Applications of State Space Models in finance

Mergner, Sascha

doi:10.17875/gup2009-216

Cited by 15 publications

(8 citation statements)

References 250 publications

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“…These are models involving stochastic volatility or, more in general, stochastic state space models. For example models of this type are used in the study of clutter in signal processing, of wave propagation in random acoustic or electromagnetic media, of noise in telecommunications, of biomedical systems in medicine, as well as in the study of other models in mathematical finance (see, for example, [9,11,13,14] and the reference therein).…”

Section: Introductionmentioning

confidence: 99%

The use of statistical tests to calibrate the normal SABR model

Fatone

Mariani

Recchioni

et al. 2013

Journal of Inverse and Ill-Posed Problems

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We investigate the idea of solving calibration problems for stochastic dynamical systems using statistical tests. We consider a specific stochastic dynamical system: the normal SABR model. The SABR model has been introduced in mathematical finance in 2002 by Hagan, Kumar, Lesniewski and Woodward. The model is a system of two stochastic differential equations whose independent variable is time and whose dependent variables are the forward prices/rates and the associated stochastic volatility. The normal SABR model is a special case of the SABR model. The calibration problem for the normal SABR model is an inverse problem that consists in determining the values of the parameters of the model from a set of data. We consider as set of data two different sets of forward prices/rates and we study the resulting calibration problems. Ad hoc statistical tests are developed to solve these calibration problems. Estimates with statistical significance of the parameters of the model are obtained. Let be a constant, we consider multiple independent trajectories of the normal SABR model associated to given initial conditions assigned at time . In the first calibration problem studied the set of the forward prices/rates observed at time in this set of trajectories is used as data sample of a statistical test. The statistical test used to solve this calibration problem is based on some new formulae for the moments of the forward prices/rates variable of the normal SABR model. The second calibration problem studied uses as data sample the observations of the forward prices/rates made on a discrete set of given time values along a single trajectory of the normal SABR model. The statistical test used to solve this second calibration problem is based on the numerical evaluation of some high-dimensional integrals. The results obtained in the study of the normal SABR model are easily extended from mathematical finance to other contexts in science and engineering where stochastic models involving stochastic volatility or stochastic state space models are used. © 2013 by Walter de Gruyter Berlin Boston 2013

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Section: Introductionmentioning

confidence: 99%

The use of statistical tests to calibrate the normal SABR model

Fatone

Mariani

Recchioni

et al. 2013

Journal of Inverse and Ill-Posed Problems

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“…P i t in week t ) is calculated as

R_{it} = \log (P_{it}) - \log (P_{i t - 1})

for t = 2,…, T (Neslihanoglu, ). The weekly risk‐free rate is calculated from 3‐month US dollar London Interbank Offered Rate (LIBOR) interest rate (Mergner, ) as follows:

R_{ft} = {((), 1 + \frac{{LIBOR}_{t}}{100})}^{\frac{1}{52}} - 1

where LIBOR t is in percentage per annum and R f t is the weekly risk‐free rate in week t .…”

Section: Datamentioning

confidence: 99%

“…for t D 2; : : : ; T (Neslihanoglu, 2014). The weekly risk-free rate is calculated from 3-month US dollar London Interbank Offered Rate (LIBOR) interest rate (Mergner, 2009) as follows: Tables II (Table 5.2 in Neslihanoglu, 2014) and III provide several descriptive statistics about the distributional form of time series data during the three different period analyses in this paper: the entire period from July 2002 to July 2012; from July 2002 to before the October 2008 financial crisis; and from after the October 2008 financial crisis to July 2012. Specifically, these tables represent information about the first four moments, and report the Jarque-Bera (JB) and Ljung-Box (LB) test statistics for the normality and the autocorrelation of asset returns, respectively.…”

Section: Datamentioning

confidence: 99%

Nonlinearities in the CAPM: Evidence from Developed and Emerging Markets

Neslihanoğlu

Sogiakas

McColl

et al. 2016

Journal of Forecasting

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This paper examines the forecasting ability of the non-linear specifications of the market model. We propose a conditional Two-moment market model with a time-varying systematic covariance (beta) risk in the form of a mean reverting process of the state space model via Kalman Filter algorithm. In addition, we account for the systematic component of co-skewness and co-kurtosis by considering 2 Serdar Neslihanoglu et al.higher moments. The analysis is implemented using data from the stock indices of several developed and emerging stock markets. The empirical findings favour the time-varying market model approaches which outperform linear model specifications, both in terms of model fit and predictability. Precisely, higher moments are necessary for datasets which involve structural changes and/or market inefficiencies which are common in most of the emerging stock markets.

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“…Note that in the SABR model the forward pr dom variable is represented as a compound random variable and that the SABR model can be seen as a stochastic state space model [5]. Compound random variables and state space models are widely used in science and engineering.…”

Section: Letmentioning

confidence: 99%

“…with ositive initial conditions (3) assumption (5). That is, we assume p , (4) and the 0 x   (with 0 probability one) and 0 0 v   (with probability one), so that Equations (9), (10) imply 0 t…”

Section: Formulae For the Moments Omentioning

confidence: 99%

Closed Form Moment Formulae for the Lognormal SABR Model and Applications to Calibration Problems

Fatone¹,

Mariani²,

Recchioni³

et al. 2013

OJAppS

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We study two calibration problems for the lognormal SABR model using the moment method and some new formulae for the moments of the logarithm of the forward prices/rates variable. The lognormal SABR model is a special case of the SABR model [1]. The acronym “SABR” means “Stochastic-αβρ” and comes from the original names of the model parameters (i.e., α,β,ρ) [1]. The SABR model is a system of two stochastic differential equations widely used in mathematical finance whose independent variable is time and whose dependent variables are the forward prices/rates and the associated stochastic volatility. The lognormal SABR model corresponds to the choice β = 1 and depends on three quantities: the parameters α,ρ and the initial stochastic volatility. In fact the initial stochastic volatility cannot be observed and can be regarded as a parameter. A calibration problem is an inverse problem that consists in determineing the values of these three parameters starting from a set of data. We consider two different sets of data, that is: i) the set of the forward prices/rates observed at a given time on multiple independent trajectories of the lognormal SABR model, ii) the set of the forward prices/rates observed on a discrete set of known time values along a single trajectory of the lognormal SABR model. The calibration problems corresponding to these two sets of data are formulated as constrained nonlinear least-squares problems and are solved numerically. The formulation of these nonlinear least-squares problems is based on some new formulae for the moments of the logarithm of the forward prices/rates. Note that in the financial markets the first set of data considered is hardly available while the second set of data is of common use and corresponds simply to the time series of the observed forward prices/rates. As a consequence the first calibration problem although realistic in several contexts of science and engineering is of limited interest in finance while the second calibration problem is of practical use in finance (and elsewhere). The formulation of these calibration problems and the methods used to solve them are tested on synthetic and on real data. The real data studied are the data belonging to a time series of exchange rates between currencies (euro/U.S. dollar exchange rates)

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Applications of State Space Models in finance

Cited by 15 publications

References 250 publications

The use of statistical tests to calibrate the normal SABR model

The use of statistical tests to calibrate the normal SABR model

Nonlinearities in the CAPM: Evidence from Developed and Emerging Markets

Closed Form Moment Formulae for the Lognormal SABR Model and Applications to Calibration Problems

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