Ecoomics 883 Sprig 205 Tauche Tradig Frictio Noise Setup Let Y be the usual cotiuous semi-martigale Y t = t 0 cs dw s We will cosider jump discotiuities later. The usual setup for modelig tradig frictio is the observed X is Y plus oise: X i = Y i + χ i 2 where χ i is a statioary mea zero process with variace Varχ i = σ 2 χ. The the icremets i X are i X = i Y + χ i χ i 3 Note X is ot a semi-martigale ad its icremets have a additioal MA piece. The oise imparts a bias o the realized variace E i X 2 = c s ds + 2σχ 2 4 i= The oise overwhelms the sigal i the limit. The mistake is samplig too fie, i.e., takig the semi-martigale model seriously at the highest frequecies a very bad idea i practice.. Coarse Samplig The simplest, ad model free, way to hadle the oise is ot to sample to fiely. Sometimes this strategy is called course samplig. Let k be a positive iteger. Course samplig is oly usig the prices X ik, i =, 2,..., /k 5 For example, if = /400 about oe miute, σ χ = 0.020, ad IV =.25, the E RV = E i= i X 2 = IV + 2σ 2 χ: 0 IV =.2500, ERV IV =.570, ERV IV /IV = 0.256 6 the -miute RV is systematically off ceter by 26 percet. O the other had, with 5-mi samplig Possible typos remai IV =.2500, ERV IV =.32, ERV IV /IV = 0.052 7
so RV is oly off by about 5 percet, a very small amout relative to the estimatio error i RV. With 0-miute samplig, IV =.2500, ERV IV =.282, ERV IV /IV = 0.0256 8 so RV is oly off by a egligible 2 percet. The message is that coarse samplig early elimiates the oise problem without puttig ay further structure o χ i. Note that X ik X i k = a j i jx, a j. 9 Thus coarse samplig is a form of pre-averagig o o-overlappig itervals. 2 R-MSE Iitial Look I the presece of tradig frictio oise the realized variace, ad other measures of variatio, ca be biased or off ceter. The time-hoored stadard for measure accuracy i the this case is the root mea squared error: R-MSE = E ˆ IV IV 2 = Var ˆ IV + E ˆ IV IV 2 0 where it is uderstood throughout that momets are coditioal o IV, E IV, Var IV, but the coditioal o IV is suppressed to reduce clutter i otatio. The bias, ad thus squared bias, ca usually straightforward to compute but the variace is more ivolved with more terms to track i all of the sums. A little classical time series comes i hady. 3 A Brief Remider from Time Series Some expressios usually used i stadard macro time series are sometimes helpful. Let x t be a discrete time covariace statioary process. Symbols like x ad t etc used i this sectio have othig to with the high-frequecy otatio. Put c j = Cov x t, x t j. Here x t is scalar but the oly thig that chages i the vector case is that c j is a matrix ad c j = c j; also, x t is defied for all periods, t = 2,, 0,, 2,... ad likewise c j is defied over j = 2,, 0,, 2,.... The covariace polyomial is cl = L j, cl = c 0 + c j L j + c j L j 2 j= j= 2
I geeral, cl is a ifiite polyomial over positive ad egative powers of the lag operator L. A coveiet result is that if bl is a polyomial i L, usually oly o-egative powers such as bl = b 0 + b L + b 2 L 2 +, ad if where ɛ t is white oise, the the covariace polyomial of x t is x t = blɛ t 3 cl = blbl σ 2 ɛ. 4 Thus if bl defies a fiite movig average, b j = 0 for j > K, K <, the cl ivolves at most powers of L ad L up to degree K: cl = K j= K The expressio for the variace of the sum is Var x j = c 0 + If c j = 0 for j > K, the j= Var x j = c 0 + j= c l L j 5 j= + K j= K 4 Back to Tradig Frictio Noise j c j 6 j c j 7 Some researchers may? argue that coarse samplig throws away data ad is therefore iefficiet for some reaso. A alterative is to use the very high frequecy data but try to average out the oise. This alterative is called pre-averagig. The idea is to form a local average of the log-price series by way of j= a j X i j 8 From the liearity it is obvious that the icremets i X are just weighted sums of the icremets i X of X. So the pre-averaged returs process, a j i jx 9 3
is the essetially but ot quite the retur o the pre-averaged price. 2 The pre-averaged returs is the cumulative geometric retur o a particular simple mechaical tradig strategy. If the effect of tradig frictios is to impart ad additive statioary error as i the the icremets are X i = Y i + χ i 20 i X = i Y + χ i χ i 2 The above is simple, sigal plus first differece of a statioary process, but etails very strog assumptios about how tradig frictios work out i actual markets. Cosider the pre-averaged retur, i a j i j X 22 I order to preserve variatio we impose /k k a2 j =. The i a j i jy + a j u i j 23 = i Ȳ + ū i What is a good choice of the pre-averagig weights, b j? At this poit it becomes plug ad grid. There is o harm i assumig that that the Y process has equal local variace C s = σ 2. Also, for the momet assume the tradig frictio oise χ i is white oise. The estimate of IV = σ 2 based o the overlappig pre-averaged data is S = k 2 i X 24 To aalyze the MSE we eed the mea ad variace of S. For the mea, E S = k E i Ȳ 2 + k E { [bl Lχ i ] 2} 25 k The first term is /k b2 j σ 2 = σ 2. To get a hadle o the secod term let cl = bl L L bl = k k c j L j 26 2 Oe might thik of averagig the price levels ad the takig logs, but we thik of pre-averagig the log-price itself vitiatig the otio of the retur o the pre-averaged price iterpretatio literally. 4
The secod term is /k c 0 where c 0 is the coefficiet of L 0 i the polyomial cl. The terms {c j } k k ca be determied from the b j but the algebra is tedious. E S = σ 2 + c 0 /k 27 Keep i mid that c 0 depeds upo the b j ad k. As for variace [ Var S = Var E i k Ȳ ] 2 + Var k k b j χ i j χ i j 2 28 The ideas are simple but the algebra tedious. 5