Volatility Estimation for Trading Strategies

Transcription

1 Volatility Estimation for Trading Strategies Benjamin Bruder Research & Development Lyxor Asset Management, Paris Thierry Roncalli Research & Development Lyxor Asset Management, Paris June 2011 Tung-Lam Dao Research & Development Lyxor Asset Management, Paris Abstract We review in this paper various techniques for estimating the volatility. We start by discussing the estimators based on the range of daily monitoring data then we consider the stochastic volatility model in order to determine the instantaneous volatility. At high trading frequency, the stock prices are fluctuated by an additional noise, so-called the micro-structure noise. This effect comes from the bid-ask bounce due to the short time scale. Within a short time interval, the trading price does not converge to the equilibrium price determined by the supply-demand equilibrium. In the second part, we discuss the effect of the micro-structure noise on the volatility estimation. It is very important topic concerning an enormous field of high-frequency trading. Examples of backtesting on index and stocks will illustrate the efficiency of considered techniques. Keywords: Volatility, voltarget strategy, range-based estimator, high-low estimator, microstructure noise. JEL classification: C0, G11, G17. 1 Introduction Measuring the volatility is one of the most important questions in finance. As stated in its name, volatility is the direct measurement of the risk for a given asset. Under the hypothesis that the realized return follows a Brownian motion, volatility is usually estimated by the standard deviation of daily price movement. As this assumption relates the stock price to the most common object of stochastic calculus, many mathematical work have been carried out on the volatility estimation. With the increasing of the trading data, we can explore more and more useful information in order to improve the precision of the volatility estimator. New class of estimators which are based on the high and low prices was invented. However, in the real world the asset price is just not a simple geometric Brownian process, different effects have been observed including the drift or the opening jump. A general correction scheme based on the combination of various estimators have been studied in order to eliminate these effects. 1

2 As far as the trading frequency increases, we expect that the precision of estimator gets better as well. However, when the trading frequency reaches certain limit 1, new phenomena due to the nonequlibrum of the market emerge and spoil the precision. It is called the micro-structure noise which is characterized by the bid-ask bounce or the transaction effect. Because of this noise, realized variance estimator overestimates the true volatility of the price process. A suggestion based on the use of two different time scales can aim to eliminate this effect. The note is organized as following. In Section II, we review the basic volatility estimator using the variance of realized return (note from B.Bruder article) then we introduce all the variation based on the range estimation. In section III, we discuss how to measure the instantaneous volatility and the effect of the lag by doing the moving-average. In section IV, we discuss the effect of the microstructure on the high frequency volatility. 2 Range-based estimators of volatility 2.1 Range based daily data In this paragraph, we discuss the general characteristics of the asset price and introduce the basic notations which will be used for the rest of the article. Let us assume that the dynamics of asset price follows the habitual Black-Scholes model. We denote the asset price S t which follows a geometric Brownian motion in continuous time: ds t S t = µ t dt + σ t db t (1) Here, µ t is the return or the drift of the process whereas σ t is the volatility. Over the period of T = 1 trading day, the evolution is divided in two time intervals: the first interval with ratio f describes the closing interval (before opening) and the second interval with ratio 1 f describes the opening interval (trading interval). On the monitoring of the data, the closing interval is unobservable and is characterized by the jumps in the opening of the market. The measure of closing interval is not given by the real closing time but the jumps in the opening of the market. If the logarithm of price follows a standard Brownian motion without drift, then the fraction f/ (1 f) is given by the square of ratio between the standard deviation of the opening jump and the daily price movement. We will see that this idea can give a first correction due to the close-open effect for all the estimators discussed below. In order to fix the notation, we define here different quantities concerning the statistics of the price evolution: T is the time interval of 1 trading day f is the fraction of closing period ˆσ 2 t is the estimator of the variance σ 2 t O ti is the closing price on a given period [t i, t i+1 [ C ti is the closing price on a given period [t i, t i+1 [ H ti = max t [ti,t i+1[ S t is the highest price on a given period [t i, t i+1 [ 1 This limit defines the optimal frequency for the classical estimator. It is more and less agreed to be one trade every 5 minutes.

3 Figure 1: Data set of 1 trading day Trading open Trading closed Trading open H ti C ti 1 O ti C ti O ti 1 Yesterday s open Yesterday s close Today s open L ti Today s close L ti = min t [ti,t i+1[ S t is the lowest price on a given period [t i, t i+1 [ o ti = ln O ti ln C ti 1 is the opening jump u ti = ln H ti ln O ti is the highest price movement during the trading open d ti = ln L ti ln O ti is the lowest price movement during the trading open c ti = ln C ti ln O ti is the daily price movement over the trading open period 2.2 Basic estimator For the sake of simplicity, let us start this paragraph by assuming that there is no opening jump f = 0. The asset price S t described by the process (1) is observed in a series of discrete dates {t 0,..., t n }. In general, this series is not necessary regular. Let R ti be the realized return in the period [t i 1, t i [, then we obtain: ti R ti = ln S ti ln S ti 1 = (σ u db u + µ u du 12 ) σ2u du t i 1 In the following, we assume that the couple (µ t, σ t ) is independent to the Brownian motion B t of the asset price evolution Estimator over a given period In appendix A, we show that the realized return R ti is related to the volatility as: E [ Rt 2 i σ, µ ] ( = (t i t i 1 )σt 2 i + (t i t i 1 ) 2 µ ti 1 1 ) 2 2 σ2 t i 1 This quantity can not be a good estimator of volatility because its standard deviation is 2 (ti+1 t i )σt 2 i which is proportional to the estimated quantity. In order to reduce the estimation error, we focus on the estimation of the average volatility over the period t n t 0. The average volatility is defined as: σ 2 = 1 t n t 0 tn t 0 σ 2 u du (2)

4 This quantity can be measured by using the canonical estimator defined as: ˆσ 2 1 = t n t 0 The variance of this estimator is approximated as var (ˆσ 2) 2σ 4 /n or the standard deviation is proportional to 2σ 2 / n. It means that the estimation error ) is small if n is large enough. Indeed the variance of the average volatility reads var( ˆσ 2 σ 2 / (2n) and the standard deviation is approximated to σ/ 2n Effect of the weight distribution In general, we can define an estimator with a weight distribution w i such as: ˆσ 2 = n n w i Rt 2 i R 2 t i then the expectation value of the estimator is given by: E [ˆσ 2 σ, µ ] = n ti t i 1 w i σ 2 u du A simple example of the general definition is the estimator with annualized return R i / t i+1 t i. In this case, our estimator becomes: for which the expectation value is: ˆσ 2 = 1 n E [ˆσ 2 σ, µ ] = n n R 2 t i t n t 1 1 t i t i 1 ti t i 1 σ 2 u du (3) We remark that if the time step (time increment) is constant t i t i 1 = T, then we obtain the same result as the canonical estimator. However, if the time step t i t i 1 is not constant, the long-term return is underweighted while the short-term return is overweighted. We will see in the next discussion on the realized volatility, the way of choosing the weight distribution can help to improve the quality of the estimator. For example, we will show that the IGARCH estimation can lead to an exponential weight distribution which is more appropriate to estimate the realized volatility Close to close, open to close estimators As discussed above, the volatility can be obtained by an using moving-average on discrete ensemble data. The standard measurement is to employ the above result of the canonical estimator for the closing prices (so-called close to close estimator): ˆσ 2 CC = 1 (n 1)T n ((o ti + c ti ) (o + c)) 2 Here, T is the time period corresponding to 1 trading day. In the rest of the paper, we user CC to denote the close to close estimator. We remark that in this formula, there are two

5 different points in comparison to the one defined above. Firstly, we have subtracted the mean value of the closing price (o + c) in order to eliminate the drift effect: o = 1 nt n o ti, c = 1 nt Secondly, the prefactor is now 1/ (n 1)T but not 1/nT. In fact, we have subtracted the mean value then maximum likehood procedure leads to the factor 1/ (n 1)T. We can define also two other volatility estimators which is open to close estimator (OC): ˆσ 2 C = and the close to open estimator (CO): ˆσ 2 O = 1 (n 1)T 1 (n 1)T n c ti n (c ti c) 2 n (o ti o) 2 We remind that o ti is the opening jump for a given trading period, c ti is the daily movement of the asset price such that the close to close return is equal to (o + c). We remark that the close to close estimator does not depend on the drift and the closing interval f. Without presence of the microstructure noise, this estimator is unbiased. Hence, it is usually used as a benchmark to judge the efficiency of other estimators ˆσ which is defined as: eff (ˆσ 2) = var(ˆσ ) CC 2 var(ˆσ 2 ) where var (ˆσ 2) = 2σ 4 /n. The quality of an estimator is determined by its high value of efficiency eff (ˆσ 2) > High-low estimators We have seen that the daily deviation can be used to define the estimator of the volatility. It comes from the fact that one has assumed that the logarithm of price follows a Brownian motion. We all know that the standard deviation in the diffusive process over an interval time t is proportional to σ t, hence using the variance to estimate the volatility is quite intuitive. Indeed, within a given time interval, if additional information of the price movement is available such as the highest value or the lowest value, this range must provide as well a good measure of the volatility. This idea is first addressed by W. Feller in Later, Parkinson (1980) has employed the first result of Feller s work to provide the first high-low estimator (so-called Parkinson estimator). If one uses close prices to estimate the volatility, one can eliminate the effect of the drift by subtracting the mean value of daily variation. By contrast, the use of high and low prices can not eliminate the drift effect in such a simple way. In addition, the high and low prices can be only observed in the opening interval, then it can not eliminate the second effect due to the opening jump. However, as demonstrated in the work of Parkinson (1980), this estimator gives a better confidence but it obviously underestimate the volatility because of the discrete observation of the price. The maximum and minimum value over a time interval t are not the true ones of the Brownian motion. They are underestimated then it is not surprising that the result will depend strongly on the frequency of the price quotation. In the high frequency market, the third effect can be negligible however we will discuss this effect in the later. Because of

6 the limitation of Parkinson s estimator, an other estimator which is also based on the work of Feller was proposed by Kunitomo (1992). In order to eliminate the drift, he construct a Brownian bridge then the deviation of this motion is again related to the diffusion coefficient. In the same line of thought, Rogers and Satchell (1991) propose an other use of high and low prices in order to obtain a drift-independent volatility estimator. In this section, we review the three techniques which are always constrained by the opening jump The Parkinson estimator Let us consider the random variable u ti d ti (namely the range of the Brownian motion over the period [t i, t i+1 [), then the Parkinson estimator is defined by using the following result (Feller 1951): [ E (u d) 2] = (4 ln 2)σ 2 T By inversing this formula, we obtain a natural estimator of volatility based on high and low prices. The Parkinson s volatility estimator is then defined as (Parkinson 1980): ˆσ 2 P = 1 nt n 1 4 ln2 (u t i d ti ) 2 In order to estimate the error of the estimator, we compute the variance of ˆσ P 2 by the following expression: var (ˆσ ( ) P 2 ) 9ζ (3) = 16 (ln 2) 2 1 σ 4 n which is given Here, ζ (x) is the Riemann function. In comparison to the benchmark estimator close to close, we have an efficiency: The Garman-Klass estimator eff (ˆσ P 2 ) 32 (ln 2) 2 = 9ζ (3) 16 (ln 2) 2 = 4.91 Another idea employing the additional information from the high and low value of the price movement within the trading day in order to increase the estimator efficiency was introduced by Garman and Klass (1980). They construct a best analytic scale estimator by proposing a quadratic form estimator and imposing the well-known invariance condition of Brownian motion on the set of variable (u, d, c). By minimizing its variance, they obtain the optimal variational form of quadratic estimator which is given by the following property: [ E (u d) (c (u + d) 2ud) 0.383c 2] = σ 2 T Then the Garman-Klass estimator is defined as: ˆσ 2 GK = 1 nt n ] [0.511 (u ti d ti ) (c ti (u ti + d ti ) 2u ti d ti ) 0.383c 2 ti The minimal value of the variance corresponding to the quadratic estimator is var ( σ 2 GK) = 0.27σ 4 /n and its efficiency is now eff ( σ 2 GK) = 7.4.

7 2.3.3 The Kunitomo estimator Let X t the logarithm of price process X t = ln S t, the Ito theorem gives us its evolution: ( ) dx t = µ t σ2 t dt + σ t db t 2 If the drift term becomes relevant in the estimation of volatility, one can eliminate it by constructing a Brownian bridge on the period T as following: W t = X t t T X T If the initial condition is normalized to X 0 = 0, then by definition we always have X T = 0. This construction eliminates automatically the drift term when its daily variation is small µ ti+1 µ ti µ ti. We define the range of the Brownian bridge D ti = M ti m ti where M ti = max t [ti,t i+1[ W t and m ti = min t [ti,t i+1[ W t. It has been demonstrated that the variance of the range of Brownian bridge is directly proportional to the volatility (Feller 1951): E [ D 2] = Tπ 2 σ 2 /6 (4) Hence, Kunimoto s estimator is defined as following: ˆσ 2 K = 1 nt n 6 π 2 (M t i m ti ) 2 Higher moment of the Brownian bridge can be also calculated analytically and is given by the formula 2.10 in Kunitomo (1992). In particular, the variance of the Kunitomo s estimator is equal to var ( σ 2 K) = σ 4 /5n which implies the efficiency of this estimator eff ( σ 2 K) = The Rogers-Satchell estimator Another way to eliminate the drift effect is proposed by Rogers and Satchell. They consider the following property of the Brownian motion: E[u (u c) + d (d c)] = σ 2 T This expectation value does not depend on the drift of the Brownian motion, hence it does provide a drift-independent estimator which can be defined as: ˆσ 2 RS = 1 nt n [u ti (u ti c ti ) + d ti (d ti c ti )] The variance of this estimator is given by var (ˆσ 2 RS) = 0.331σ 4 /n which gives an efficiency eff (ˆσ 2 RS) = 6. Like the other techniques based on the range high-low, this estimator underestimates the volatility due to the fact that the maximum of a discretized Brownian motion is smaller than the true value. Rogers and Satchell have also proposed a correction scheme which can be generalized for other technique. Let M be the number of quoted price, then h = T/M is the step of the discretization, then the corrected estimator taking account of the finite step error is give by the root of the following equation: ˆσ 2 h = 2bhˆσ 2 h + 2 (u d)a hˆσ h + ˆσ 2 RS where a = 2π [ 1/4 ( 2 1 ) /6 ] and b = (1 + 3π/4)/12.

8 2.4 How to eliminate both drift and opening effects? A common way to eliminate both effects coming from the drift and the opening jump is to combine the various available volatility estimators. The general scheme is to form a linear combination of opening estimator σ O and close estimator σ C or a high-low estimator σ HL. The coefficients of this combination are determined by a minimization procedure on the variance of the result estimator. Given the faction of closing interval f, we can improve all high-low estimators discussed above by introducing the combination: ˆσ 2 = α ˆσ2 O f + (1 α) ˆσ2 HL 1 f Here, the trivial choice is α = f and the estimator becomes independent of the opening jump. However, the optimal value of the coefficient is chosen as α = 0.17 for Parkinson and Kunimoto estimators whereas it value is α = 0.12 for Garman-Klass estimator (Garman and Klass 1980). This technique can eliminate the effect of the opening jump for all estimator but only Kunimoto estimator can avoid both effects. Applying the same idea, Yang and Zhang (2000) have proposed another combination which can also eliminate both effect as Kunimoto estimator. They choose the following combination: ˆσ Y 2 Z = α ˆσ2 O f + 1 α ( κˆσ 2 1 f C + (1 κ) ˆσ HL) 2 In the work of Yang ans Zhang, they have used ˆσ RS 2 as high-low estimator because it is drift independent estimator. The coefficient α will be chosen as α = f and κ is given by optimizing the variance of estimator. The minimization procedure gives the optimal value of the parameter κ: κ o = β 1 β + n+1 n 1 [ where β = E (u (u c) + d (d c)) 2] /σ 4 (1 f) 2. As the numerator is proportional to (1 f) 2, β is in dependent of f. Indeed, the value of β varies not too much (from to 1.5) when the drift is changed. In practice, the value of β is chosen as Numerical simulations Simulation with constant volatility We test various volatility estimators via a simulation of a geometric Brownian motion with constant annualized drift µ = 30% and constant annualized volatility σ = 15%. We realize the simulation based on N = 1000 trading days with M = 50 or 500 intra-day observations in order to illustrate the effect of the discrete price on the family of high-low estimators. Effect of the discretization We first test the effect of the discretization on the various estimators. Here, we take M = 50 or 500 intraday observations with µ = 0 and f = 0. In Figure 2, we present the simulation results for M = 50 price quotation in a trading day. All the high-low estimators are weakly biased due the discretization effect. They all underestimate the volatility as the range of estimator is small than the true range of Brownian motion. We remark that the close-to-close is unbiased however its variance is too large. The correction scheme proposed by Roger and Satchell can eliminate the discretization effect. When the number of observation is larger, the discretization effect is negligible and all estimators are unbiased (see Figure 3).

9 Figure 2: Volatility estimators without drift and opening effects (M = 50) Simulated σ, CC, OC, P, K, GK, RS, RS h, YZ σ (%) Figure 3: Volatility estimators without drift and opening effect (M = 500) Simulated σ, CC, OC, P, K, GK, RS, RS h, YZ σ (%)

10 Effect of the non-zero drift We consider now the case with non-zero annual drift µ = 30%. Here, we take M = 500 intraday observations. In Figure 4, we observe that the Parkinson estimator and the Garman-Klass estimator are strongly dependent on the drift of Brownian motion. Kunimoto estimator and Rogers-Satchell estimator are not dependent on the drift. Figure 4: Volatility estimators with µ = 30% and without opening effect (M = 500) 26 Simulated σ, CC, OC, P, K, GK, RS, RS h, YZ σ (%) Effect of the opening jump For the effect of the opening jump, we simulate data with f = 0.3. In Figure 4, we take M = 500 intraday observations with zero drift µ = 0. We observe that with the effect of the opening jump, all high-low estimator underestimate the volatility except for the YZ estimator. By using the combination between the open volatility estimator ˆσ O 2 with the other estimators, the effect of the opening can be completely eliminated (see Figure 6) Simulation with stochastic volatility We consider now the simulation with stochastic volatility which is described by the following model: { dst = µ t S t dt + σ t S t db t dσt 2 = ξσt 2 dbt σ (5) in which B σ t is a Brownian motion independent to the one of asset process. We will first estimate the volatility with all the proposed estimators then verify the quality of these estimators via a backtest using the voltarget strategy 2. For the simulation of the volatility, we take the same parameters as above with f = 0, µ = 0, N = 5000, 2 The detail description of voltarget strategy is presented in Section Backtest

11 Figure 5: Volatility estimators with opening effect f = 0.3 and without drift (M = 500) Simulated σ, CC, OC, P, K, GK, RS, RS h, YZ σ (%) Figure 6: Volatility estimators with correction of the opening jump (f = 0.3) Simulated σ, CC, OC, P, K, GK, RS, RS h, YZ σ (%)

12 M = 500, ξ = 0.01 and σ 0 = 0.4. In Figure 7, we present the result corresponding to different estimators. We remark that the group of high-low estimators gives a better result for volatility estimation. We can estimate the error committed for each estimator by the Figure 7: Volatility estimators on stochastic volatility simulation Simulated σ, CC, OC, P, K, GK, RS, RS h, YZ σ (%) following formula: ɛ = N (ˆσ t σ t ) 2 The errors obtained for various estimators are summarized in the below Table 1. t=1 Table 1: Estimation error for various estimators Estimator ˆσ 2 CC ˆσ 2 P ˆσ 2 K ˆσ 2 GK ˆσ 2 RS ˆσ 2 Y Z N t=1 (ˆσ σ) We now apply the estimation of the volatility to perform the voltarget strategies. The result of the this test is presented in Figure 8. In order to control the quality of the voltarget strategy, we compute the volatility of the voltarget strategy obtained by each estimator. We remark that the calculation of the volatility on the voltarget strategies is effectuated by the close-to-close estimator with the same averaging window of 3 months (or 65 trading days). The result is reported in Figure 9. As shown in the figure, all estimators give more and less the same results. If we compute the error committed by these estimators, we obtain ɛ CC = , ɛ P = , ɛ K = , ɛ GK = , ɛ RS = , ɛ Y Z = This result may comes form the fact that we have used the close-to-close estimator to calculate the volatility of all voltarget strategies. Hence, we consider another check of the estimation quality. We compute the realized return of the voltarget strategies: R V (t i ) = ln V ti ln V ti 1

13 Figure 8: Test of voltarget strategy with stochastic volatility simulation Benchmark, CC, OC, P, GK, RS, YZ t Figure 9: Test of voltarget strategy with stochastic volatility simulation 25 CC, OC, P, K, GK, RS, YZ 20 σ (%)

14 where V ti is the wealth of the voltarget portfolio. We expect that this quantity follows a Gaussian probability distribution with volatility σ = 15%. Figure 10 shows the probability density function (Pdf) of the realized returns corresponding to all considered estimators. In order to have a more visible result, we compute the different between the cumulative distribution function (Cdf) of each estimator and the expected Cdf (see Figure 11). Both results confirm that the Parkinson and the Kunitomo estimators improve the quality of the volatility estimation. Figure 10: Comparison between different probability density functions Expected Pdf, CC, OC, P, K, GK, RS, YZ Pdf R V 2.6 Backtest Volatility estimations of S&P 500 index We now employ the estimators discussed above for the S&P 500 index. Here, we do not have all tick-by-tick intraday data, hence the Kunimoto s estimator and the Rogers-Satchell correction can not be applied. We remark that the effect of the drift is almost negligible which is confirmed by Parkinson and Garman-Klass estimators. The spontaneous opening jump is estimated simply by: ( ( ) ) 2 ˆσC f t = 1 + ˆσ O We then employ the exponential-average technique to obtain a filter of this quantity. We obtain the average value of closing interval over the considered data for S&P 500 f = and for BBVA SQ Equity f = In the following, we use different estimators in order to extract the signal f t. The trivial one is using f t as the prediction of the opening jump, we denote ˆf t, then we contruct the habitual ones like the moving-average ˆf ma, the

15 Figure 11: Comparison between the different cumulative distribution functions 0.07 CC, OC, P, K, GK, RS, YZ Cdf R V Figure 12: Volatility estimators on S& P 500 index 100 CO, CC, OC, P, GK, RS, YZ σ (%) / / / / / /20011

16 Figure 13: Volatility estimators on on BHI UN Equity 80 CO, CC, OC, P, GK, RS, YZ σ (%) / / / / / /2011 exponential moving-average ˆf exp and the cumulated average ˆf c. In Figure 15, we show result corresponding to different filtered f on the BHI UN Equity data. Figure 13 shows that the family of high-low estimator give a better result than the calissical close-to-close estimator. In order to check the quality of these estimators on the prediction of the volatility, we checke the value of the Likehood function corresponding to each estimator. Assuming that the observable signal follows the Gaussian distribution, the likehood function is defined as: l(σ) = n 2 ln 2π 1 n ln σi n ( ) 2 Ri+1 2 where R is the future realized return. In Figure 17, we present the result of the likehood function for different estimators. This function reaches its maximal value for the Roger- Satchell estimator Backtest on voltarget strategy We now backtest the efficiency of various volatility estimators with vol-target strategy on S&P 500 index and an individual stock. Within the vol-target strategy, the exposition to the risky asset is determined by the following expression: α t = σ ˆσ t where σ is the expected volatility of the strategy and ˆσ t is the prediction of the volatility given by the estimators above. In the backtest, we take the annualized volatility σ = 15% with historical data since 01/01/2001 to 31/12/2011. We present the results for two cases: Backtest on S&P 500 index with moving-average equal to 1 month (n = 21) of historical data. We remark in this case that the volatility of the index is small then the error on σ i

17 Figure 14: Estimation of the closing interval for S&P 500 index 0.15 Realized closing ratio Moving average Exponential average Cummulated average Average 0.1 f / / / / / /2011 Figure 15: Estimation of the closing interval for BHI UN Equity 0.8 Realized closing ratio Moving average Exponential average Cummulated average Average 0.6 f / / / / / /2011

18 Figure 16: Likehood function for various estimators on S&P x CC OC P GK RS YZ Figure 17: Likehood function for various estimators on BHI UN Equity x CC OC P GK RS YZ

19 the volatility estimation causes less effect. However, the high-low estimators suffer the effect of discretization then they underestimate the volatility. For the index, this effect is more important therefore the close-to-close estimator gives the best performance. Backtest on single asset with moving-average equal to 1 month (n = 21) of historical data. In the case with a particular asset such as the BBVA SQ Equity, the volatility is important hence the error due the efficiency of volatility estimators are important. High-low estimators now give better results than the classical one. Figure 18: Backtest of voltarget strategy on S&P 500 index S&P 500, CC, OC, P, GK, RS, YZ / / / / / /2011 In order to illustrate the efficiency of the range-based estimators, we realize a ranking between high-low estimator and the benchmark estimator close-to-close. We apply the voltarget strategy for close-to-close estimator ˆσ CC 2 and a high-low estimator ˆσ2 HL. Then we compare the Sharpe ratio obtained by these two estimators and compute the number of times where the high-low estimator gives better performance over the ensemble of stocks. The result over S&P 500 index and its first 100 compositions is summarized in Table 3. Table 2: Performance of ˆσ 2 HL versus ˆσ2 CC for different averaging windows Estimator ˆσ P 2 ˆσ GK 2 ˆσ RS 2 ˆσ Y 2 Z 6 month 56.2% 52.8% 52.8% 57.3% 3 month 52.8% 49.4% 51.7% 53.9% 2 month 60.7% 60.7% 60.7% 56.2% 1 month 65.2% 64.0% 64.0% 64.0%

20 Figure 19: Backtest of voltarget strategy on BHI UN Equity 3 Benchmark, CC, OC, P, GK, RS, YZ / / / / / /2011 Table 3: Performance of ˆσ 2 H L versus ˆσ2 CC for different filters of f Estimator ˆσ P 2 ˆσ GK 2 ˆσ RS 2 ˆσ Y 2 Z ˆf c 65.2% 64.0% 64.0% 64.0% ˆf ma 64.0% 61.8% 61.8% 64.0% ˆf exp 64.0% 61.8% 60.7% 64.0% ˆf t 64.0% 61.8% 60.7% 64.0%

21 3 Estimation of realized volatility The common way to estimate the realized volatility is to estimate the expectation value of the variance over an observed windows. Then we compute the corresponding volatility. However, to do so we encounter a great dilemma: taking a long historical window can help to decrease the estimation error as discussed in the last paragraph or taking a short historical data allows an estimation of volatility closer to the present volatility. In order to overcome this dilemma, we need to have an idea about the dynamics of the variance σ 2 t that we would like to measure. Combining this knowledge on the dynamics of σ2 t with the committed error on the long historical window, we can find out an optimal windows for the volatility estimator. We assume that the variance follows a simplified dynamics which has been used in the last numerical simulation: { dst = µ t S t dt + σ t S t db t dσ 2 t = ξσ 2 t db σ t in which B σ t is a Brownian motion independent to the one of asset process. 3.1 Moving-average estimator In this section, we show how the optimal window of the moving-average estimator is obtained via a simple example. Let us consider the canonical estimator: ˆσ 2 = 1 nt Here, the time increment is chosen to be constant t i t i 1 = T, then the variance of this estimator at instant t n is: var (ˆσ 2) 2σ4 T tn = 2σ4 tn t n t 0 n On another hand, σt 2 is now itself a stochastic process, hence its conditional variance to σt 2 n gives us the error due to the use of historical observations. We rewrite: 1 t n t 0 tn t 0 n σt 2 dt = 1 σ2 t n t n t 0 R 2 t i tn t 0 (t t 0 )σ 2 t ξ dbσ t then the error due to the stochastic volatility is given by: ( 1 tn ) var σt 2 dt t n t 0 σ2 t n t n t 0 σt 4 3 n ξ 2 = ntσ4 t n ξ 2 3 t 0 The total error of the canonical estimator is simply the sum of these errors due to the fact that the two considered Brownian motions are supposed to be independent. We define the function of total estimation errors as following: e (ˆσ 2) = 2σ4 t n n + ntσ4 t n ξ 2 3 In order to obtain the optimal window for volatility estimation, we minimize the error function e (ˆσ 2) with respect to nt which leads to the following equation: σ 4 t n ξ 2 3 2σ4 t n n 2 T = 0

22 This equation provides a very simple solution nt = 6T/ξ with the optimal error is now e (ˆσ 2 opt) 2 2T/3σ 4 tn ξ. The major difficulty of this estimator is to calibrate the parameter ξ which is not trivial because ξ 2 t is an unobservable process. Different techniques can be considered such as the maximum likehood which will be discussed later. 3.2 IGARCH estimator We discuss now another approach for estimating the realized volatility based on the IGARCH model. The detail theoretical derivation of the method is given in Drost F.C. et al. (1993, 1999) It consists of a volatility estimator of the form: ˆσ t 2 = βˆσ t T β T R2 t where T is a constant increment of estimation. In integrating the recurrence relation above, we obtain the estimator of the variance IGARCH in function of the return observed in the past: ˆσ t 2 = 1 β n β i Rt it 2 + β nˆσ t nt 2 (6) T We remark that the contribution of the last term tends to 0 when n tends to infinity. This estimator again has the form of a weighted average then similar approach as in the canonical estimator is applicable. Assuming that the volatility follows the lognormal dynamics described by Equation 3, therefore the optimal value of β is given by: β = ξ 8T ξ 2 T 2 4 ξ 2 T 4 We encounter here again the same question as the canonical case that is how to calibrate the parameter ξ of the lognormal dynamics. In practice, we proceed in the inverse way. We seek first the optimal value β of the IGARCH estimator then use the inverse relation of equation 7 to determine the value of ξ: 4 (1 β ξ = ) 2 T 1 + β 2 (7) Remark 1 Finally, as insisted at the beginning of this discussion, we would like to point out that IGARCH estimator can be considered as an exponential weighted average. We begin first with a IGARCH estimator with constant time increment. The expectation value of this estimator is: E [ [ ] ˆσ t 2 ] 1 β + σ = E β i Rt it 2 T σ = 1 β T = = Tβi t it+t β i σu 2 du TeiTλ t it + t it+t β i σu 2 du t it t it+t e itλ σu 2 du t it

23 with λ = lnβ/t. In this present form, we conclude that the IGARCH estimator is a weighted-average of the variance σt 2 with an exponential weight distribution. The annualized estimator of the volatility can be written as: E [ ˆσ t 2 σ ] = + e itλ t it+t t it + Te itλ This expression admits a continuous limit when T 0. σ 2 u du 3.3 Extension to range-based estimators The estimation of the optimal window in the last discussion can be also generalized to the case of range-based estimators. The main idea is to obtain the trade-off between the estimator error (variance of the estimator) and the dynamic volatility described by the model (3). The equation that determines the total error of the estimator is given by: e(ˆσ 2 ) = var (ˆσ 2) + nt 3 σ4 t n ξ 2 Here, we remind that the first term in this expression is the estimator error coming from the discrete sum whereas the second term is the error of the stochastic volatility. In fact, the first term is already given by the study of various estimators in the last section. The second term is typically dependent on the choice of volatility dynamics. Using the notation of the estimator efficiency, we rewrite the above expression as: e(ˆσ 2 ) = 1 2σt 4 n eff(ˆσ 2 ) n + nt 3 σ4 t n ξ 2 The minimization procedure of the total error is exactly the same as the last example on the canonical estimator, then we obtain the following result of the optimal averaging window: 6T nt = eff(ˆσ 2 )ξ 2 (8) The IGARCH estimator can also be applied for various type of high-low estimator, the extension consists of performing an exponential moving average in stead of the simple average. The parameter of the exponential moving average β will be determined again by the maximum likehood method as shown in the discussion below. 3.4 Calibration procedure of the estimators of realized volatility As discussed above, the estimators of realized volatility depend on the choice of the underlying dynamics of the volatility. In order to obtain the best estimation of the realized volatility, we must estimate the parameter which characterizes this dynamics. Two possible approaches to obtain the optimal value of the these estimators are: using the least square problem which consists to minimize the following objective function: n ( R 2 ti+t T ˆσ t 2 ) 2 i

24 or using the maximum likehood problem which consists to maximize the log-likehood objective function: n 2 ln 2π n i=0 1 2 ln( T ˆσ t 2 ) n R t 2 i+t 2T ˆσ 2 i=0 t i We remark here that the moving-average estimator depends only on the averaging window whereas the IGARCH estimator depends only on the parameter β. In general, there is no way to compare these two estimators if we do not use a specific dynamics. By this way, the optimal values of both parameters are obtained by the optimal value of ξ and that offers a direct comparison between the quality of these two estimators Example of realized volatility We illustrate here how the realized volatility is computed by the two methods discussed above. In order to illustrate how the optimal value of the averaging window nt or β are calibrated, we plot the likehood functions of these two estimator for one value of volatility at a given date. In Figure 20, we present the logarithm of likehood functions for different value of ξ. The maximal value of the function l(ξ) gives us the optimal value ξ which will be used to evaluate the volatility for the two methods. We remark that the IGARCH estimator is better to estimate the global maximum because its logarithm likehood is a concave function. For the the moving-average method, its logarithm likehood function is not smooth and presents complicated structure with local maximums which is less efficient for the optimization procedure. Figure 20: Comparison between IGARCH estimator and CC estimator CC optimal IGARCH l(ξ) ξ We now test the implementation of IGARCH estimators for various high-low estimators. As we have demonstrated that the IGARCH estimator is equivalent to exponential moving-

25 average, then the implementation for high-low estimators can be set up in the same way as the case of close-to-close estimator. In order to determine the optimal parameter β, we perform an optimization scheme on the logarithm likehood function. In Figure 21, we present the comparison of the logarithm likehood function between different estimators in function of the parameter β. The optimal parameter β of each estimator corresponds to the maximum of the logarithm likehood function. In order to have a clear idea about the Figure 21: Likehood function of high-low estimators versus filtered parameter β 1490 CC, OC, P, GK, RS, YZ l(β) β corresponding size of the moving-average window to the optimal parameter β, we use the formula (7) to effectuate the conversion. The result is reported in the Figure 22 below Backtest on the voltarget strategy We take historical data of S&P 500 index over the period since 01/2001 to 12/2011 and the averaging window of the close-to-close estimator is chosen as n = 25. In Figure23, we show the different estimations of realized volatility. In order to test the efficiency of these realized estimators (moving-average and IGARCH), we first evaluate the likehood function for the close-to-close estimator and realized estimators then apply these estimators for the voltarget strategy as performed in the last section. In Figure 25, we present the value of likehood function over the period from 01/2001 to 12/2010 for three estimators: CC, CC optimal (moving-average) and IGARCH. The estimator corresponding to the highest value of the likehood function is the one that gives the best prediction of the volatility. In Figure 27, the result of the backtest on voltarget strategy is performed for the three considered estimators. The estimators which dynamical choice of averaging parameters always give better result than a simple close-to-close estimator with fixed averaging window n = 25. We next backtest on the IGARCH estimator applied on the high-low price data, the

26 Figure 22: Likehood function of high-low estimators versus effective moving window 1485 CC, OC, P, GK, RS, YZ l(n) n Figure 23: IGARCH estimator versus moving-average estimator for close-to-close prices 100 CC CC optimal IGARCH σ (%) / / / / / /2011

27 Figure 24: Comparison between different IGARCH estimators for high-low prices 90 CC, CO, P, GK, RS, YZ σ (%) / / / / / /2011 Figure 25: Daily estimation of the likehood function for various close-to-close estimators CC CC optimal CC IGARCH l(ˆσ) / / / / / /20011

28 Figure 26: Daily estimation of the likehood function for various high-low estimators 1900 CC, OC, P, GK, RS, YZ l(ˆσ) / / / / / /2011 comparison with IGARCH applied on close-to-close data is shown in Figure 28. We observe that the IGARCH estimator for close-to-close price is one of the estimators which produce the best backtest. 4 High-frequency volatility estimators We have discussed in the previous sections how to measure the daily volatility based on the range of the observed prices. If more information is available in the trading data like having all the real-time quotation, can one estimate more accurately the volatility? As far as the trading frequency increases, we expect that the precision of estimator get better as well. However, when the trading frequency reaches certain limit, new phenomenon coming from the non-equilibrium of the market emerges and spoils the precision. This limit defines the optimal frequency for the classical estimator. In the literature, it is more and less agree to be at the frequency of one trade every 5 minutes. This phenomenon is called the microstructure noise which are characterized by the bid-ask spread or the transaction effect. In this section, we will summarize and test some recent proposals which attempt to eliminate the micro-structure noise. 4.1 Microstructure effect It has been demonstrated in the financial literature that the realized return estimator is not robust when the sampling frequency is too high. Two possible explanations of this effect the following. In the probabilistic point of view, this phenomenon comes from the fact that the cumulated return (or the logarithm of price) is not a semimartingal as we assumed in the last section. However, it emerges only in the short time scale when the trading frequency is high enough. In the financial point of view, this effect is explained by the existence of the

29 Figure 27: Backtest for close-to-close estimator and realized estimators 1.4 S&P 500 CC CC optimal CC IGARCH / / / / / /2011 Figure 28: Backtest for IGARCH high-low estimators comparing to IGARCH close-to-close estimator 1.4 S&P 500, CC, OC, P, GK, RS, YZ / / / / / /2011

30 so-called market microstructure noises. These noises come from the existence of the bid-ask spread. We now discuss the simplest model which includes the mircrostruture noise as an independent noise to the underlying Brownian motion. We assume that the true cumulated return is an unobservable process and follows a Brownian motion: ( ) dx t = µ t σ2 t dt + σ t db t 2 The observed signal Y t is the cumulated return which is perturbed by the microstructure noise ɛ t : Y t = X t + ɛ t For the sake of symplicity, we use the following assumptions: (i) ɛ ti is iid with E[ɛ ti ] = 0 and E [ ɛ 2 t i ] = E [ ɛ 2 ] (ii) ɛ t B t From these assumptions, we see immediately that the volatility estimator based on historical data Y ti is biased: var(y ) = var(x) + E [ ɛ 2] The first term var(x) is scaled as t (estimation horizon) and E [ ɛ 2] is constant, this estimator can be considered as unbiased if the time horizon is large enough (t > E [ ɛ 2] /σ 2 ). At high frequency, the second term is not negligible and better estimator must be able to eliminate this term. 4.2 Two time-scale volatility estimator Using different time scales to extract the true volatility of the hidden price process (without noise) is both independently proposed by Zhang et al. (2005) and Bandi et al. (2004). In this paragraph, we employ the approach in the first reference to define the intra-day volatility estimator. We prefer here discussing the main idea of this method and its practical implementation rather than all the detail of stochastic calculus concerning the expectation value and the variance of the realized return Definitions and notations In order to fix the notations, let us consider a time-period [0, T] which is divided in to M 1 intervals (M can be understood as the frequency). The quadratic variation of the Bronian motion over this period is denoted: X, X T = T 0 σ 2 t dt For the discretized version of the quadratic variation, we employ the [.,.] notation: ( ) 2 [X, X] T = Xti+1 X ti t i,t i+1 [0,T] Then the habitual estimator of realized return over the interval [0, T] is given by: ( ) 2 [Y, Y ] T = Yti+1 Y ti t i,t i+1 [0,T] 3 Detail of the derivation of this technique can be found in Zhang et al. (2005).

31 We remark that the number of points in the interval [0, T] can be changed. In fact, the expectation value of the quadratic variation should not depend on the distribution of points in this interval. Let us define the ensemble of points in one period as a grid G: Then a subgrid H is defined as: G = {t 0,...,t M } H = {t k1,..., t km } where (t kj ) with j = 1,...m is a subsequence of (t i ) with i = 1,...M. The number of increments is denoted as: H = card(h) 1 With these notations, the quadratic variation over a subgrid H reads: [Y, Y ] H T = ) 2 (Y tki+1 Y tki t ki,t ki+1 H The realized volatility estimator over the full grid If we compute the quadratic variation over the full grid G which means that at highest frequency. As discussed above, it is not surprising that it will suffer the most effect of the microstructure noise: [Y, Y ] G T = [X, X]G T + 2 [X, ɛ]g T + 2 [ɛ, ɛ]g T Under the hypothesis of the microstructure noise, the conditional expectation value of this estimator is equal to: [ E [Y, Y ] G T and the variation of the estimator: ( ) var [Y, Y ] G T X = 4ME [ ɛ 4] + ] X = [X, X] GT + 2ME[ ɛ 2] ( 8 [X, X] G T E[ ɛ 2] 2var ( ɛ 2)) + O(n 1/2 ) In these two expressions above, the sums are arranged order by order. In the limit M, we obtain the habitual result of central limit theorem: M 1/2 ( [Y, Y ] G T 2ME[ ɛ 2]) L 2 ( E [ ɛ 4 ]) 1/2 N(0, 1) Hence, as M increases, [Y, Y ] G T becomes a good estimator of the microstructure noise and we denote: Ê[ɛ 2 ] = 1 2M [Y, Y ]G T The central limit theorem for this estimator states: M 1/2 ( Ê[ɛ 2 ] E [ ɛ 2]) L ( E [ ɛ 4 ]) 1/2 N(0, 1) as M The realized volatility estimator over subgrid As we mentioned in the last discussion, increasing the frequency will spoil the estimation of the volatility due to the presence of the microstructure noise. The naive solution is to reduce the number of point in the grid or to consider only a subgrid, then one can take the average over a number choice of subgrids. Let us consider a subgrid H with H = m 1, then the same result as for the full grid can be obtained in replacing M by m: E [ [Y, Y ] H T X ] = [X, X] HT + 2mE[ ɛ 2]

32 Let us now consider a sequence of subgrids H (k) with k = 1...K which satisfies G = K k=1 H(k) and H (k) H (l) = with k l. By averaging over these K subgrid, we obtain the result: [ E [Y, Y ] avg T ] X = 1 K K k=1 [Y, Y ] H(k) T We define the average length of the subgrid m = (1/K) K k=1 m k, then the final expression is: [ ] E [Y, Y ] avg T X = [X, X] avg T + 2mE [ ɛ 2] This estimator of volatility is still biased and the precision depends strongly on the choice of the length of subgrid and the number of subgrids. In the paper of Zhang et al., the authors have demonstrated that there exists an optimal value K for which we can reach the best performance of estimator Two time-scale estimator As the full-grid averaging estimator and the subgrid averaging estimator both contain the same component coming from the microstructure noise to a factor, we can employ both estimators to have a new one where the microstructure noise can be completely eliminated. Let us consider the following estimator: ˆσ 2 ts = ( 1 m ) 1 ( [Y, Y ] avg T m ) M M [Y, Y ]G T This estimator now is an unbiased estimator with its precision determined by the choice of K and m. In the theoretical framework, this optimal value is given as a function of the noise variance and the forth moment of the volatility. In practice, we employ a scan over the number of the subgrid of size m M/K in order to look for the optimal estimator. 4.3 Numerical implementation and backtesting We now backtest the proposed technique on the S&P 500 index with the choice of the sub grid as following. The full grid is defined by the ensemble of data every minute from the opening to the close of trading days (9h to 17h30). Data is taken since the 1st February 2011 to the 6th June We denote the full grid for each trading day period: and the subgrid is chosen as following: G = {t 0,...,t M } H (k) = {t k 1, t k 1+K...,t k 1+nk K} where the indice k = 1,...,K and n k is the integer making t k 1+nk K the last element in H (k). As we can not compute exactly the value of the optimal value K for each trading period, we employ an iterative scheme which tends to converge to the optimal value. Analytical expression of K is given by Zhang et al.: K = ( ( [ ]) ) 12 E ɛ 2 2 1/3 M 2/3 T Eη 2