46 research outputs found
Comparison of Field Theory Models of Interest Rates with Market Data
We calibrate and test various variants of field theory models of the interest
rate with data from eurodollars futures. A model based on a simple
psychological factor are seen to provide the best fit to the market. We make a
model independent determination of the volatility function of the forward rates
from market data.Comment: 9 figure
A Path Integral Approach to Option Pricing with Stochastic Volatility: Some Exact Results
The Black-Scholes formula for pricing options on stocks and other securities
has been generalized by Merton and Garman to the case when stock volatility is
stochastic. The derivation of the price of a security derivative with
stochastic volatility is reviewed starting from the first principles of
finance. The equation of Merton and Garman is then recast using the path
integration technique of theoretical physics. The price of the stock option is
shown to be the analogue of the Schrodinger wavefuction of quantum mechacnics
and the exact Hamiltonian and Lagrangian of the system is obtained. The results
of Hull and White are generalized results for pricing stock options for the
general correlated case are derived.Comment: Needs subeqnarray.sty. To appear in J. de Phys. I (Dec 97
A Common Market Measure for Libor and Pricing Caps, Floors and Swaps in a Field Theory of Forward Interest Rates
The main result of this paper that a martingale evolution can be chosen for
Libor such that all the Libor interest rates have a common market measure; the
drift is fixed such that each Libor has the martingale property. Libor is
described using a field theory model, and a common measure is seen to be emerge
naturally for such models. To elaborate how the martingale for the Libor
belongs to the general class of numeraire for the forward interest rates, two
other numeraire's are considered, namely the money market measure that makes
the evolution of the zero coupon bonds a martingale, and the forward measure
for which the forward bond price is a martingale. The price of an interest rate
cap is computed for all three numeraires, and is shown to be numeraire
invariant. Put-call parity is discussed in some detail and shown to emerge due
to some non-trivial properties of the numeraires. Some properties of swaps, and
their relation to caps and floors, are briefly discussed.Comment: 28 pages, 4 figure
Superstrings, Gauge Fields and Black Holes
There has been spectacular progress in the development of string and
superstring theories since its inception thirty years ago. Development in this
area has never been impeded by the lack of experimental confirmation. Indeed,
numerous bold and imaginative strides have been taken and the sheer elegance
and logical consistency of the arguments have served as a primary motivation
for string theorists to push their formulations ahead. In fact the development
in this area has been so rapid that new ideas quickly become obsolete. On the
other hand, this rapid development has proved to be the greatest hindrance for
novices interested in this area. These notes serve as a gentle introduction to
this topic. In these elementary notes, we briefly review the RNS formulation of
superstring theory, GSO projection, -branes, bosonic strings, dualities,
dynamics of -branes and the microscopic description of Bekenstein entropy of
a black hole.Comment: Lecture notes for talk delivered at NUS in 1997-1998. Some recent
updates added. The material may be somewhat outdated but it could still be
useful for physicists new to the fiel
Maximum temperature for an Ideal Gas of Kac-Moody Fermions
A lagrangian for gauge fields coupled to fermions with the Kac-Moody group as
its gauge group yields, for the pure fermions sector, an ideal gas of Kac-Moody
fermions. The canonical partition function for the case is shown to
have a maximum temperature , where is the
coupling of the super charge operator to the fermions. This result is
similar to the case of strings but unlike strings the result is obtained from a
well-defined lagrangian.Comment: Needs subeqnarray.sty; To be published in Phys. Rev. D, Dec 15, 1995.
Some typographical errors have been corrected in the revised versio
Quantum Field Theory of Forward Rates with Stochastic Volatility
In a recent formulation of a quantum field theory of forward rates, the
volatility of the forward rates was taken to be deterministic. The field theory
of the forward rates is generalized to the case of stochastic volatility. Two
cases are analyzed, firstly when volatility is taken to be a function of the
forward rates, and secondly when volatility is taken to be an independent
quantum field. Since volatiltiy is a positive valued quantum field, the full
theory turns out to be an interacting nonlinear quantum field theory in two
dimensions. The state space and Hamiltonian for the interacting theory are
obtained, and shown to have a nontrivial structure due to the manifold moving
with a constant velocity. The no arbitrage condition is reformulated in terms
of the Hamiltonian of the system, and then exactly solved for the nonlinear
interacting case.Comment: 7 Figure
Market dynamics immediately before and after financial shocks: quantifying the Omori, productivity and Bath laws
We study the cascading dynamics immediately before and immediately after 219
market shocks. We define the time of a market shock T_{c} to be the time for
which the market volatility V(T_{c}) has a peak that exceeds a predetermined
threshold. The cascade of high volatility "aftershocks" triggered by the "main
shock" is quantitatively similar to earthquakes and solar flares, which have
been described by three empirical laws --- the Omori law, the productivity law,
and the Bath law. We analyze the most traded 531 stocks in U.S. markets during
the two-year period 2001-2002 at the 1-minute time resolution. We find
quantitative relations between (i) the "main shock" magnitude M \equiv \log
V(T_{c}) occurring at the time T_{c} of each of the 219 "volatility quakes"
analyzed, and (ii) the parameters quantifying the decay of volatility
aftershocks as well as the volatility preshocks. We also find that stocks with
larger trading activity react more strongly and more quickly to market shocks
than stocks with smaller trading activity. Our findings characterize the
typical volatility response conditional on M, both at the market and the
individual stock scale. We argue that there is potential utility in these three
statistical quantitative relations with applications in option pricing and
volatility trading.Comment: 16 pages, double column, 13 figures, 1 Table; Changes made in Version
2 in response to referee comment
The Kinetic Interpretation of the DGLAP Equation, its Kramers-Moyal Expansion and Positivity of Helicity Distributions
According to a rederivation - due to Collins and Qiu - the DGLAP equation can
be reinterpreted (in leading order) in a probabilistic way. This form of the
equation has been used indirectly to prove the bound
between polarized and unpolarized distributions, or positivity of the helicity
distributions, for any . We reanalize this issue by performing a detailed
numerical study of the positivity bounds of the helicity distributions. To
obtain the numerical solution we implement an x-space based algorithm for
polarized and unpolarized distributions to next-to-leading order in ,
which we illustrate. We also elaborate on some of the formal properties of the
Collins-Qiu form and comment on the underlying regularization, introduce a
Kramers-Moyal expansion of the equation and briefly analize its Fokker-Planck
approximation. These follow quite naturally once the master version is given.
We illustrate this expansion both for the valence quark distribution and
for the transverse spin distribution .Comment: 38 pages, 27 figures, Dedicated to Prof. Pierre Ramond for his 60th
birthda
Generalized pricing formulas for stochastic volatility jump diffusion models applied to the exponential Vasicek model
Path integral techniques for the pricing of financial options are mostly
based on models that can be recast in terms of a Fokker-Planck differential
equation and that, consequently, neglect jumps and only describe drift and
diffusion. We present a method to adapt formulas for both the path-integral
propagators and the option prices themselves, so that jump processes are taken
into account in conjunction with the usual drift and diffusion terms. In
particular, we focus on stochastic volatility models, such as the exponential
Vasicek model, and extend the pricing formulas and propagator of this model to
incorporate jump diffusion with a given jump size distribution. This model is
of importance to include non-Gaussian fluctuations beyond the Black-Scholes
model, and moreover yields a lognormal distribution of the volatilities, in
agreement with results from superstatistical analysis. The results obtained in
the present formalism are checked with Monte Carlo simulations.Comment: 9 pages, 2 figures, 1 tabl
Brownian markets
Financial market dynamics is rigorously studied via the exact generalized
Langevin equation. Assuming market Brownian self-similarity, the market return
rate memory and autocorrelation functions are derived, which exhibit an
oscillatory-decaying behavior with a long-time tail, similar to empirical
observations. Individual stocks are also described via the generalized Langevin
equation. They are classified by their relation to the market memory as heavy,
neutral and light stocks, possessing different kinds of autocorrelation
functions