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

Dynamic Risk Profile of the US Term Structure by Wavelet MRA

A careful examination of interest rate time series from different U.S. Treasury maturities by Wavelet Multiresolution Analysis (MRA) suggests that the first differences of the term structure of interest rate series are periodic or, at least, cyclic, non-stationary, long-term dependent, in particular, anti-persistent. Each nodal time series from a particular maturity has its own uniqueness and accordingly supports the Market Segmentation theory. The findings also imply that affine models are insufficient to describe the dynamics of the interest rate diffusion processes and call for more intensive research that might provide better, most likely fractal or nonlinear, term structure models for each maturity. If this is correct, empirical term structure models may describe chaotic, i.e., diffusion processes with non-unique dynamic equilibria.Wavelet, Interest rates, Hurst exponent, Term structure, Yield curve

Multifractal Modeling of the US Treasury Term Structure and Fed Funds Rate

This paper identifies the Multifractal Models of Asset Return (MMARs) from the eight nodal term structure series of US Treasury rates as well as the Fed Funds rate and, after proper synthesis, simulates those MMARs. We show that there is an inverse persistence term structure in the sense that the short term interest rates show the highest persistence, while the long term rates are closer to the GBM's neutral persistence. The simulations of the identified MMAR are compared with the original empirical time series, but also with the simulated results from the corresponding Brownian Motion and GARCH processes. We find that the eight different maturity US Treasury and the Fed Funds rates are multifractal processes. Moreover, using wavelet scalograms, we demonstrate that the MMAR outperforms both the GBM and GARCH(1,1) in time-frequency comparisons, in particular in terms of scaling distribution preservation. Identified distributions of all simulated processes are compared with the empirical distributions in snapshot and over time-scale (frequency) analyses. The simulated MMAR can replicate all attributes of the empirical distributions, while the simulated GBM and GARCH(1,1) processes cannot preserve the thick-tails, high peaks and proper skewness. Nevertheless, the results are somewhat inconclusive when the MMAR is applied on the Fed Funds rate, which has globally a mildly anti-persistent and possibly chaotic diffusion process completely different from the other nodal term structure rates.MMAR, multifractal spectrum, long memory, scaling, term stucture, persistence, Brownian motion, GARCH, time-frequency analysis

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