1 On the Board Week 7
If”N(0;1),then”2 2(1):E ”2=1=Var(“). In volatility models, try to predict “2:
2t =Et ”2t+1:
2t+1 =v+ 2t v+!t+1;
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whatís the process for “2t+1? Recall: “2t+1 = 2t + noise. We showed in an earlier HW that in this case “2t+1 is an ARMA(1,1) process.
To see this it is easiest to consider the special case where the shocks to realized and expected returns are perfectly (negatively) correlated. Let
rt+1 = xt+”t+1; xt+1 = xt x”t+1:
Here, expected returns xt follows an AR(1). Letís Önd process for rt+1:
= xt 1 x”t + “t+1
= (rt ”t) x”t +”t+1
= rt (+x)”t +”t+1
= 1rt 1″t + “t+1;
where 1 = , 1 = + x: So, returns follow an ARMA(1,1) process in this case. This is true also for the more general case where the correlation between shocks to realized and expected returns is not perfect. This case is covered in Homework 3.
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