameter λ 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 $36 and $53,

update the correlation estimate.

Answer 1

Answer to a)

According to question,

Correlation(A,B) = 0.26

Standard Deviation(A) = 1.65%

Standard Deviation(B) = 2.32%

Parameter  λ = 0.95

Yesterday trading closing price A = $35

Yesterday trading closing price, B = $52

Correlation (A,B)

= Covariance(A,B) / (Standard Deviation(A)*Standard Deviation(B))

Hence , Covariance(A,B)

= Correlation(A,B)*Standard Deviation(A)*Standard Deviation(B)

= 0.26 * 0.0165 * 0.0232

Covariance (A,B) = 0.00009952

Answer to b)

If we assume that the prices of the assets closes today are $36 and $53 for asset A and asset B then the respective proportionate change in its assets value are as follows :-

Asset A

Proportionate change = (36-35)/35 = 0.0286

Asset B

Proportionate change = (53-52)/52 = 0.0192

As per the EWMA model today's variance is a function of prior day's variance and the formula is as mentioned below :-

σ_n²(EWMA)=λσ_n²+(1−λ)u²_n−1

λ=Degree of Weighting decrease

σ²=Value of Variance at time period n

u²=Value of EWMA at time period n

Covariance (A,B) =

(0.95*0.000099528) + (1-0.95)*0.0286*0.0192

=0.000122

Variance estimate for Asset A

σ_n²(EWMA)

= 0.95*0.0165^2 + (1-0.95)*0.0165^2*0.0286^2

= 0.00026

Standard Deviation (A) = Square root(Variance A)

= 0.00026^(1/2)

= 0.01612

Variance estimate for Asset B

=σ_n²(EWMA)

=0.95*0.0232^2 + (1-0.95)*0.0232^2*0.0192^2

= 0.0005

Standard Deviation(B) = Square Root(Variance B)

= 0.0005^(1/2)

= 0.02236

Correlation estimate :-

Correlation(A,B) = Covariance(A,B) / (Standard Deviation(A)*Standard Deviation(B))

Correlation(A,B)

= 0.000122 / (0.01612*0.02236)

Hence, Correlation(A,B) = 0.338