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

(a) Given: Corr(A,B) = 0.26

Std Dev(A) = 1.65%

Std Dev(B) = 2.32%

parameter  λ = 0.95

Yesterday closing price, A = $35

Yesterday closing price, B = $52

correlation (A,B) = Covariance(A,B) / (Std dev(A)*Std Dev(B))

so, cov(A,B) = Corr(A,B)*Std dev(A)*Std Dev(B)

Cov(A,B) = 0.26*0.0165*0.0232

Cov(A,B) = 0.000099528

(b) If the prices of the assets closes todays are $36 and $53 respectively, then the proportionate changes in the asset values are:

for asset A, proportionate change = (36-35)/35 = 0.0286

for asset B, proportionate change = (53-52)/52 = 0.0192

According to EWMA model: Today's variance is a function of prior day's variance

σ_n​²(EWMA)=λσ_n²​+(1−λ)u²_n−1​

where:

λ=the degree of weighting decrease

σ²=value of variance at time period n

u²=value of EWMA at time period n​

The new covariance estimate is:

Cov(A,B) = (0.95*0.000099528) + (1-0.95)*0.0286*0.0192

cov(A,B) = 0.000122

The new variance estimate for asset A: σ_n​²(EWMA) = 0.95*0.0165^2 + (1-0.95)*0.0165^2*0.0286^2

σ_n​²(EWMA) = 0.00026

Std dev(A) =  sqrt(variance A) = 0.00026^(1/2) = 0.01612

The new variance estimate for asset B: σ_n​²(EWMA) = 0.95*0.0232^2 + (1-0.95)*0.0232^2*0.0192^2

σ_n​²(EWMA) for asset B = 0.0005

Std dev(B) =  sqrt(variance B) = 0.0005^(1/2) = 0.02236

The new correlation estimate is:

Corr(A,B) = Cov(A,B) / (Std dev(A)*Std Dev(B))

Corr(A,B) = 0.000122 / (0.01612*0.02236)

Corr(A,B) = 0.338