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Volatility Measurement Eduardo Rossi University of Pavia December 2013 Rossi Volatility Measurement Financial Econometrics - 2012 1 / 53

Outline 1 Volatility definitions Continuous-Time No-Arbitrage Price Processes Return decomposition Quadratic variation Notional volatility The expected volatility 2 Volatility modeling and measurement 3 RV Rossi Volatility Measurement Financial Econometrics - 2012 2 / 53

Introduction Stylized facts: Actual realizations of financial returns volatility are not directly observable. Financial volatility changes through time. Both ex-ante and ex-post volatility measures are in common use. Rossi Volatility Measurement Financial Econometrics - 2012 3 / 53

Continuous-Time No-Arbitrage Price Processes Univariate risky log-price process: p(t) defined on (Ω, F, P). The price process evolves in continuous time over the interval [0, T ], T finite integer. Natural filtration: (F t ) t [0,T ] F. F t contains the full history (up to time t) of the realized values of the asset price and other relevant (possibly latent) state variables. Information set generated by the asset price history alone: (F t ) t [0,T ] F F T, by definition F t F t. Rossi Volatility Measurement Financial Econometrics - 2012 4 / 53

Continuous-Time No-Arbitrage Price Processes The continuously compounded return over the time interval [t h, t] is r(t, h) = p(t) p(t h) 0 h t T r(t) r(t, t) = p(t) p(0) r(t, h) = p(t) p(0) + p(0) p(t h) = r(t) (p(t h) p(0)) = r(t) r(t h) Rossi Volatility Measurement Financial Econometrics - 2012 5 / 53

Continuous-Time No-Arbitrage Price Processes Maintained assumption: the asset price process is almost surely strictly positive and finite, so that p(t) and r(t) are well defined over [0, T ] a.s. r(t) has only countably many jumps points over [0, T ]. Càdlàg version of the process for which r(t ) r(t+) r(t) = r(t+) lim r(τ) τ t,τ<t lim r(τ) τ t,τ>t a.s. The jumps in the cumulative price and return process are r(t) r(t) r(t ) 0 t T At continuity points for r(t): r(t) = 0. Rossi Volatility Measurement Financial Econometrics - 2012 6 / 53

Continuous-Time No-Arbitrage Price Processes A jump occurrence is unusual in the sense that we generically have: Pr( r(t) 0) = 0 t [0, T ] This does not imply that jumps necessarily are rare. There is the possibility of a (countably) infinite number of jumps over any discrete interval - a phenomenon referred to as an explosion. Regular Processes: Jump processes that do not explode. Prices as semi-martingales In a frictionless word, no-arbitrage opportunities and finite expected returns imply log-price process must constitute a semi-martingale (Black, 1991). Rossi Volatility Measurement Financial Econometrics - 2012 7 / 53

Return decomposition The unique canonical return decomposition (Protter, 1992) Any arbitrage-free logarithmic price process (subject to regularity conditions) may be uniquely represented as r(t) p(t) p(0) = µ(t) + M(t) = µ(t) + M C (t) + M J (t) instantaneous return decomposed into an expected return component µ(t) and a martingale innovation, M(t): µ(t) is a predictable and finite variation process M(t) local martingale M C (t) a continuous sample path, infinite variation local martingale component. M J (t) a compensated jump martingale. This provides a unique decomposition of the instantaneous return into an expected return component and a (martingale) innovation. Rossi Volatility Measurement Financial Econometrics - 2012 8 / 53

Assumption: normalized initial conditions Return decomposition µ(0) M(0) M C (0) M J (0) 0 r(t) = p(t) Over discrete intervals, r(t, h) = r(t) r(t h) = µ(t) µ(t h) + M(t) M(t h) = µ(t, h) + M(t, h) where µ(t, h) µ(t) µ(t h) 0 < h t T. is the expected returns over [t h, t]. M(t, h) = M(t) M(h) Expected returns m(t, h) E[r(t, h) F t h ] = E[µ(t, h) F t h ] 0 < h t T Rossi Volatility Measurement Financial Econometrics - 2012 9 / 53

Return decomposition The return innovation takes the form r(t, h) m(t, h) = (µ(t, h) m(t, h)) + M(t, h) The expected return process, even though it is (locally) predictable, may evolve stochastically over the [t h, t] interval. If µ(t, h) is measurable with respect to F t h, and thus known at time t h, then the discrete time return innovation reduces to M(t, h) M(t) M(t h). Although the discrete-time return innovation incorporates two distinct terms, the martingale component, M(t, h), is generally the dominant contributor to the return variation over short intervals, i.e., for h small. Rossi Volatility Measurement Financial Econometrics - 2012 10 / 53

Return decomposition Let s decompose the expected return process into: 1 a purely continuous predictable finite variation part, µ c (t), 2 a purely predictable jump part, µ j (t). Because the continuous component, µ c (t), is of finite variation it is locally an order of magnitude smaller than the corresponding contribution from the continuous component of the innovation term, M c (t). The reason is that an asset earning, say, a positive expected return over the risk-free rate must have innovations that are an order of magnitude larger than the expected return over infinitesimal intervals. Otherwise, a sustained long position (infinitely many periods over any interval) in the risky asset will tend to be perfectly diversified due to a Law of Large Numbers, as the martingale part is uncorrelated. Thus, the risk-return relation becomes unbalanced. Rossi Volatility Measurement Financial Econometrics - 2012 11 / 53

Return decomposition The presence of a non-trivial M j (t) component may similarly serve to eliminate arbitrage and retain a balanced risk-return trade-off relationship. Analogous considerations apply to the jump component for the expected return process, µ j (t), if this factor is present. There cannot be a predictable jump in the mean - i.e., a perfectly anticipated jump in terms of both time and size - unless it is accompanied by large jump innovation risk as well, so that Pr{M(t) 0} > 0. Again - intuitively - if there was a known, say, positive jump then this induces arbitrage (by going long the asset) unless there is offsetting (jump) innovation risk. Rossi Volatility Measurement Financial Econometrics - 2012 12 / 53

Return decomposition Stochastic Volatility Jump Diffusion with Non-Zero Mean Jumps dp(t) = µ + βσ 2 (t) + σ(t)dw (t) + κ(t)dq(t) 0 t T Assumptions: σ(t) is a strictly positive continuous sample path process (a.s.) W (t) denotes a standard Bm, q(t) is a pure jump process with q(t) = { 1 jump in t 0 otherwise κ(t) refers to the size of the corresponding jumps. The jump size distribution : E[κ(t)] = µ κ and Var[κ(t)] = σ 2 k The jump intensity is assumed constant (and finite) at a rate λ per unit time. Rossi Volatility Measurement Financial Econometrics - 2012 13 / 53

Return decomposition The return components: t µ(t) = µ c (t) = µ t + β t M c (t) = σ(s)dw (s) 0 M j (t) = κ(s)q(s) λ µ κ t 0 s t 0 σ 2 (s)ds + λ µ κ t Rossi Volatility Measurement Financial Econometrics - 2012 14 / 53

Return decomposition The concept of an instantaneous return employed in continuous-time models (SDE form) is pure short-hand notation that is formally defined only through the corresponding integral representation. Real-time price data are not available at every instant and, due to pertinent microstructure features, prices are invariably constrained to lie on a discrete grid, both in the price and time dimension. There is no real-world counterpart to the notion of a continuous sample path martingale with infinite variation over arbitrarily small time intervals (say, less than a second). It is only feasible to measure return (and volatility) realizations over discrete time intervals. Sensible measures can only be constructed over much longer horizons than given by the minimal interval length for which consecutive trade prices or quotes are recorded. Rossi Volatility Measurement Financial Econometrics - 2012 15 / 53

Return decomposition Volatility seeks to capture the strength of the (unexpected) return variation over a given period of time. Two distinct features importantly differentiate the construction of all (reasonable) volatility measures. 1 Forecasts of future return volatility. The focus is on ex-ante expected volatility. Search for a model that may be used to map the current information set into a volatility forecast 2 Given a set of actual return observations, the emphasis is on ex-post measurement of the volatility. The (ex-post) realized volatility may be computed (or approximated) without reference to any specific model (nonparametric procedure). Rossi Volatility Measurement Financial Econometrics - 2012 16 / 53

Return decomposition Focus on the behavior of M(t) process (the martingale component in the return decomposition). Prerequisite for observing the M(t) is that we have access to a continuous record of price data. Such data are simply not available, even for extremely liquid markets. The presence of microstructure effects (discrete price grids, bid-ask bounce effects, etc.) prevents from ever getting really close to a true continuous sample path realization. We focus on measures that represent the (average) volatility over a discrete time interval, rather than the instantaneous (point-in-time) volatility. Rossi Volatility Measurement Financial Econometrics - 2012 17 / 53

Quadratic variation Quadratic variation General notion of volatility based on the quadratic variation process for the local martingale component in the unique semi-martingale return decomposition. Quadratic variation Let X (t) denote any semi-martingale. The unique quadratic variation process, [X, X ] t, t [0, T ], associated with X (t) is then formally defined by t [X, X ] t X (t) 2 2 X (s )dx (s), 0 where the stochastic integral of the adapted càglàd process, X (s ), with respect to the càdlàg semi-martingale, X (s), is well-defined. The quadratic variation, [X, X ] t, is an increasing stochastic process. Also, jumps in the sample path of the quadratic variation process necessarily occur concurrent with the jumps in the underlying semimartingale process, [X, X ] = ( X ) 2. Rossi Volatility Measurement Financial Econometrics - 2012 18 / 53

Quadratic variation Local Martingale If M is a locally square integrable martingale, then the associated (M 2 [M, M]) process is a local martingale, E[M(t, h) 2 ([M, M] t [M, M] t h ) F t h ] = 0 0 < h t T. Rossi Volatility Measurement Financial Econometrics - 2012 19 / 53

Quadratic variation Let a sequence of possible random partitions of [0, T ], τ m, be given s.t. τ m {τ m,j } j 0, m = 1, 2,... where satisfy a.s. for m, τ m,0 τ m,1 τ m,2... τ m,0 0; Then, for t [0, T ], sup τ m,j T ; j 1 sup (τ m,j+1 τ m,j ) 0. j 1 lim { (X (t τ m,j ) X (t τ m,j 1 )) 2 } [X, X ] t m j 1 where t τ min (t, τ), and the convergence is uniform in probability. The quadratic variation process represents the (cumulative) realized sample path variability of X (t) over the [0, t] time interval. Rossi Volatility Measurement Financial Econometrics - 2012 20 / 53

Notional volatility Theoretical notion of ex-post return variability: Notional volatility (general) or Actual volatility (Barndorff-Nielsen and Shephard (2002)) over [t h, t], 0 < h t T : υ 2 (t, h) [M, M] t [M, M] t h = [M c, M c ] t [M c, M c ] t h + t h<s t M 2 (s). Rossi Volatility Measurement Financial Econometrics - 2012 21 / 53

Notional volatility Notional volatility Under the maintained assumption of no predictable jumps in the return process, and noting that the quadratic variation of any finite variation process, such as µ c (t), is zero, we also have υ 2 (t, h) [r, r] t [r, r] t h = [M c, M c ] t [M c, M c ] t h + r 2 (s). t h<s t the notional volatility equals (the increment to) the quadratic variation for the return series. Ex-post it is possible to approximate the notional volatility arbitrarily well through the accumulation of ever finely sampled high-frequency squared return. This approach remains consistent independent of the expected return process. Rossi Volatility Measurement Financial Econometrics - 2012 22 / 53

Notional volatility The notional volatility captures the sample path variability of the log-price process over the [t h, t] time interval. incorporates the effect of (realized) jumps in the price process: jumps contribute to the realized return variability and forecasts of volatility must account for the potential occurrence of such jumps The expected notional volatility E[υ 2 (t, h) F t h ] = E[M(t, h) 2 F t h ] = E[M(t) 2 F t h ] M(t h) 2 represents the expected future (cumulative) squared return innovation. this component is typically the dominant determinant of the expected return volatility. Rossi Volatility Measurement Financial Econometrics - 2012 23 / 53

The expected volatility The expected volatility The Expected Volatility over [t h, t], 0 < h t T, is defined by ϑ 2 (t, h) E[{r(t, h) E(µ(t, h) F t h )} 2 F t h ] = E[{r(t, h) m(t, h)} 2 F t h ] If the µ(t, h) process is not measurable with respect to F t h, the expected volatility will typically differ from the expected notional volatility. The future return variability reflects both genuine return innovations, as in the expected notional volatility, and intra-period innovations to the conditional mean process. For models with an assumed constant mean return, or for one-period-ahead discrete-time volatility forecasts with given conditional mean representation, the two concepts coincide. Rossi Volatility Measurement Financial Econometrics - 2012 24 / 53

The expected volatility The expected volatility may generally be expressed as ϑ 2 (t, h) = E[{r(t, h) E(µ(t, h) F t h )} 2 F t h ] = E[{r(t, h) µ(t, h) + m(t, h) µ(t, h)} 2 F t h ] = E[(r(t, h) µ(t, h)) 2 F t h ] + E[(µ(t, h) m(t, h)) 2 F t h ] + 2E[(r(t, h) µ(t, h))(µ(t, h) m(t, h)) F t h ] but since E[(r(t, h) µ(t, h)) 2 F t h ] = E[M(t, h) 2 F t h ] = E[υ 2 (t, h) F t h ] ϑ 2 (t, h) = E[υ 2 (t, h) F t h ] + Var[µ(t, h) F t h ] + 2 Cov[M(t, h), µ(t, h) F t h ] Rossi Volatility Measurement Financial Econometrics - 2012 25 / 53

The expected volatility the total expected return volatility involves the expected notional volatility (quadratic variation) as well as two terms induced by future within-forecast-period variation in the conditional mean. The random variation in the mean component is a direct source of future return variation, and covariation between the return and conditional mean innovations will further impact the return variability. However, under standard conditions and moderate forecast horizons, the dominant factor is indisputably the expected notional volatility, as the innovations to the mean return process generally will be very small relative to the cumulative return innovations. This does not rule out asymmetric effects from current return innovations to future return volatility, as the leverage and volatility feedback effects operate, respectively, exclusively or primarily through the impact on the notional volatility process. Rossi Volatility Measurement Financial Econometrics - 2012 26 / 53

The expected volatility Instantaneous Volatility [ σt 2 lim E h 0 { [M c, M c ] t [M c, M c ] t h h } F t h ] This definition is consistent with the terminology commonly employed in the literature on continuous-time parametric stochastic volatility models. Spot Volatility (Barndorff-Nielsen & Shephard, 2002) Although the instantaneous volatility is a natural concept, practical volatility measurement invariably takes place over discrete time intervals. Rossi Volatility Measurement Financial Econometrics - 2012 27 / 53

The expected volatility Discrete-Time Stochastic Volatility (ARCH) Model Consider the discrete-time process for p(t) defined over the unit time interval, p(t) = p(t 1) + µ + βσ 2 (t) + σ(t)z(t) t = 1, 2,..., T z(t) denotes a m.d.s. with unit variance, while σ(t) is a (possibly latent) positive (a.s.) stochastic process that is measurable with respect to F t 1. In this situation the continuous-time no-arbitrage arguments based on infinitely many long-short positions over fixed length time intervals are no longer operative. Rossi Volatility Measurement Financial Econometrics - 2012 28 / 53

The expected volatility In this specific model we have, for t = 1, 2,..., T T µ(t) = µ t + β σ 2 (s) s=1 T M(t) M J (t) = σ(s)z(s) s=1 The one-period notional volatility measure is then υ 2 (t, 1) = σ 2 (t)z 2 (t) the expected notional volatility is E[υ 2 (t, 1) F t 1 ] = σ 2 (t) the expected return is m(t, 1) = µ + βσ 2 (t) Rossi Volatility Measurement Financial Econometrics - 2012 29 / 53

The expected volatility Although direct comparison of the order of magnitude of the mean return relative to the return innovation is not feasible here, empirical studies based on daily and weekly returns invariably find the mean parameters, such as µ and β to be very small relative to the expected notional volatility, σ 2 (t). In practice, the contribution of innovations in the mean process to the overall return variability is negligible over such horizons. Rossi Volatility Measurement Financial Econometrics - 2012 30 / 53

The expected volatility When volatility is stochastic - the ex-post (realized) notional volatility will not correspond to the ex-ante expected volatility. The ex-ante expected notional volatility generally is not identical to the usual notion of return volatility as an ex-ante characterization of future return variability over a discrete holding period. The fact that the latter quantity is highly relevant for financial decision making motivates the standard discrete-time expected volatility. Rossi Volatility Measurement Financial Econometrics - 2012 31 / 53

The expected volatility Black and Scholes model The simplest possible case is provided by the time-invariant diffusion (continuous-time random walk), dp(t) = µdt + σdw (t) which underlies the Black-Scholes option pricing formula. This process has a deterministic mean return ( µ ) so the expected return volatility trivially equals the expected notional volatility E[ϑ 2 t F t h ] = E[υ 2 t F t h ] Because the volatility is also constant, the expected notional volatility is identical to the notional volatility. The notional volatility is equal to υ 2 t = h 0 σ 2 (t h + s)ds = h σ 2 Rossi Volatility Measurement Financial Econometrics - 2012 32 / 53

The expected volatility Black and Scholes model This model is also straightforward to estimate from discretely sampled data by, e.g., maximum likelihood, as the returns are i.i.d. and normally distributed. Suppose that observations are only available at n + 1 equally spaced points over the [t h, t] time interval, where 0 h < t T ; i.e., t h, t h + (h/n),..., t h + (n 1)(h/n), t. By the definition of the process, the corresponding sequence of i = 1, 2,..., n discrete (h/n)-period returns, ( r t h + i h n, h ) ( = p t h + i h ) ( p t h + (i 1) h ) n n n ( r t h + i h n, h ) ( i.i.d.n µ h n n, h ) σ2. n Rossi Volatility Measurement Financial Econometrics - 2012 33 / 53

The expected volatility Black and Scholes model The MLE of the drift is simply given by the sample mean of the (scaled) returns, ˆµ n = 1 n n ( h ) 1r ( t h + i h n n, h ) = n i=1 r(t, h) h = p(t) p(t h) h It follows immediately that E[ˆµ n ] = µ For a fixed interval the in-fill asymptotics, obtained by increasing the number of intraday observations, are irrelevant for estimating the expected return. The estimator of the drift is independent of the sampling frequency, given by n, and depends only on the span of the data, h. Rossi Volatility Measurement Financial Econometrics - 2012 34 / 53

The expected volatility Black and Scholes model The variance of the estimator [( ) 2 ] Var[ˆµ n ] = E ˆµ n µ [( p(t) p(t h) ) 2 ] = E µ h [( µ h + σ(w (t) W (t h)) = E h = σ2 h ) 2 ] µ Thus, although ˆµ n is an unbiased estimator for µ, it is not consistent as n. Rossi Volatility Measurement Financial Econometrics - 2012 35 / 53

The expected volatility Black and Scholes model Consider now the (unadjusted) estimator for σ 2 defined by the sum of the (scaled) squared returns, ˆσ n 2 1 n ( h ) 1r ( t h + i h n n n, h ) 2 n Because it follows that i=1 [ ( E r t h + i h n, h ) 2 ] n = µ 2( h n E[ˆσ n] 2 = σ 2 + µ 2( h ) n ) ( 2 + σ 2 h ) n Hence, the drift induces only a second order bias, or O(n 1 ) term, in the estimation of σ 2 as n. Rossi Volatility Measurement Financial Econometrics - 2012 36 / 53

The expected volatility Black and Scholes model This estimator for the diffusion coefficient is consistent as n [ ( E r t h + i h n, h ) 3 ] = 3µσ 2( h ) ( 2 + µ 3 h ) 3 n n n [ ( E r t h + i h n, h ) 4 ] = 3σ 4( h ) 2 + 6µ 2 σ 2( h ) ( 3 + µ 4 h ) 4 n n n n The terms involving the drift coefficient are an order of magnitude smaller, for n large, than those that pertain only to the diffusion coefficient. This feature allows us to estimate the return variation with a high degree of precision even without specifying the underlying mean drift. Rossi Volatility Measurement Financial Econometrics - 2012 37 / 53

The expected volatility Black and Scholes model This (along with the second moment given above) and the fact that the returns are i.i.d., implies that Var[ˆσ n] 2 = E[(ˆσ n) 2 2 ] E[ˆσ n] 2 2 = 2σ4 n + 4µ2 σ 2 n 2 By a standard Law of Large Numbers, p lim ˆσ n 2 = σ 2. n The realized variation measure is a biased but consistent estimator of the underlying (squared) volatility coefficient. Increasing the number of (scaled) squared return observations over the interval then produces an increasing number of unbiased and uncorrelated measures of σ 2, and simply averaging these yields a consistent estimator. Rossi Volatility Measurement Financial Econometrics - 2012 38 / 53

Volatility modeling and measurement Two approaches for empirically quantifying volatility: Procedures based on estimation of parametric models. Alternative parametric models differentiate through: 1 different assumptions regarding the expected volatility, ϑ 2 (t, h) 2 distinct functional forms 3 the nature of the variables in the information set, F t h. Nonparametric measurements, that typically quantify the notional volatility, υ 2 (t, h), directly. Both set of procedures differ importantly in terms of the choice of time interval for which the volatility measure applies: a discrete interval, h > 0 a point-in-time (instantaneous) measure, obtained as the limiting case for h 0. Rossi Volatility Measurement Financial Econometrics - 2012 39 / 53

Volatility modeling and measurement Parametric Volatility Models Discrete-Time Parametric Models explicitly parameterize the expected volatility, ϑ 2 (t, h), h > 0, as a non-trivial function of the time t-h information set, F t h. ARCH models: F t h depends on past returns and other directly observable variables only. Stochastic Volatility (SV) models: F t h explicitly incorporates past returns as well as latent state variables. Continuous-Time Volatility Models: explicit parametrization of the instantaneous volatility, σt 2 as a (non-trivial) function of the F t information set, with additional volatility dynamics possibly introduced through time variation in the process governing jumps in the price path. Rossi Volatility Measurement Financial Econometrics - 2012 40 / 53

Volatility modeling and measurement Nonparametric Volatility Measurement Nonparametric measurement utilizes the ex-post returns, or F τ, in extracting measures of the notational volatility: ARCH Filters and Smoothers rely on continuous sample paths, or M j 0, in measuring the instantaneous volatility. Filters: information up to τ = t Smoothers: information up to τ > t. Realized Volatility measures directly quantify the notional volatility over (non-trivial) fixed-length time intervals. Rossi Volatility Measurement Financial Econometrics - 2012 41 / 53

Volatility modeling and measurement In contrast to the parametric procedures, the nonparametric volatility measurements are generally void of any specific functional form assumptions about the stochastic process(es) governing the local martingale, M(t), as well as the predictable and finite variation process, µ(t), in the unique return decomposition. These procedures also differ importantly from the parametric models in their focus on providing measures of the notional volatility, υ 2 (t, h), rather than the expected volatility, ϑ 2 (t, h). Rossi Volatility Measurement Financial Econometrics - 2012 42 / 53

RV The data-driven, or nonparametric volatility measurements afford direct ex-post empirical appraisals of the notional volatility, υ 2 (t, h), without any specific functional form assumptions. The nonparametric measurements more generally achieve consistency by measuring the volatility as (weighted) sample averages of increasingly finer sampled squared (or absolute) returns over (and possible outside) the [t h, t] interval. The realized volatility (RV ) measures build on the idea of an increasing number of observations over fixed length time intervals. Rossi Volatility Measurement Financial Econometrics - 2012 43 / 53

RV Continuous Sample Path Diffusions The continuous-time models in the theoretical asset and derivatives pricing literature frequently assume that the sample paths are continuous, with the corresponding diffusion processes given in the form of stochastic differential equations. Set up No price jumps and frictionless market. The asset s logarithmic price process p(t) must be a semi-martingale to rule out arbitrage opportunities: dp(t) = µ(t)dt + σ(t)dw (t) 0 t T µ(t) and σ(t) are predictable processes, µ(t) is of finite variation, σ(t) is strictly positive and square integrable, i.e., E [ ] t 0 σ2 (s)ds < the processes µ(t) and σ(t) signify the instantaneous conditional mean and volatility of the return. Rossi Volatility Measurement Financial Econometrics - 2012 44 / 53

RV Quadratic variation The continuously compounded return over the time interval from t h to t, t t r(t, h) = p(t) p(t h) = µ(s)ds + σ(s)dw (s) t h t h and its notional volatility υ 2 (t, h) (or quadratic variation QV (t, h)) is υ 2 (t, h) = t t h σ 2 (s)ds the innovations to the mean component µ(t) do not affect the sample path variation of the return. Intuitively, this is because the mean term, µ(t)dt, is of lower order in terms of second order properties than the diffusive innovations, σ(t)dw (t). When cumulated across many high-frequency returns over a short time interval of length h they can effectively be neglected. Rossi Volatility Measurement Financial Econometrics - 2012 45 / 53

RV Realized Volatility The diffusive sample path variation over [t h, t] is also known as the integrated variance IV (t, h), In this setting, the quadratic and integrated variation coincide. This is however no longer true for more general return process like, e.g., the stochastic volatility jump-diffusion model. Absent microstructure noise and measurement error, the return quadratic variation can be approximated arbitrarily well by the corresponding cumulative squared return process. Rossi Volatility Measurement Financial Econometrics - 2012 46 / 53

RV Realized Volatility Consider a partition {t h + j, j = 1,..., n h} n The corresponding sequence of i = 1, 2,..., n discrete (h/n)-period returns, ( r t h + i h n ; h ) = p(t h + i h n n ) p(t h + (i 1)h n )). The RV is simply the second (uncentered) sample moment of the return process over a fixed interval of length h, scaled by the number of observations n (corresponding to the sampling frequency 1/n), so that it provides a volatility measure calibrated to the h-period measurement interval. RV (t, h; n) = n r i=1 ( t h + h i n ; h ) 2 n Rossi Volatility Measurement Financial Econometrics - 2012 47 / 53

RV Realized Volatility The RV provides a consistent nonparametric measure of the Notional Volatility. Although the definition is stated in terms of equally spaced observations, most results carry over to situations in which the RV is based on the sum of unevenly, but increasingly finely sampled squared returns. Rossi Volatility Measurement Financial Econometrics - 2012 48 / 53

RV Stochastic Volatility Jump Diffusion with Non-Zero Mean Jumps dp(t) = µ + βσ 2 (t) + σ(t)dw (t) + κ(t)dq(t) 0 t T The notional volatility is υ 2 (t, h) = h 0 σ 2 (t h + s)ds + t h<s t κ 2 (s) The expected notional volatility involves taking the conditional expectation of this expression. Without an explicit model for the volatility process, this cannot be given in closed form. However, for small h the (expected) notional volatility is typically very close to the value attained if volatility is constant. Rossi Volatility Measurement Financial Econometrics - 2012 49 / 53

RV In particular, to a first-order approximation, E[υ 2 (t, h) F t h ] σ 2 (t h) h + λ h (µ 2 k + σ 2 k) [σ 2 (t h) + λ (µ 2 k + σ 2 k)] h while m(t, h) [µ + βσ 2 (t h) + λµ k ] h Thus, the expected notional volatility is of order h, the expected return is of order h (and the variation of the mean return of order h 2 ), whereas the martingale (return innovation) is of the order h 1/2, and hence an order of magnitude larger for small h. Rossi Volatility Measurement Financial Econometrics - 2012 50 / 53

RV Realized Volatility Realized Volatility as an Unbiased Volatility Estimator If the return process is square-integrable and µ(t) = 0 then for any value of n 1 and h > 0, ϑ 2 (t, h) = E[υ 2 (t, h) F t h ] = E[RV (t, h; n) F t h ] the ex-post RV is an unbiased estimator of ex-ante expected volatility. The result remains approximately true for a stochastically evolving mean return process over relevant horizons under weak auxiliary conditions, as long as the underlying returns are sampled at sufficiently high frequencies. The RV approach interprets RV (t, h; n) as a measure of the overall volatility for the [t h, t] time interval. Since RV is approximately unbiased for the corresponding unobserved QV, the RV measure is the natural benchmark against which to gauge the performance of volatility forecasts. The quadratic variation is directly related to the actual return variance and to the expected return variance. Rossi Volatility Measurement Financial Econometrics - 2012 51 / 53

RV Realized Volatility The ex-ante expected notional volatility (E[υ 2 (t, h) F t h ]) is also the critical determinant of expected volatility, expressed as ϑ 2 (t, h) = E[{r(t, h) E(µ(t, h) F t h )} 2 F t h ] = E[υ 2 (t, h) F t h ] + Var[µ(t, h) F t h ] + 2 Cov[M(t, h), µ(t, h) F t h ] Any empirical measures of (ex-ante expected) notional volatility will necessarily depend on the assumed parametric model structure. Rossi Volatility Measurement Financial Econometrics - 2012 52 / 53

RV Realized Volatility The theoretical properties of RV have been discussed from different perspectives in a number of paper: Andersen, T.G., T. Bollerslev, F.X. Diebold and P. Labys (2001a), "The Distribution of Realized Exchange Rate Volatility," Journal of the American Statistical Association, 96, 42-55. Andersen, T.G., Bollerslev, T., Diebold, F.X. and Labys, P. (2003), "Modeling and forecasting realized volatility," Econometrica 71, 579-625. Andersen, T.G., T. Bollerslev, F.X. Diebold and P. Labys (2000b), "Exchange Rate Returns Standardized by Realized Volatility are (Nearly) Gaussian," Multinational Financ Journal, 4, 159-179. Barndorff-Nielsen, O.E. and N. Shephard (2001a), "Non-Gaussian Ornstein-Uhlenbeck-Based Models and Some of Their Uses in Financial Economics," Journal of the Royal Statistical Society, Series B, 63, 167-241. Barndorff-Nielsen, O.E. and N. Shephard (2002a), "Econometric Analysis of Realised Volatility and its Use in Estimating Stochastic Volatility Models," Journal of the Royal Statistical Society, Series B, 64, 253-280. Rossi Volatility Measurement Financial Econometrics - 2012 53 / 53