Accurate asset price modeling and related statistical problems under microstructure noise
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1 Accurate asset prce modelng and related statstcal problems under mcrostructure nose José E. Fgueroa-López 1 1 Department of Statstcs Purdue Unversty Explorng Statstcal Scence Research Semnar December 8, 2009
2 Outlne 1 Motvaton Asset prce modelng Stochastc models for asset prces 2 Hgh-frequency based statstcs General dea Examples 3 Market Mcrostructure 4 Open problems
3 Motvaton Asset prce modelng Outlne 1 Motvaton Asset prce modelng Stochastc models for asset prces 2 Hgh-frequency based statstcs General dea Examples 3 Market Mcrostructure 4 Open problems
4 Motvaton Asset prce modelng How does the prce of a stock behaves n tme?
5 Motvaton Asset prce modelng How does the prce of a stock behaves n tme?
6 Motvaton Asset prce modelng How does the prce of a stock behaves n tme?
7 Motvaton Asset prce modelng How does the prce of a stock behaves n tme?
8 Motvaton Asset prce modelng How does the prce of a stock behaves n tme?
9 Motvaton Asset prce modelng Stylzed emprcal features of stock prces 1 Sudden bg" changes n the prce levels (Jumps) 2 Volatlty clusterng (ntermttency) 3 Log returns wth heavy-tals and hgh-kurtoss dstrbutons 4 Leverage phenomenon
10 Motvaton Stochastc models for asset prces Outlne 1 Motvaton Asset prce modelng Stochastc models for asset prces 2 Hgh-frequency based statstcs General dea Examples 3 Market Mcrostructure 4 Open problems
11 Motvaton Stochastc models for asset prces What are good models for the prce process {S t } t 0?
12 Motvaton Stochastc models for asset prces What are good models for the prce process {S t } t 0? Black-Scholes model (1973); Samuelson (1965): R (δ) = log S (+1)δ S δ = bδ + δv Z, where Z..d. N (0, 1). Jgglng moton, no jump-lke changes, no clusterng or leverage
13 Motvaton Stochastc models for asset prces What are good models for the prce process {S t } t 0? Stochastc volatlty Heston model (1993): R (δ) = bδ + δ v 1 δ Z, v δ = v δ 1 + α(m v δ 1)δ + γ δv δ 1 Z, ρ = Corr(Z, Z ).
14 Motvaton Stochastc models for asset prces What are good models for the prce process {S t } t 0? Stochastc volatlty wth jumps: R (δ) = bδ + δ v δ v δ where 0 < β < 2 and J (β) Remarks: 1 Z + θδ 1/β J (β) = v δ 1 + α(m v δ 1)δ + γ, δv δ 1 Z. are..d. symmetrc wth heavy tals: P(J (β) x) cx β, as x. P(Z x) e x /x; hence, J feels lke jumps compared to Z ; The larger β, the lghter the tals, and the smaller J (β) β s called the ndex of jump actvty. wll tend to be;
15 Motvaton Stochastc models for asset prces What are good models for the prce process {S t } t 0? Stochastc volatlty wth jumps: R (δ) = bδ + δ v δ v δ, 1 Z + θδ 1/β J (β) = v δ 1 + α(m v δ 1)δ + γ δv δ 1 Z.
16 Motvaton Stochastc models for asset prces What are good models for the prce process {S t } t 0? Stochastc volatlty wth jumps: R (δ) = bδ + δ v δ v δ, 1 Z + θδ 1/β J (β) = v δ 1 + α(m v δ 1)δ + γ δv δ 1 Z.
17 Motvaton Stochastc models for asset prces What are good models for the prce process {S t } t 0? Stochastc volatlty wth jumps: R (δ) = bδ + δ v δ v δ 1 Z + θδ 1/β J (β) = v δ 1 + α(m v δ 1)δ + γ, δv δ 1 Z.
18 Hgh-frequency based statstcs General dea Outlne 1 Motvaton Asset prce modelng Stochastc models for asset prces 2 Hgh-frequency based statstcs General dea Examples 3 Market Mcrostructure 4 Open problems
19 Hgh-frequency based statstcs General dea Estmaton methods based on hgh-frequency data Set-up: We collect n log returns at ntervals of sze δ: for = 1,..., n. Overnght returns are fltered out; R (δ) := log S δ S ( 1)δ, The samplng tmes are then t = δ and the tme horzon s T = nδ. What s t? Any estmator ˆθ δ,n whch s consstent for a parameter" θ when δ 0: ˆθ δ,n P θ, as δ 0, keepng T fxed!
20 Hgh-frequency based statstcs General dea Estmaton methods based on hgh-frequency data Set-up: We collect n log returns at ntervals of sze δ: for = 1,..., n. Overnght returns are fltered out; R (δ) := log S δ S ( 1)δ, The samplng tmes are then t = δ and the tme horzon s T = nδ. What s t? Any estmator ˆθ δ,n whch s consstent for a parameter" θ when δ 0: ˆθ δ,n P θ, as δ 0, keepng T fxed!
21 Hgh-frequency based statstcs General dea Estmaton methods based on hgh-frequency data Set-up: We collect n log returns at ntervals of sze δ: for = 1,..., n. Overnght returns are fltered out; R (δ) := log S δ S ( 1)δ, The samplng tmes are then t = δ and the tme horzon s T = nδ. What s t? Any estmator ˆθ δ,n whch s consstent for a parameter" θ when δ 0: ˆθ δ,n P θ, as δ 0, keepng T fxed!
22 Hgh-frequency based statstcs Examples Outlne 1 Motvaton Asset prce modelng Stochastc models for asset prces 2 Hgh-frequency based statstcs General dea Examples 3 Market Mcrostructure 4 Open problems
23 Hgh-frequency based statstcs Examples Examples 1 Quadratc varaton of the contnuous component: v t := lm v δ 0 1δ, δ 0 t T. :δ t Then, for any α > 0 and 0 > ω < 1/2, v δ,n,α,ω t := v δ t 2 Index of jump actvty β: β δ,n,α,α,ω := β δ := := log :δ t R (δ) 2 1 n o P v R (δ) αδ ω t, ( P:δ t R(δ) P:δ t R(δ) 1 { R (δ) 1 { R (δ) log(α ) αδ ω } α αδ ω } ) P β.
24 Hgh-frequency based statstcs Examples Index of jump actvty for Heston wth stable jumps Stochastc volatlty wth jumps: R (δ) = bδ + δ v δ v δ 1 Z + θδ 1/β J (β) = v δ 1 + α(m v δ 1)δ + γ, δv δ 1 Z.
25 Hgh-frequency based statstcs Examples Index of jump actvty for Heston wth stable jumps
26 Hgh-frequency based statstcs Examples Index of jump actvty for Heston model Stochastc volatlty Heston model: R (δ) = bδ + δ v 1 δ Z, v δ = v δ 1 + α(m v δ 1)δ + γ δv δ 1 Z.
27 Hgh-frequency based statstcs Examples Index of jump actvty for Heston model
28 Hgh-frequency based statstcs Examples Jump ndex for Heston wth tme-changed NIG jumps Model: R (δ) = bδ + δ v δ v δ 1 ), 1 Z + J (δv δ = v δ 1 + α(m v δ 1)δ + γ δv δ 1 Z, Var(J (t) ) = t, J (t)..d. "Heavy tal dstrbuton", β = 1.
29 Hgh-frequency based statstcs Examples Jump ndex for Heston wth tme-changed NIG jumps
30 Hgh-frequency based statstcs Examples Jump ndex for pure tme-changed NIG jumps Model: R (δ) v δ = bδ + J (δv δ 1 ), = v δ 1 + α(m v δ 1)δ + γ δv δ 1 Z, Var(J (t) ) = t, J (t)..d. "Heavy tal dstrbuton", β = 1.
31 Hgh-frequency based statstcs Examples Jump ndex for pure tme-changed NIG jumps
32 Hgh-frequency based statstcs Examples Index of jump actvty for INTC wth 5-sec returns
33 Hgh-frequency based statstcs Examples Index of jump actvty for INTC wth 5-sec returns
34 Hgh-frequency based statstcs Examples Index of jump actvty for INTC wth 15-sec returns
35 Hgh-frequency based statstcs Examples Index of jump actvty for INTC wth 15-sec returns
36 Market Mcrostructure Market mcrostructure features 1 Hgh-frequency estmaton depend heavly on an accurate descrpton of the stock prce evoluton at a very small-tme scale. 2 However, real stock prces exhbt several features nherted from the way tradng takes place n the market: () Nontradng effects () Clusterng nose e.g. Prces tend to fall more often on whole-dollar multples than on half-dollar multples, or than 1/4-dollar multples, etc. () Bd/ask bounce effect Recorded stock prces can be at the bd or at the ask prces. Bd/ask prce bouncng creates spurous correlaton n returns. 3 One modelng approach: Mcrostructure nose
37 Market Mcrostructure Market mcrostructure features 1 Hgh-frequency estmaton depend heavly on an accurate descrpton of the stock prce evoluton at a very small-tme scale. 2 However, real stock prces exhbt several features nherted from the way tradng takes place n the market: () Nontradng effects () Clusterng nose e.g. Prces tend to fall more often on whole-dollar multples than on half-dollar multples, or than 1/4-dollar multples, etc. () Bd/ask bounce effect Recorded stock prces can be at the bd or at the ask prces. Bd/ask prce bouncng creates spurous correlaton n returns. 3 One modelng approach: Mcrostructure nose
38 Market Mcrostructure Market mcrostructure features 1 Hgh-frequency estmaton depend heavly on an accurate descrpton of the stock prce evoluton at a very small-tme scale. 2 However, real stock prces exhbt several features nherted from the way tradng takes place n the market: () Nontradng effects () Clusterng nose e.g. Prces tend to fall more often on whole-dollar multples than on half-dollar multples, or than 1/4-dollar multples, etc. () Bd/ask bounce effect Recorded stock prces can be at the bd or at the ask prces. Bd/ask prce bouncng creates spurous correlaton n returns. 3 One modelng approach: Mcrostructure nose
39 Market Mcrostructure Market mcrostructure features 1 Hgh-frequency estmaton depend heavly on an accurate descrpton of the stock prce evoluton at a very small-tme scale. 2 However, real stock prces exhbt several features nherted from the way tradng takes place n the market: () Nontradng effects () Clusterng nose e.g. Prces tend to fall more often on whole-dollar multples than on half-dollar multples, or than 1/4-dollar multples, etc. () Bd/ask bounce effect Recorded stock prces can be at the bd or at the ask prces. Bd/ask prce bouncng creates spurous correlaton n returns. 3 One modelng approach: Mcrostructure nose
40 Market Mcrostructure Market mcrostructure features 1 Hgh-frequency estmaton depend heavly on an accurate descrpton of the stock prce evoluton at a very small-tme scale. 2 However, real stock prces exhbt several features nherted from the way tradng takes place n the market: () Nontradng effects () Clusterng nose e.g. Prces tend to fall more often on whole-dollar multples than on half-dollar multples, or than 1/4-dollar multples, etc. () Bd/ask bounce effect Recorded stock prces can be at the bd or at the ask prces. Bd/ask prce bouncng creates spurous correlaton n returns. 3 One modelng approach: Mcrostructure nose
41 Market Mcrostructure Market mcrostructure features 1 Hgh-frequency estmaton depend heavly on an accurate descrpton of the stock prce evoluton at a very small-tme scale. 2 However, real stock prces exhbt several features nherted from the way tradng takes place n the market: () Nontradng effects () Clusterng nose e.g. Prces tend to fall more often on whole-dollar multples than on half-dollar multples, or than 1/4-dollar multples, etc. () Bd/ask bounce effect Recorded stock prces can be at the bd or at the ask prces. Bd/ask prce bouncng creates spurous correlaton n returns. 3 One modelng approach: Mcrostructure nose
42 Market Mcrostructure Market mcrostructure features 1 Hgh-frequency estmaton depend heavly on an accurate descrpton of the stock prce evoluton at a very small-tme scale. 2 However, real stock prces exhbt several features nherted from the way tradng takes place n the market: () Nontradng effects () Clusterng nose e.g. Prces tend to fall more often on whole-dollar multples than on half-dollar multples, or than 1/4-dollar multples, etc. () Bd/ask bounce effect Recorded stock prces can be at the bd or at the ask prces. Bd/ask prce bouncng creates spurous correlaton n returns. 3 One modelng approach: Mcrostructure nose R Obs = R (δ) + ε, where ε s a statonary sequence.
43 Market Mcrostructure Market mcrostructure features 1 Hgh-frequency estmaton depend heavly on an accurate descrpton of the stock prce evoluton at a very small-tme scale. 2 However, real stock prces exhbt several features nherted from the way tradng takes place n the market: () Nontradng effects () Clusterng nose e.g. Prces tend to fall more often on whole-dollar multples than on half-dollar multples, or than 1/4-dollar multples, etc. () Bd/ask bounce effect Recorded stock prces can be at the bd or at the ask prces. Bd/ask prce bouncng creates spurous correlaton n returns. 3 One modelng approach: Mcrostructure nose R Obs = Quantzaton(R (δ) + ε ), where ε s a statonary sequence.
44 Open problems Estmaton under mcrostructure nose" 1 Ultra hgh-frequency samplng wll eventually recover the tck-by-tck data. 2 How frequently to sample? The hgher samplng frequency, the smaller the theoretcal standard error of the estmaton methods (under absence of nose), but the hgher the mcrostructure nose. 3 Need to analyze the performance of the methods towards mcrostructure nose (or other knd of nose). 4 Need to devse methods that are robust aganst marker mcrostructure nose. 5 There are some models for tck-by-tck data, but the brdge between these models and semmartngale models s not well-understood yet.
45 Open problems Estmaton under mcrostructure nose" 1 Ultra hgh-frequency samplng wll eventually recover the tck-by-tck data. 2 How frequently to sample? The hgher samplng frequency, the smaller the theoretcal standard error of the estmaton methods (under absence of nose), but the hgher the mcrostructure nose. 3 Need to analyze the performance of the methods towards mcrostructure nose (or other knd of nose). 4 Need to devse methods that are robust aganst marker mcrostructure nose. 5 There are some models for tck-by-tck data, but the brdge between these models and semmartngale models s not well-understood yet.
46 Open problems Estmaton under mcrostructure nose" 1 Ultra hgh-frequency samplng wll eventually recover the tck-by-tck data. 2 How frequently to sample? The hgher samplng frequency, the smaller the theoretcal standard error of the estmaton methods (under absence of nose), but the hgher the mcrostructure nose. 3 Need to analyze the performance of the methods towards mcrostructure nose (or other knd of nose). 4 Need to devse methods that are robust aganst marker mcrostructure nose. 5 There are some models for tck-by-tck data, but the brdge between these models and semmartngale models s not well-understood yet.
47 Open problems Estmaton under mcrostructure nose" 1 Ultra hgh-frequency samplng wll eventually recover the tck-by-tck data. 2 How frequently to sample? The hgher samplng frequency, the smaller the theoretcal standard error of the estmaton methods (under absence of nose), but the hgher the mcrostructure nose. 3 Need to analyze the performance of the methods towards mcrostructure nose (or other knd of nose). 4 Need to devse methods that are robust aganst marker mcrostructure nose. 5 There are some models for tck-by-tck data, but the brdge between these models and semmartngale models s not well-understood yet.
48 Open problems Estmaton under mcrostructure nose" 1 Ultra hgh-frequency samplng wll eventually recover the tck-by-tck data. 2 How frequently to sample? The hgher samplng frequency, the smaller the theoretcal standard error of the estmaton methods (under absence of nose), but the hgher the mcrostructure nose. 3 Need to analyze the performance of the methods towards mcrostructure nose (or other knd of nose). 4 Need to devse methods that are robust aganst marker mcrostructure nose. 5 There are some models for tck-by-tck data, but the brdge between these models and semmartngale models s not well-understood yet.
49 Open problems Estmaton under mcrostructure nose" 1 Ultra hgh-frequency samplng wll eventually recover the tck-by-tck data. 2 How frequently to sample? The hgher samplng frequency, the smaller the theoretcal standard error of the estmaton methods (under absence of nose), but the hgher the mcrostructure nose. 3 Need to analyze the performance of the methods towards mcrostructure nose (or other knd of nose). 4 Need to devse methods that are robust aganst marker mcrostructure nose. 5 There are some models for tck-by-tck data, but the brdge between these models and semmartngale models s not well-understood yet.
50 Appendx Graphs Emprcal dstrbuton of returns Return durng a gven tme perod = log Fnal prce Intal prce. Back
51 Appendx Graphs Dynamcs of the prce process Back
52 Appendx Graphs Tmes seres of returns Fgures taken from Cont (2001) Back
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