1,867 research outputs found
Long Memory Options: Valuation
This paper graphically demonstrates the significant impact of the observed financial market persistence, i.e., long term memory or dependence, on European option valuation. Many empirical researchers have observed non-Fickian degrees of persistence or long memory in the financial markets different from the Fickian neutral independence (i.i.d.) of the returns innovations assumption of Black-Scholes' geometric Brownian motion assumption. Moreover, Elliott and van der Hoek (2003) have now also provided a theoretical framework for incorporating these findings in the Black-Scholes risk-neutral valuation framework. This paper provides the first graphical demonstration why and how such long term memory phenomena change European option values and provides thereby a basis for informed long term memory arbitrage. Risk-neutral valuation is equivalent to valuation by real world probabilities. By using a mono-fractional Brownian motion, it is easy to incorporate the various degrees of persistence into the binomial and Black-Scholes pricing formulas. Long memory options are of considerable importance in Corporate remuneration packages, since warrants are written on a company's own shares for long expiration periods. Therefore, we recommend that for a proper valuation of such warrants, the degrees of persistence of the companies' share markets are measured and properly incorporated in the warrant valuation.Options, Long Memory, Persistence, Hurst Exponent, Executive Remuneration
Blocks adjustment -- reduction of bias and variance of detrended fluctuation analysis using Monte Carlo simulation
The length of minimal and maximal blocks equally distant on log-log scale
versus fluctuation function considerably influences bias and variance of DFA.
Through a number of extensive Monte Carlo simulations and different fractional
Brownian motion/fractional Gaussian noise generators, we found the pair of
minimal and maximal blocks that minimizes the sum of mean-squared error of
estimated Hurst exponents for the series of length N=2^p, p=7,...,15.
Sensitivity of DFA to sort-range correlations was examined using ARFIMA(p,d,q)
generator. Due to the bias of the estimator for anti-persistent processes, we
narrowed down the range of Hurst exponent to 1/2<=H< 1.Comment: 20 pages, 14 figures, accepted for publication in Physica A: August
9, 200
Long Memory Options: LM Evidence and Simulations
This paper demonstrates the impact of the observed financial market persistence or long term memory on European option valuation by simple simulation. Many empirical researchers have observed the non-Fickian degrees of persistence or long memory in the financial markets different from the Fickian neutral independence (i.i.d.) of the returns innovations assumption of Black-Scholes' geometric Brownian motion assumption. Moreover, Elliott and van der Hoek (2003) provide a theoretical framework for incorporating these findings into the Black- Scholes risk-neutral valuation framework. This paper provides the first graphical demonstration why and how such long term memory phenomena change European option values and provides thereby a basis for informed long term memory arbitrage. By using a simple mono-fractal Fractional Brownian motion, it is easy to incorporate the various degrees of persistence into the Black-Scholes pricing formula. Long memory options are of considerable importance in corporate remuneration packages, since stock options are written on a company's own shares for long expiration periods. It makes a significant difference in the valuation when an option is 'blue' or when it is 'red.' For a proper valuation of such stock options, the degrees of persistence of the companies' share markets must be precisely measured and properly incorporated in the warrant valuation, otherwise substantial pricing errors may result.Options, Long Memory, Persistence, Hurst Exponent, Identification, Simulation, Executive Remuneration
On the relationship between the Hurst exponent, the ratio of the mean square successive difference to the variance, and the number of turning points
The long range dependence of the fractional Brownian motion (fBm), fractional
Gaussian noise (fGn), and differentiated fGn (DfGn) is described by the Hurst
exponent . Considering the realisations of these three processes as time
series, they might be described by their statistical features, such as half of
the ratio of the mean square successive difference to the variance,
, and the number of turning points, . This paper investigates
the relationships between and , and between and . It is
found numerically that the formulae in case of
fBm, and for fGn and DfGn, describe well the
relationship. When is considered, no simple formula is
found, and it is empirically found that among polynomials, the fourth and
second order description applies best. The most relevant finding is that when
plotted in the space of , the three process types form
separate branches. Hence, it is examined whether and may
serve as Hurst exponent indicators. Some real world data (stock market indices,
sunspot numbers, chaotic time series) are analyzed for this purpose, and it is
found that the 's estimated using the relations (expressed
as inverted functions) are consistent with the 's extracted
with the well known wavelet approach. This allows to efficiently estimate the
Hurst exponent based on fast and easy to compute and , given
that the process type: fBm, fGn or DfGn, is correctly classified beforehand.
Finally, it is suggested that the relation for fGn and DfGn
might be an exact (shifted) power-law.Comment: 20 pages in one-column format, 7 figures; matches the version
accepted for publicatio
Wavelet Multiresolution Analysis of High-Frequency Asian FX Rates, Summer 1997
FX pricing processes are nonstationary and their frequency characteristics are time-dependent. Most do not conform to geometric Brownian motion, since they exhibit a scaling law with a Hurst exponent between zero and 0.5 and fractal dimensions between 1.5 and 2. This paper uses wavelet multiresolution analysis, with Haar wavelets, to analyze the nonstationarity (time-dependence) and self-similarity (scale-dependence) of intra-day Asian currency spot exchange rates. These are the ask and bid quotes of the currencies of eight Asian countries (Japan, Hong Kong, Indonesia, Malaysia, Philippines, Singapore, Taiwan, Thailand), and of Germany for comparison, for the crisis period May 1, 1998 - August 31, 1997, provided by Telerate (U.S. dollar is the numeraire). Their time-scale dependent spectra, which are localized in time, are observed in wavelet based scalograms. The FX increments can be characterized by the irregularity of their singularities. This degrees of irregularity are measured by homogeneous Hurst exponents. These critical exponents are used to identify the fractal dimension, relative stability and long term dependence of each Asian FX series. The invariance of each identified Hurst exponent is tested by comparing it at varying time and scale (frequency) resolutions. It appears that almost all FX markets show anti-persistent pricing behavior. The anchor currencies of the D-mark and Japanese Yen are ultra-efficient in the sense of being most anti-persistent. The Taiwanese dollar is the most persistent, and thus unpredictable, most likely due to administrative control. FX markets exhibit these non- linear, non-Gaussian dynamic structures, long term dependence, high kurtosis, and high degrees of non-informational (noise) trading, possibly because of frequent capital flows induced by non-synchronized regional business cycles, rapidly changing political risks, unexpected informational shocks to investment opportunities, and, in particular, investment strategies synthesizing interregional claims using cash swaps with different duration horizons.foreign exchange markets, anti-persistence, long-term dependence, multi-resolution analysis, wavelets, time-scale analysis, scaling laws, irregularity analysis, randomness, Asia
Wavelet Multiresolution Analysis of High-Frequency FX Rates, Summer 1997
FX pricing processes are nonstationary and their frequency characteristics are time-dependent. Most do not conform to geometric Brownian motion, since they exhibit a scaling law with a Hurst exponent between zero and 0.5 and fractal dimensions between 1.5 and 2. This paper uses wavelet multiresolution analysis, with Haar wavelets, to analyze the nonstationarity (time-dependence) and self-similarity (scale-dependence) of intra-day Asian currency spot exchange rates.foreign exchange, anti-persistence, multi-resolution analysis, wavelets, Asia
Identification of the multiscale fractional Brownian motion with biomechanical applications
In certain applications, for instance biomechanics, turbulence, finance, or
Internet traffic, it seems suitable to model the data by a generalization of a
fractional Brownian motion for which the Hurst parameter is depending on
the frequency as a piece-wise constant function. These processes are called
multiscale fractional Brownian motions. In this contribution, we provide a
statistical study of the multiscale fractional Brownian motions. We develop a
method based on wavelet analysis. By using this method, we find initially the
frequency changes, then we estimate the different parameters and afterwards we
test the goodness-of-fit. Lastly, we give the numerical algorithm.
Biomechanical data are then studied with these new tools
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