Volatility Measurement
|
|
- Mark Jessie King
- 5 years ago
- Views:
Transcription
1 Volatility Measurement Eduardo Rossi University of Pavia December 2013 Rossi Volatility Measurement Financial Econometrics / 53
2 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 / 53
3 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 / 53
4 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 / 53
5 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 / 53
6 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 / 53
7 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 / 53
8 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 / 53
9 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 / 53
10 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 / 53
11 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 / 53
12 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 / 53
13 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 / 53
14 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 / 53
15 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 / 53
16 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 / 53
17 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 / 53
18 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 / 53
19 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 / 53
20 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 / 53
21 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 / 53
22 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 / 53
23 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 / 53
24 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 / 53
25 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 / 53
26 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 / 53
27 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 / 53
28 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 / 53
29 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 / 53
30 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 / 53
31 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 / 53
32 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 / 53
33 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 / 53
34 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 / 53
35 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 / 53
36 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 / 53
37 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 / 53
38 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 / 53
39 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 / 53
40 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 / 53
41 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 / 53
42 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 / 53
43 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 / 53
44 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 / 53
45 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 / 53
46 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 / 53
47 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 / 53
48 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 / 53
49 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 / 53
50 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 / 53
51 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 / 53
52 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 / 53
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 / 53
54 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 / 53
55 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 / 53
56 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 / 53
57 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 / 53
58 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, Andersen, T.G., Bollerslev, T., Diebold, F.X. and Labys, P. (2003), "Modeling and forecasting realized volatility," Econometrica 71, 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, 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, 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, Rossi Volatility Measurement Financial Econometrics / 53
Realized Measures. Eduardo Rossi University of Pavia. November Rossi Realized Measures University of Pavia / 64
Realized Measures Eduardo Rossi University of Pavia November 2012 Rossi Realized Measures University of Pavia - 2012 1 / 64 Outline 1 Introduction 2 RV Asymptotics of RV Jumps and Bipower Variation 3 Realized
More informationUniversité de Montréal. Rapport de recherche. Empirical Analysis of Jumps Contribution to Volatility Forecasting Using High Frequency Data
Université de Montréal Rapport de recherche Empirical Analysis of Jumps Contribution to Volatility Forecasting Using High Frequency Data Rédigé par : Imhof, Adolfo Dirigé par : Kalnina, Ilze Département
More informationTECHNICAL WORKING PAPER SERIES PARAMETRIC AND NONPARAMETRIC VOLATILITY MEASUREMENT. Torben G. Andersen Tim Bollerslev Francis X.
TECHNICAL WORKING PAPER SERIES PARAMETRIC AND NONPARAMETRIC VOLATILITY MEASUREMENT Torben G. Andersen Tim Bollerslev Francis X. Diebold Technical Working Paper 279 http://www.nber.org/papers/t0279 NATIONAL
More informationOn modelling of electricity spot price
, Rüdiger Kiesel and Fred Espen Benth Institute of Energy Trading and Financial Services University of Duisburg-Essen Centre of Mathematics for Applications, University of Oslo 25. August 2010 Introduction
More informationVolatility. Roberto Renò. 2 March 2010 / Scuola Normale Superiore. Dipartimento di Economia Politica Università di Siena
Dipartimento di Economia Politica Università di Siena 2 March 2010 / Scuola Normale Superiore What is? The definition of volatility may vary wildly around the idea of the standard deviation of price movements
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Simulating Stochastic Differential Equations Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationAbsolute Return Volatility. JOHN COTTER* University College Dublin
Absolute Return Volatility JOHN COTTER* University College Dublin Address for Correspondence: Dr. John Cotter, Director of the Centre for Financial Markets, Department of Banking and Finance, University
More informationM5MF6. Advanced Methods in Derivatives Pricing
Course: Setter: M5MF6 Dr Antoine Jacquier MSc EXAMINATIONS IN MATHEMATICS AND FINANCE DEPARTMENT OF MATHEMATICS April 2016 M5MF6 Advanced Methods in Derivatives Pricing Setter s signature...........................................
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
More informationAsymptotic Theory for Renewal Based High-Frequency Volatility Estimation
Asymptotic Theory for Renewal Based High-Frequency Volatility Estimation Yifan Li 1,2 Ingmar Nolte 1 Sandra Nolte 1 1 Lancaster University 2 University of Manchester 4th Konstanz - Lancaster Workshop on
More informationOn the Forecasting of Realized Volatility and Covariance - A multivariate analysis on high-frequency data 1
1 On the Forecasting of Realized Volatility and Covariance - A multivariate analysis on high-frequency data 1 Daniel Djupsjöbacka Market Maker / Researcher daniel.djupsjobacka@er-grp.com Ronnie Söderman,
More information1 The continuous time limit
Derivative Securities, Courant Institute, Fall 2008 http://www.math.nyu.edu/faculty/goodman/teaching/derivsec08/index.html Jonathan Goodman and Keith Lewis Supplementary notes and comments, Section 3 1
More informationI Preliminary Material 1
Contents Preface Notation xvii xxiii I Preliminary Material 1 1 From Diffusions to Semimartingales 3 1.1 Diffusions.......................... 5 1.1.1 The Brownian Motion............... 5 1.1.2 Stochastic
More informationUsing MCMC and particle filters to forecast stochastic volatility and jumps in financial time series
Using MCMC and particle filters to forecast stochastic volatility and jumps in financial time series Ing. Milan Fičura DYME (Dynamical Methods in Economics) University of Economics, Prague 15.6.2016 Outline
More informationModeling the extremes of temperature time series. Debbie J. Dupuis Department of Decision Sciences HEC Montréal
Modeling the extremes of temperature time series Debbie J. Dupuis Department of Decision Sciences HEC Montréal Outline Fig. 1: S&P 500. Daily negative returns (losses), Realized Variance (RV) and Jump
More informationOptimum Thresholding for Semimartingales with Lévy Jumps under the mean-square error
Optimum Thresholding for Semimartingales with Lévy Jumps under the mean-square error José E. Figueroa-López Department of Mathematics Washington University in St. Louis Spring Central Sectional Meeting
More informationLimit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies
Limit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies George Tauchen Duke University Viktor Todorov Northwestern University 2013 Motivation
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets Liuren Wu ( c ) The Black-Merton-Scholes Model colorhmoptions Markets 1 / 18 The Black-Merton-Scholes-Merton (BMS) model Black and Scholes (1973) and Merton
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets (Hull chapter: 12, 13, 14) Liuren Wu ( c ) The Black-Scholes Model colorhmoptions Markets 1 / 17 The Black-Scholes-Merton (BSM) model Black and Scholes
More informationWeierstrass Institute for Applied Analysis and Stochastics Maximum likelihood estimation for jump diffusions
Weierstrass Institute for Applied Analysis and Stochastics Maximum likelihood estimation for jump diffusions Hilmar Mai Mohrenstrasse 39 1117 Berlin Germany Tel. +49 3 2372 www.wias-berlin.de Haindorf
More informationThere are no predictable jumps in arbitrage-free markets
There are no predictable jumps in arbitrage-free markets Markus Pelger October 21, 2016 Abstract We model asset prices in the most general sensible form as special semimartingales. This approach allows
More informationEconomics 201FS: Variance Measures and Jump Testing
1/32 : Variance Measures and Jump Testing George Tauchen Duke University January 21 1. Introduction and Motivation 2/32 Stochastic volatility models account for most of the anomalies in financial price
More informationParameters Estimation in Stochastic Process Model
Parameters Estimation in Stochastic Process Model A Quasi-Likelihood Approach Ziliang Li University of Maryland, College Park GEE RIT, Spring 28 Outline 1 Model Review The Big Model in Mind: Signal + Noise
More informationEstimation of High-Frequency Volatility: An Autoregressive Conditional Duration Approach
Estimation of High-Frequency Volatility: An Autoregressive Conditional Duration Approach Yiu-Kuen Tse School of Economics, Singapore Management University Thomas Tao Yang Department of Economics, Boston
More informationVariance derivatives and estimating realised variance from high-frequency data. John Crosby
Variance derivatives and estimating realised variance from high-frequency data John Crosby UBS, London and Centre for Economic and Financial Studies, Department of Economics, Glasgow University Presentation
More informationOperational Risk. Robert Jarrow. September 2006
1 Operational Risk Robert Jarrow September 2006 2 Introduction Risk management considers four risks: market (equities, interest rates, fx, commodities) credit (default) liquidity (selling pressure) operational
More informationOptimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models
Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models José E. Figueroa-López 1 1 Department of Statistics Purdue University University of Missouri-Kansas City Department of Mathematics
More informationHedging Credit Derivatives in Intensity Based Models
Hedging Credit Derivatives in Intensity Based Models PETER CARR Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU Stanford
More informationEstimation of dynamic term structure models
Estimation of dynamic term structure models Greg Duffee Haas School of Business, UC-Berkeley Joint with Richard Stanton, Haas School Presentation at IMA Workshop, May 2004 (full paper at http://faculty.haas.berkeley.edu/duffee)
More informationARCH and GARCH models
ARCH and GARCH models Fulvio Corsi SNS Pisa 5 Dic 2011 Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 1 / 21 Asset prices S&P 500 index from 1982 to 2009 1600 1400 1200 1000 800 600 400 200
More informationVOLATILITY FORECASTING IN A TICK-DATA MODEL L. C. G. Rogers University of Bath
VOLATILITY FORECASTING IN A TICK-DATA MODEL L. C. G. Rogers University of Bath Summary. In the Black-Scholes paradigm, the variance of the change in log price during a time interval is proportional to
More informationComments on Hansen and Lunde
Comments on Hansen and Lunde Eric Ghysels Arthur Sinko This Draft: September 5, 2005 Department of Finance, Kenan-Flagler School of Business and Department of Economics University of North Carolina, Gardner
More informationFinancial Econometrics and Volatility Models Estimating Realized Variance
Financial Econometrics and Volatility Models Estimating Realized Variance Eric Zivot June 2, 2010 Outline Volatility Signature Plots Realized Variance and Market Microstructure Noise Unbiased Estimation
More informationCourse information FN3142 Quantitative finance
Course information 015 16 FN314 Quantitative finance This course is aimed at students interested in obtaining a thorough grounding in market finance and related empirical methods. Prerequisite If taken
More informationMath 416/516: Stochastic Simulation
Math 416/516: Stochastic Simulation Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 13 Haijun Li Math 416/516: Stochastic Simulation Week 13 1 / 28 Outline 1 Simulation
More informationContinuous Time Finance. Tomas Björk
Continuous Time Finance Tomas Björk 1 II Stochastic Calculus Tomas Björk 2 Typical Setup Take as given the market price process, S(t), of some underlying asset. S(t) = price, at t, per unit of underlying
More informationAsymptotic Methods in Financial Mathematics
Asymptotic Methods in Financial Mathematics José E. Figueroa-López 1 1 Department of Mathematics Washington University in St. Louis Statistics Seminar Washington University in St. Louis February 17, 2017
More informationShort-Time Asymptotic Methods in Financial Mathematics
Short-Time Asymptotic Methods in Financial Mathematics José E. Figueroa-López Department of Mathematics Washington University in St. Louis Probability and Mathematical Finance Seminar Department of Mathematical
More informationRisk Neutral Valuation
copyright 2012 Christian Fries 1 / 51 Risk Neutral Valuation Christian Fries Version 2.2 http://www.christian-fries.de/finmath April 19-20, 2012 copyright 2012 Christian Fries 2 / 51 Outline Notation Differential
More informationStochastic Differential Equations in Finance and Monte Carlo Simulations
Stochastic Differential Equations in Finance and Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH China 2009 Outline Stochastic Modelling in Asset Prices 1 Stochastic
More informationTangent Lévy Models. Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford.
Tangent Lévy Models Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford June 24, 2010 6th World Congress of the Bachelier Finance Society Sergey
More informationParametric Inference and Dynamic State Recovery from Option Panels. Nicola Fusari
Parametric Inference and Dynamic State Recovery from Option Panels Nicola Fusari Joint work with Torben G. Andersen and Viktor Todorov July 2012 Motivation Under realistic assumptions derivatives are nonredundant
More informationIntroduction to Stochastic Calculus With Applications
Introduction to Stochastic Calculus With Applications Fima C Klebaner University of Melbourne \ Imperial College Press Contents Preliminaries From Calculus 1 1.1 Continuous and Differentiable Functions.
More informationCentral Limit Theorem for the Realized Volatility based on Tick Time Sampling. Masaaki Fukasawa. University of Tokyo
Central Limit Theorem for the Realized Volatility based on Tick Time Sampling Masaaki Fukasawa University of Tokyo 1 An outline of this talk is as follows. What is the Realized Volatility (RV)? Known facts
More informationRough volatility models: When population processes become a new tool for trading and risk management
Rough volatility models: When population processes become a new tool for trading and risk management Omar El Euch and Mathieu Rosenbaum École Polytechnique 4 October 2017 Omar El Euch and Mathieu Rosenbaum
More informationAsset Pricing Models with Underlying Time-varying Lévy Processes
Asset Pricing Models with Underlying Time-varying Lévy Processes Stochastics & Computational Finance 2015 Xuecan CUI Jang SCHILTZ University of Luxembourg July 9, 2015 Xuecan CUI, Jang SCHILTZ University
More informationTo apply SP models we need to generate scenarios which represent the uncertainty IN A SENSIBLE WAY, taking into account
Scenario Generation To apply SP models we need to generate scenarios which represent the uncertainty IN A SENSIBLE WAY, taking into account the goal of the model and its structure, the available information,
More informationThe Impact of Microstructure Noise on the Distributional Properties of Daily Stock Returns Standardized by Realized Volatility
The Impact of Microstructure Noise on the Distributional Properties of Daily Stock Returns Standardized by Realized Volatility Jeff Fleming, Bradley S. Paye Jones Graduate School of Management, Rice University
More informationImplementing the HJM model by Monte Carlo Simulation
Implementing the HJM model by Monte Carlo Simulation A CQF Project - 2010 June Cohort Bob Flagg Email: bob@calcworks.net January 14, 2011 Abstract We discuss an implementation of the Heath-Jarrow-Morton
More informationA new approach for scenario generation in risk management
A new approach for scenario generation in risk management Josef Teichmann TU Wien Vienna, March 2009 Scenario generators Scenarios of risk factors are needed for the daily risk analysis (1D and 10D ahead)
More informationParametric Inference and Dynamic State Recovery from Option Panels. Torben G. Andersen
Parametric Inference and Dynamic State Recovery from Option Panels Torben G. Andersen Joint work with Nicola Fusari and Viktor Todorov The Third International Conference High-Frequency Data Analysis in
More informationTrading Durations and Realized Volatilities. DECISION SCIENCES INSTITUTE Trading Durations and Realized Volatilities - A Case from Currency Markets
DECISION SCIENCES INSTITUTE - A Case from Currency Markets (Full Paper Submission) Gaurav Raizada Shailesh J. Mehta School of Management, Indian Institute of Technology Bombay 134277001@iitb.ac.in SVDN
More informationOn Market Microstructure Noise and Realized Volatility 1
On Market Microstructure Noise and Realized Volatility 1 Francis X. Diebold 2 University of Pennsylvania and NBER Diebold, F.X. (2006), "On Market Microstructure Noise and Realized Volatility," Journal
More informationBeyond the Black-Scholes-Merton model
Econophysics Lecture Leiden, November 5, 2009 Overview 1 Limitations of the Black-Scholes model 2 3 4 Limitations of the Black-Scholes model Black-Scholes model Good news: it is a nice, well-behaved model
More informationIndian Institute of Management Calcutta. Working Paper Series. WPS No. 797 March Implied Volatility and Predictability of GARCH Models
Indian Institute of Management Calcutta Working Paper Series WPS No. 797 March 2017 Implied Volatility and Predictability of GARCH Models Vivek Rajvanshi Assistant Professor, Indian Institute of Management
More informationLecture 17. The model is parametrized by the time period, δt, and three fixed constant parameters, v, σ and the riskless rate r.
Lecture 7 Overture to continuous models Before rigorously deriving the acclaimed Black-Scholes pricing formula for the value of a European option, we developed a substantial body of material, in continuous
More informationReading: You should read Hull chapter 12 and perhaps the very first part of chapter 13.
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Asset Price Dynamics Introduction These notes give assumptions of asset price returns that are derived from the efficient markets hypothesis. Although a hypothesis,
More informationLecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing
Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing We shall go over this note quickly due to time constraints. Key concept: Ito s lemma Stock Options: A contract giving
More informationMSc Financial Engineering CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL. To be handed in by monday January 28, 2013
MSc Financial Engineering 2012-13 CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL To be handed in by monday January 28, 2013 Department EMS, Birkbeck Introduction The assignment consists of Reading
More informationDefinition 9.1 A point estimate is any function T (X 1,..., X n ) of a random sample. We often write an estimator of the parameter θ as ˆθ.
9 Point estimation 9.1 Rationale behind point estimation When sampling from a population described by a pdf f(x θ) or probability function P [X = x θ] knowledge of θ gives knowledge of the entire population.
More informationSimulating Stochastic Differential Equations
IEOR E4603: Monte-Carlo Simulation c 2017 by Martin Haugh Columbia University Simulating Stochastic Differential Equations In these lecture notes we discuss the simulation of stochastic differential equations
More informationEconomics 883: The Basic Diffusive Model, Jumps, Variance Measures, and Noise Corrections. George Tauchen. Economics 883FS Spring 2014
Economics 883: The Basic Diffusive Model, Jumps, Variance Measures, and Noise Corrections George Tauchen Economics 883FS Spring 2014 Main Points 1. The Continuous Time Model, Theory and Simulation 2. Observed
More informationBROWNIAN MOTION Antonella Basso, Martina Nardon
BROWNIAN MOTION Antonella Basso, Martina Nardon basso@unive.it, mnardon@unive.it Department of Applied Mathematics University Ca Foscari Venice Brownian motion p. 1 Brownian motion Brownian motion plays
More informationProperties of Bias Corrected Realized Variance in Calendar Time and Business Time
Properties of Bias Corrected Realized Variance in Calendar Time and Business Time Roel C.A. Oomen Department of Accounting and Finance Warwick Business School The University of Warwick Coventry CV 7AL,
More informationIntraday and Interday Time-Zone Volatility Forecasting
Intraday and Interday Time-Zone Volatility Forecasting Petko S. Kalev Department of Accounting and Finance Monash University 23 October 2006 Abstract The paper develops a global volatility estimator and
More informationExploring Volatility Derivatives: New Advances in Modelling. Bruno Dupire Bloomberg L.P. NY
Exploring Volatility Derivatives: New Advances in Modelling Bruno Dupire Bloomberg L.P. NY bdupire@bloomberg.net Global Derivatives 2005, Paris May 25, 2005 1. Volatility Products Historical Volatility
More informationAnalysis of Realized Volatility for Nikkei Stock Average on the Tokyo Stock Exchange
Journal of Physics: Conference Series PAPER OPEN ACCESS Analysis of Realized Volatility for Nikkei Stock Average on the Tokyo Stock Exchange To cite this article: Tetsuya Takaishi and Toshiaki Watanabe
More informationPractical example of an Economic Scenario Generator
Practical example of an Economic Scenario Generator Martin Schenk Actuarial & Insurance Solutions SAV 7 March 2014 Agenda Introduction Deterministic vs. stochastic approach Mathematical model Application
More informationDynamic Replication of Non-Maturing Assets and Liabilities
Dynamic Replication of Non-Maturing Assets and Liabilities Michael Schürle Institute for Operations Research and Computational Finance, University of St. Gallen, Bodanstr. 6, CH-9000 St. Gallen, Switzerland
More informationS t d with probability (1 p), where
Stochastic Calculus Week 3 Topics: Towards Black-Scholes Stochastic Processes Brownian Motion Conditional Expectations Continuous-time Martingales Towards Black Scholes Suppose again that S t+δt equals
More informationData Sources. Olsen FX Data
Data Sources Much of the published empirical analysis of frvh has been based on high hfrequency data from two sources: Olsen and Associates proprietary FX data set for foreign exchange www.olsendata.com
More informationThe stochastic calculus
Gdansk A schedule of the lecture Stochastic differential equations Ito calculus, Ito process Ornstein - Uhlenbeck (OU) process Heston model Stopping time for OU process Stochastic differential equations
More informationUltra High Frequency Volatility Estimation with Market Microstructure Noise. Yacine Aït-Sahalia. Per A. Mykland. Lan Zhang
Ultra High Frequency Volatility Estimation with Market Microstructure Noise Yacine Aït-Sahalia Princeton University Per A. Mykland The University of Chicago Lan Zhang Carnegie-Mellon University 1. Introduction
More informationMATH3075/3975 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS
MATH307/37 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS School of Mathematics and Statistics Semester, 04 Tutorial problems should be used to test your mathematical skills and understanding of the lecture material.
More informationTwo hours. To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER
Two hours MATH20802 To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER STATISTICAL METHODS Answer any FOUR of the SIX questions.
More informationFE570 Financial Markets and Trading. Stevens Institute of Technology
FE570 Financial Markets and Trading Lecture 6. Volatility Models and (Ref. Joel Hasbrouck - Empirical Market Microstructure ) Steve Yang Stevens Institute of Technology 10/02/2012 Outline 1 Volatility
More informationThe Implied Volatility Index
The Implied Volatility Index Risk Management Institute National University of Singapore First version: October 6, 8, this version: October 8, 8 Introduction This document describes the formulation and
More informationOptimal Hedging of Variance Derivatives. John Crosby. Centre for Economic and Financial Studies, Department of Economics, Glasgow University
Optimal Hedging of Variance Derivatives John Crosby Centre for Economic and Financial Studies, Department of Economics, Glasgow University Presentation at Baruch College, in New York, 16th November 2010
More informationReplication and Absence of Arbitrage in Non-Semimartingale Models
Replication and Absence of Arbitrage in Non-Semimartingale Models Matematiikan päivät, Tampere, 4-5. January 2006 Tommi Sottinen University of Helsinki 4.1.2006 Outline 1. The classical pricing model:
More informationHigh-Frequency Data Analysis and Market Microstructure [Tsay (2005), chapter 5]
1 High-Frequency Data Analysis and Market Microstructure [Tsay (2005), chapter 5] High-frequency data have some unique characteristics that do not appear in lower frequencies. At this class we have: Nonsynchronous
More informationVolatility estimation with Microstructure noise
Volatility estimation with Microstructure noise Eduardo Rossi University of Pavia December 2012 Rossi Microstructure noise University of Pavia - 2012 1 / 52 Outline 1 Sampling Schemes 2 General price formation
More informationFinancial Risk Forecasting Chapter 9 Extreme Value Theory
Financial Risk Forecasting Chapter 9 Extreme Value Theory Jon Danielsson 2017 London School of Economics To accompany Financial Risk Forecasting www.financialriskforecasting.com Published by Wiley 2011
More informationValuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model
Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 1(23) Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility
More informationSlides for Risk Management
Slides for Risk Management Introduction to the modeling of assets Groll Seminar für Finanzökonometrie Prof. Mittnik, PhD Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik,
More informationLarge Deviations and Stochastic Volatility with Jumps: Asymptotic Implied Volatility for Affine Models
Large Deviations and Stochastic Volatility with Jumps: TU Berlin with A. Jaquier and A. Mijatović (Imperial College London) SIAM conference on Financial Mathematics, Minneapolis, MN July 10, 2012 Implied
More informationChanges of the filtration and the default event risk premium
Changes of the filtration and the default event risk premium Department of Banking and Finance University of Zurich April 22 2013 Math Finance Colloquium USC Change of the probability measure Change of
More informationMarket Timing Does Work: Evidence from the NYSE 1
Market Timing Does Work: Evidence from the NYSE 1 Devraj Basu Alexander Stremme Warwick Business School, University of Warwick November 2005 address for correspondence: Alexander Stremme Warwick Business
More informationDrunken Birds, Brownian Motion, and Other Random Fun
Drunken Birds, Brownian Motion, and Other Random Fun Michael Perlmutter Department of Mathematics Purdue University 1 M. Perlmutter(Purdue) Brownian Motion and Martingales Outline Review of Basic Probability
More informationTEST OF BOUNDED LOG-NORMAL PROCESS FOR OPTIONS PRICING
TEST OF BOUNDED LOG-NORMAL PROCESS FOR OPTIONS PRICING Semih Yön 1, Cafer Erhan Bozdağ 2 1,2 Department of Industrial Engineering, Istanbul Technical University, Macka Besiktas, 34367 Turkey Abstract.
More informationEconomics 883: The Basic Diffusive Model, Jumps, Variance Measures. George Tauchen. Economics 883FS Spring 2015
Economics 883: The Basic Diffusive Model, Jumps, Variance Measures George Tauchen Economics 883FS Spring 2015 Main Points 1. The Continuous Time Model, Theory and Simulation 2. Observed Data, Plotting
More informationStochastic Differential equations as applied to pricing of options
Stochastic Differential equations as applied to pricing of options By Yasin LUT Supevisor:Prof. Tuomo Kauranne December 2010 Introduction Pricing an European call option Conclusion INTRODUCTION A stochastic
More informationAn Introduction to Stochastic Calculus
An Introduction to Stochastic Calculus Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 2-3 Haijun Li An Introduction to Stochastic Calculus Week 2-3 1 / 24 Outline
More informationVolatility Estimation
Volatility Estimation Ser-Huang Poon August 11, 2008 1 Introduction Consider a time series of returns r t+i,i=1,,τ and T = t+τ, thesample variance, σ 2, bσ 2 = 1 τ 1 τx (r t+i μ) 2, (1) i=1 where r t isthereturnattimet,
More informationBox-Cox Transforms for Realized Volatility
Box-Cox Transforms for Realized Volatility Sílvia Gonçalves and Nour Meddahi Université de Montréal and Imperial College London January 1, 8 Abstract The log transformation of realized volatility is often
More informationBeta Estimation Using High Frequency Data*
Beta Estimation Using High Frequency Data* Angela Ryu Duke University, Durham, NC 27708 April 2011 Faculty Advisor: Professor George Tauchen Abstract Using high frequency stock price data in estimating
More informationFinancial Risk Management
Financial Risk Management Professor: Thierry Roncalli Evry University Assistant: Enareta Kurtbegu Evry University Tutorial exercices #3 1 Maximum likelihood of the exponential distribution 1. We assume
More informationFinancial Derivatives Section 5
Financial Derivatives Section 5 The Black and Scholes Model Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un. of
More informationRisk management. Introduction to the modeling of assets. Christian Groll
Risk management Introduction to the modeling of assets Christian Groll Introduction to the modeling of assets Risk management Christian Groll 1 / 109 Interest rates and returns Interest rates and returns
More informationFinancial Engineering. Craig Pirrong Spring, 2006
Financial Engineering Craig Pirrong Spring, 2006 March 8, 2006 1 Levy Processes Geometric Brownian Motion is very tractible, and captures some salient features of speculative price dynamics, but it is
More informationAn Introduction to Market Microstructure Invariance
An Introduction to Market Microstructure Invariance Albert S. Kyle University of Maryland Anna A. Obizhaeva New Economic School HSE, Moscow November 8, 2014 Pete Kyle and Anna Obizhaeva Market Microstructure
More information