Volatility. Roberto Renò. 2 March 2010 / Scuola Normale Superiore. Dipartimento di Economia Politica Università di Siena

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1 Dipartimento di Economia Politica Università di Siena 2 March 2010 / Scuola Normale Superiore

2 What is? The definition of volatility may vary wildly around the idea of the standard deviation of price movements Let X t be a stochastic process (log-price or interest rate) Representation theorem + smoothness conditions : Ito s semimartingales: (see Jacod and Shiraev, Theorem I.4.18) dx t = µ t dt + σ t dw t + dj t W t is a standard Brownian motion J t is a jump process with jump measure µ(dx, dt) and compensator ν(dt) is the stochastic process σ t

3 What is? The definition of volatility may vary wildly around the idea of the standard deviation of price movements Let X t be a stochastic process (log-price or interest rate) Representation theorem + smoothness conditions : Ito s semimartingales: (see Jacod and Shiraev, Theorem I.4.18) dx t = µ t dt + σ t dw t + dj t W t is a standard Brownian motion J t is a jump process with jump measure µ(dx, dt) and compensator ν(dt) is the stochastic process σ t

4 What is? The definition of volatility may vary wildly around the idea of the standard deviation of price movements Let X t be a stochastic process (log-price or interest rate) Representation theorem + smoothness conditions : Ito s semimartingales: (see Jacod and Shiraev, Theorem I.4.18) dx t = µ t dt + σ t dw t + dj t W t is a standard Brownian motion J t is a jump process with jump measure µ(dx, dt) and compensator ν(dt) is the stochastic process σ t

5 What is? The definition of volatility may vary wildly around the idea of the standard deviation of price movements Let X t be a stochastic process (log-price or interest rate) Representation theorem + smoothness conditions : Ito s semimartingales: (see Jacod and Shiraev, Theorem I.4.18) dx t = µ t dt + σ t dw t + dj t W t is a standard Brownian motion J t is a jump process with jump measure µ(dx, dt) and compensator ν(dt) is the stochastic process σ t

6 Financial Returns SPX500 index, , frequency: one minute 4 x x 10 3

7 Financial Returns are not Normal 1 Number of returns analyzed: Unconditional returns Normal distribution

8 Standardized Financial Returns are not Normal? 1 Number of returns analyzed: Conditional (standardized) returns Unconditional returns Normal distribution

9 Standardized Financial Returns are not Normal? Number of returns analyzed: Conditional (standardized) returns Unconditional returns Normal distribution

10 Evenly sampled returns Consider an interval [0, T] with T fixed. Consider a real number = T/n. We define the evenly sampled returns as: j X = X j X (j 1), j = 1,...,n The case with unevenly sampled returns is similar (even if times are stochastic, but not in the case they are not independent from price)

11 Evenly sampled returns Consider an interval [0, T] with T fixed. Consider a real number = T/n. We define the evenly sampled returns as: j X = X j X (j 1), j = 1,...,n The case with unevenly sampled returns is similar (even if times are stochastic, but not in the case they are not independent from price)

12 Evenly sampled returns Consider an interval [0, T] with T fixed. Consider a real number = T/n. We define the evenly sampled returns as: j X = X j X (j 1), j = 1,...,n The case with unevenly sampled returns is similar (even if times are stochastic, but not in the case they are not independent from price)

13 Which are the Difficulties in Estimating In this seminars, we show that volatility estimates can be heavily distorted by three factors: Jumps Microstructure noise Intraday effects We discuss how to get rid of this three effects

14 Infill Asymptotics We can estimate σ t using infill asymptotics: µ t dt is of order if J t is a Levy process, dj t is of order (stochastic continuity) σ t dw t is of order σ t is the leading order as 0. We can estimate it in a fixed time span [0, T] with observations at times sup i (t i+1 t i ) 0 We need high-frequency data

15 Infill Asymptotics We can estimate σ t using infill asymptotics: µ t dt is of order if J t is a Levy process, dj t is of order (stochastic continuity) σ t dw t is of order σ t is the leading order as 0. We can estimate it in a fixed time span [0, T] with observations at times sup i (t i+1 t i ) 0 We need high-frequency data

16 Infill Asymptotics We can estimate σ t using infill asymptotics: µ t dt is of order if J t is a Levy process, dj t is of order (stochastic continuity) σ t dw t is of order σ t is the leading order as 0. We can estimate it in a fixed time span [0, T] with observations at times sup i (t i+1 t i ) 0 We need high-frequency data

17 Integrated volatility From the above discussion, it is clear that estimating σ t is similar to the problem of numerical differentiation A possible solution is concentrating on integrated volatility: T 0 σ 2 sds Integrated volatility plays a fundamental role in: option pricing portfolio selection volatility forecasting

18 Integrated volatility From the above discussion, it is clear that estimating σ t is similar to the problem of numerical differentiation A possible solution is concentrating on integrated volatility: T 0 σ 2 sds Integrated volatility plays a fundamental role in: option pricing portfolio selection volatility forecasting

19 Integrated volatility From the above discussion, it is clear that estimating σ t is similar to the problem of numerical differentiation A possible solution is concentrating on integrated volatility: T 0 σ 2 sds Integrated volatility plays a fundamental role in: option pricing portfolio selection volatility forecasting

20 Realized Variance Realized variance is defined as: RV (X) t = [t/ ] j=1 ( j X ) 2 It is the cumulative sum of intraday returns It depends on ; the smaller the, the closer the approximation to quadratic variation For financial data, cannot be smaller than the average time between transactions.

21 Realized Variance Realized variance is defined as: RV (X) t = [t/ ] j=1 ( j X ) 2 It is the cumulative sum of intraday returns It depends on ; the smaller the, the closer the approximation to quadratic variation For financial data, cannot be smaller than the average time between transactions.

22 Convergence of Realized Variance As 0, if J = 0, we have: ( t ) 1 2 RV (X) t σsds 2 L t 2 σsdw 2 s 0 0 where W is a Brownian motion uncorrelated with W. If J 0 a different result applies (assume J is Poisson): RV (X) t t p 0 N t σsds 2 + i=1 c 2 i

23 Convergence of Realized Variance As 0, if J = 0, we have: ( t ) 1 2 RV (X) t σsds 2 L t 2 σsdw 2 s 0 0 where W is a Brownian motion uncorrelated with W. If J 0 a different result applies (assume J is Poisson): RV (X) t t p 0 N t σsds 2 + i=1 c 2 i

24 Disentangling diffusion from jumps When we observe variations in the level of any state variable, are these variations continouous or discontinuous? Two methods: Bipower Variation (Barndorff-Nielsen and Shephard, 2006) Threshold method (Mancini, 2009)

25 Disentangling diffusion from jumps When we observe variations in the level of any state variable, are these variations continouous or discontinuous? Two methods: Bipower Variation (Barndorff-Nielsen and Shephard, 2006) Threshold method (Mancini, 2009)

26 Power Variation The realized power variation of order γ is defined as: PV (X) γ t [t/ ] = 1 γ/2 j=1 j X γ the normalization term 1 γ/2 disappears when γ = 2, goes to infinity when γ > 2 and goes to zero when γ < 2. when γ = 2 PV coincides with RV.

27 Power Variation The realized power variation of order γ is defined as: PV (X) γ t [t/ ] = 1 γ/2 j=1 j X γ the normalization term 1 γ/2 disappears when γ = 2, goes to infinity when γ > 2 and goes to zero when γ < 2. when γ = 2 PV coincides with RV.

28 Power variation of continuous semimartingales When J = 0 and σ, µ are independent of W, power variation converges to the integrated power-volatility: (J = 0) p lim 0 PV (X) γ t = µ γ t 0 σ γ s ds where. µ γ = E( u γ ) = 2 u N(0, 1) p+1 γ/2γ( 2 ) Γ(1/2)

29 Power variation in the general case In the more general case, we have: p lim 0 µ 1 γ PV (X) γ t = t 0 σγ s ds if γ < 2 t 0 σ2 sds + N t i=1 c2 i if γ = 2 + if γ > 2 However, it is impossible to get an estimate of the integrated volatility using power variation.

30 Bipower Variation The [1, 1]-order bipower variation, when it exists, is defined as: [t/ ] BPV(X) [1,1] t = p lim j 1 X j X 0 j=2 In the case µ = 0 and σ independent from W t, we have: BPV(X) [1,1] t t p µ 2 1 σsds 2 0 where µ 1 = 2 π This result can be extended to the case µ 0

31 Bipower Variation The [1, 1]-order bipower variation, when it exists, is defined as: [t/ ] BPV(X) [1,1] t = p lim j 1 X j X 0 j=2 In the case µ = 0 and σ independent from W t, we have: BPV(X) [1,1] t t p µ 2 1 σsds 2 0 where µ 1 = 2 π This result can be extended to the case µ 0

32 The Modulus of Continuity of the Brownian Motion The second idea to disentangle diffusion from jumps is based on the modulus of continuity of the Brownian motion: r(δ) = 2δ log 1 δ which has the following property, as established by Lévy: P lim sup δ 0 max W(t) W(s) t s δ r(δ) = 1 = 1 It measures the speed at which the Brownian motion shrinks to zero.

33 The intuition When δ 0, diffusive variations go to zero, while jumps do not. Moreover, we know the rate at which the diffusive variations shrink to zero: the modulus of continuity. Thus, we can identify the jumps as those variations which are larger than a suitable threshold ϑ(δ) which goes to zero, as δ 0, slower than r(δ).

34 The intuition When δ 0, diffusive variations go to zero, while jumps do not. Moreover, we know the rate at which the diffusive variations shrink to zero: the modulus of continuity. Thus, we can identify the jumps as those variations which are larger than a suitable threshold ϑ(δ) which goes to zero, as δ 0, slower than r(δ).

35 The intuition When δ 0, diffusive variations go to zero, while jumps do not. Moreover, we know the rate at which the diffusive variations shrink to zero: the modulus of continuity. Thus, we can identify the jumps as those variations which are larger than a suitable threshold ϑ(δ) which goes to zero, as δ 0, slower than r(δ).

36 The Result of Mancini, 2009 Suppose X = Y + J Where Y is a Brownian martingale plus drift and J is a jump process with counting process N with E[N T ] < and time horizon T <. If ϑ(δ) is a real deterministic function such that δ log 1 δ lim ϑ (δ) = 0 and lim δ 0 δ 0 ϑ (δ) = 0 then for P-almost all ω, δ(ω) such that δ < δ(ω) we have i = 1,..., n, I { N=0} (ω) = I {( X) 2 ϑ(δ)} (ω).

37 Threshold Bipower Variation Inspired by the results of Mancini (2009), Corsi, Pirino and Renó (2008) proposed the Threshold Bipower Variation: TBPV t = 2 n 2 t,j X t,j+1 X I π { t,j X 2 I ϑ j 1 } { t,j+1 X 2 ϑ j } j=0 where ϑ t is a threshold function proportional to the local variance. It combines threshold estimation for large jumps and bipower variation for smaller jumps (twin blade). It is expected to be not too sensitive to the threshold.

38 CLT for TMPV in presence of jumps Theorem (Corsi, Pirino and Renò, 2008) If J 0, assume that θ t = c t Θ(δ), δ log δ where Θ(δ) is a real function satisfying Θ(δ) 0 and 0, and c t is a stochastic process on [0, T] Θ t which is a.s. bounded and with a strictly positive lower bound. Then, as δ 0: TMPV δ (X) [γ 1,...,γ M ] t 0 1 p MY Z µ γk A k=1 t σ γ γ M s ds (1) where the above convergence is in probability, and δ 1 2 TMPV δ (X) t Z t+t t «σ γ γ M s ds c γ Z t+t t σ γ γ M s dw s (2) where W is a Brownian motion independent on W.

39 Simulation study we simulate a one-factor jump-diffusion model with stochastic volatility: SV1FJ dx t = µ dt + v t dw x,t + dj t, d log v t = (α β log v t ) dt + ηdw v,t, W x and W v are standard Brownian motions with corr(dw y, dw v ) = ρ, dj is a compound Poisson process with constant activity λ and random jump size which is Normally distributed with zero mean and standard deviation σ J. We use model parameters estimated on S&P500 by Andersen, Benzoni and Lund (2002) Results: Bias Estimator R R σs γ ds σ γ s ds jumps conditional to the presence of

40 Simulation Results Estimator Relative bias (%) no jumps one jump two jumps two jumps consecutive BPV ( 0.53) ( 1.74) ( 3.36) ( 21.07) stag BPV ( 0.53) ( 1.72) ( 6.32) ( 2.43) TRV ( 0.49) ( 0.52) ( 0.53) ( 0.52) CTRV ( 0.46) 9.40 ( 0.55) ( 0.61) ( 0.61) TBPV ( 0.56) ( 0.60) ( 0.58) ( 0.58) CTBPV ( 0.53) 7.87 ( 0.62) ( 0.66) ( 0.74) min-rv ( 0.65) 3.65 ( 0.73) ( 2.55) ( 26.21) med-rv ( 0.58) 4.25 ( 0.64) ( 4.74) ( 27.29) QPV ( 1.33) ( 5.41) ( 22.79) ( 88.71) TQV ( 0.91) ( 0.94) ( 1.00) ( 1.00) CTQV ( 1.01) ( 1.53) ( 2.00) ( 2.06) TQPV ( 1.28) ( 1.36) ( 1.32) ( 1.36) CTQPV ( 1.33) ( 1.71) ( 1.94) ( 2.88) TriPV ( 1.24) ( 11.64) ( 94.69) ( ) TTriPV ( 1.21) ( 1.28) ( 1.25) ( 1.28) CTTriPV ( 1.25) ( 1.69) ( 1.95) ( 3.16)

41 The C Tz test (CPR, 2009) We use a correction under the null to test for jumps Then the jump test we use is: CTz = δ 1 2 (RV δ (X) T CTBPV δ (X) T ) RV δ (X) 1 T { }, θ max 1, CTTriPV δ (X) T (CTBPV δ (X) T ) 2 As c ϑ, C Tz z.

42 Is that a jump? Day 04/12/1990, C-Tz=4.505, z=

43 Microstructure noise Financial prices are contaminated by observation noise, also known as microstructure noise. We model it as: Y t is the latent efficient true X t = Y t + ε t price ε t is the noise (rounding, bid-ask, transaction costs, volume effects) We are interested in the volatility of Y t, but we only observe X t

44 Financial Returns Intraday Time Series SPX500 index, , frequency: all data 1.5 x

45 Estimating with Microstructure Noise Consider q non-overlapping subsamples. Two scale estimator (Zhang et al., 2005): V ts = 1 q V subsamples }{{} q R σ 2 s ds+ξ q 1 all data V q }{{} Ξ q σ 2 sds The optimal rate of convergence is n 1 6. See also the kernel estimator of Barndorff-Nielsen et al (2009). Both are not robust to jumps.

46 Estimating with Microstructure Noise Consider q non-overlapping subsamples. Two scale estimator (Zhang et al., 2005): V ts = 1 q V subsamples }{{} q R σ 2 s ds+ξ q 1 all data V q }{{} Ξ q σ 2 sds The optimal rate of convergence is n 1 6. See also the kernel estimator of Barndorff-Nielsen et al (2009). Both are not robust to jumps.

47 Combining both: preaveraged estimator Preaveraging (Jacod et al., 2008) is a two-step procedure for getting rid of microstructure noise First step: compute averaged returns: r i = with k n but k n /n 0 n k n i=1 ( ) i g r i n Second step: apply bipower variation Optimal rate of convergence: n 1/4

48 Combining both: preaveraged estimator Preaveraging (Jacod et al., 2008) is a two-step procedure for getting rid of microstructure noise First step: compute averaged returns: r i = with k n but k n /n 0 n k n i=1 ( ) i g r i n Second step: apply bipower variation Optimal rate of convergence: n 1/4

49 Combining both: preaveraged estimator Preaveraging (Jacod et al., 2008) is a two-step procedure for getting rid of microstructure noise First step: compute averaged returns: r i = with k n but k n /n 0 n k n i=1 ( ) i g r i n Second step: apply bipower variation Optimal rate of convergence: n 1/4

50 Preaveraging SPX500 index, , frquency: all data 1.5 x 10 3 Original returns Preaveraged returns

51 From Integrated to Spot Numerical differentiation (Bandi and Renò, 2008): σ 2 t = φn 0 σ 2 sds φ n with φ n 0 Localization (Kristensen, 2009, Mattiussi and Renò, 2009): σ 2 t = n i=1 1 h n K ( t ti h n ) ( i X) 2 with h n 0 and nh n. It is enough that: ( ) 1 t ti K δ(t) h n h n

52 From Integrated to Spot Numerical differentiation (Bandi and Renò, 2008): σ 2 t = φn 0 σ 2 sds φ n with φ n 0 Localization (Kristensen, 2009, Mattiussi and Renò, 2009): σ 2 t = n i=1 1 h n K ( t ti h n ) ( i X) 2 with h n 0 and nh n. It is enough that: ( ) 1 t ti K δ(t) h n h n

53 Intraday Effect intraday factor :30 12:00 13:30 15:00 Intraday time

54 Intraday Spot Trend Spot volatility

55 Correcting for the Intraday Effect Number of returns analyzed: Conditional (standardized) returns Unconditional returns Normal distribution

56 Correcting for the Intraday Effect 10 0 Number of returns analyzed: Conditional (standardized) returns Unconditional returns Normal distribution Corrected for intraday effect

57 Conclusions To measure volatility, we need to take into account jumps, microstructure noise and intraday effect A complete theory of the joint contribution of the three effects is still missing Neglecting one of them can lead to wrong conclusions in the economic analysis appears to be the most important financial quantity to be modelled.