In this note we describe briefly our HJM(H&W) model for DI Future Options which models the underl... more In this note we describe briefly our HJM(H&W) model for DI Future Options which models the underlying overnight CDI rate (represented as an equivalent continuously compounded short rate) as mean-reverting normal process embedded into the general no-arbitrage HJM framework. We present the final pricing formula and sketch its derivation using standard no-arbitrage arguments and some idealizations. I. NOTATION Notation. Lets start with establishing some notation. Today's date expressed as an offset in years from start of some "asof date" Expiry of the option expressed as time offset in years Futures expiration time last reserve date before futures expiration date as time (close) expressed as time offset in years Number of reserve days between option expiry date (including) and future expiration date (excluded) Market price of a DI futures contract at time t Closing price of the DI future on day DI Future's rate. Futures rate and not price is the standard market quote for DI Future DI Futures Option strike is typically expressed as futures rate quote DI Futures Option strike expressed as futures price quote Call price per one "dollar" of the underlying DI future notional Put price per one "dollar" of the underlying DI future notional Price of discount bond maturing at Volatility of the discount bond maturing at Volatility of continuously compounded forward rate for accrual period [ at time II. DIF OPTION UNDERLYING-DI FUTURE At the highest level the DI Futures Option underlying is the DI Futures contract and the payoff of the option depends on the difference between the future's price at the option's expiry and an effective strike expressed as futures price. A call option payoff of is given by the familiar canonical Call payoff expression: (1) where is DI futures price at option's expiry time, , and is a strike expressed as futures price. Because DI future is traded based on the futures rate quote and not the price quote, the strike of DIF option is contractually specified as futures rate strike, , the relationship between the two being:
In this note we describe the HJM(LLM) model for pricing Mid-Curve Money Market Future Options. Th... more In this note we describe the HJM(LLM) model for pricing Mid-Curve Money Market Future Options. The model is based on assuming a lognormal process for the relevant forward money market (“LIBOR”) rates and imbedding it into the general no-arbitrage HJM interest rate modeling framework (understood in the most general sense as a manifestly self-consistent framework for formulating interest rate models in terms of any type of interest rate, including the observable market rates [2, 3, 4]). The relevant convexity adjustment factor is calculated by leveraging on the known functional form for the discount bond price volatility in any no-arbitrage lognormal “LIBOR” rate model, and approximating it with a leading order deterministic approximation. This makes it possible to derive an analytic expression for the HJM(LLM) model Discount Propagator using the path integral techniques and leads to the explicit analytic pricing formulas for the Mid-Curve futures options.
We consider the general nonstationary passive systems modeled by linearRCG (G = gyrator) netw... more We consider the general nonstationary passive systems modeled by linearRCG (G = gyrator) networks. Such networks in general contain conventional topological degeneracies (i.e., capacitor-only loops and/or inductor-only cutsets) as well as the topological degeneracies due to gyrator positioning in the network. Our central result is the explict demonstration that the existence of topological degeneracies does not impose any obstruction to the existence of the explicit state model, which we derive under no restriction on the topology of the network. We also discuss the conditions for the existence of the state model, its unique solutions, and the continuity of the state vector. The nature of the degeneracies inherent in the formulation is highlighted and it is shown that a gyrator-only subnetwork is accountable for algebraic degeneracies. Since the nature and the existence of topological degeneracies does not have anything to do with whether the element characteristics are linear or nonlinear, passive or active, our results are easy to extend to a large class of nonlinear and/or activeRCG networks.
Petar Simic was supported by a Fellowship from the Rockefeller University.
The Rockefeller University - Dissertation Abstracts International
This thesis consists of two parts. In the first, we develop a correct canonical formulation of lo... more This thesis consists of two parts. In the first, we develop a correct canonical formulation of local field theory in the colored monopole sector of realistic, spontaneously broken non-Abelian theories with fermions. Then, we study the charge properties and excitations of the ground state in the soliton (monopole) sector of ordinary Grand Unified Theories (GUTs) as well as in GUTs in which the Peccei -Quinn symmetry is used to solve the strong CP problem. In the second part, we study the problem of constructing the effective low energy theory of the light mesons in QCD starting from first principles. We show that the partition functional of QCD in the limit of large number of colors (N) and extremely low energies can be written in terms of (massive) quarks and (light) meson degrees of freedom. Integrating out the quarks, we derive a somewhat extended version of the Skyrme model; it contains the anomaly term (Wess-Zumino term) as well as non-topological terms, the coefficients of which depend on a classical scalar meson background. Soliton stability is discussed and an interpretation of our results as the link between QCD and the topological soliton-bag model is proposed.
A systematic application of the collective coordinate method to quantization of spontaneously bro... more A systematic application of the collective coordinate method to quantization of spontaneously broken gauge theories with fermions, such as GUTs, in the monopole sector is presented. Usual and unusual charge properties of the ground state GUT monopole induced by CP violation, and in particular by the fact that thetaQCD != thetaEM, are studied. In the presence of axions, semiclassically the ground state of GUT monopole is an actual realization of a spherical color-capacitor with the capacitance controlled by the magnitude of the Peccei-Quinn symmetry breaking scale. The "vacuum" around it is characterized by strongly enhanced (strong) CP violation and extends considerably far outside the monopole core. The effect of light fermions is studied in some detail. The possibility that even in the presence of axions (i.e. thetaQCD=0) there might be a non-vanishing and, in general, order-one color electric charge on the monopole ground state is pointed out.
A framework for extracting the low-energy dynamics of pseudoscalar mesons from QCD at large N is ... more A framework for extracting the low-energy dynamics of pseudoscalar mesons from QCD at large N is developed and, to leading order in the decoupling of heavy mesons, the pure pseudoscalar theory is calculated truncated to four derivatives. The soliton is found not to be manifestly stabilized by the four derivative terms. Under a plausible assumption about the classical solution for
In the limit of extremely low energies QCD describes essentially the interactions between the app... more In the limit of extremely low energies QCD describes essentially the interactions between the approximately massless pseudoscalars. They are the bound states of the ''chiral'' quarks. Equivalently they are collective Goldstone modes of dynamically broken chiral symmetry, and interact in a nonlinear way with the ''constituent'' quarks. These two pictures are related by going into the ''constituent gauge,'' which we define in this paper as a QCD analog of the unitary gauge in theories with Higgs scalars. We develop a framework for extracting the low-energy dynamics of pions directly from QCD in the limit of a large number of colors, and under some additional assumptions we calculate the pure pion theory truncated to four derivatives. The model obtained is a somewhat extended Skyrme model, and contains the anomaly term (Wess-Zumino term) as well as the nontopological terms, the coefficients of which depend on a classical scalar-meson-field background. Stability of the soliton is discussed. We show that in the limit in which symmetry breaking is turned off, the coefficients in front of the pseudoscalar interactions vanish. Under a plausible assumption about the behavior of a scalar-meson background, we interpret this as a natural realization of the space cutoff entering the topological soliton bag model.
The onset of dynamical chaos is studied numerically in (2+1)-dimensional non-Abelian field theory... more The onset of dynamical chaos is studied numerically in (2+1)-dimensional non-Abelian field theory with the Chern-Simons topological term. In the limit of strong fields, slowly varying in space (spatially homogeneous fields), this theory is an analog to a system of three charged particles moving in a plane in an orthogonal magnetic field and under the influence of a quartic potential. The ‘‘phase transition’’ (order chaos) is observed within a narrow energy range. The threshold of the transition depends on the sign of the angular momentum of the field reflecting parity violation in the underlying field theory. The transition region is investigated in some detail and the hyperfine structure of order-chaos-order-... transitions is observed suggesting the necessity of probabilistic description.
There is an interesting connection between two, recently popular, methods for finding good approx... more There is an interesting connection between two, recently popular, methods for finding good approximate solutions to hard optimisation problems, the ‘neural’ approach of Hopfield and Tank and the elastic-net method of Durbin and Willshaw. They both have an underlying statistical mechanics foundation and can be derived as the leading approximation to the thermodynamic free energy of related physical models. The apparent difference in the form of the two algorithms comes from different handling of constraints when evaluating the thermodynamic partition function. If all the constraints are enforced ‘softly’, the ‘mean-field’ approximation to the thermodynamic free energy is just the neural network Lyapunov function. If, on the other hand, half of the constraints are enforced ‘strongly’, the leading approximation to the thermodynamic free energy is the elastic-net Lyapunov function. Our results have interesting implications for the general problem of mapping optimisation problems to ‘neural’ and ‘elastic’ networks, and suggest a natural and systematic way to generalise the elastic net and ‘neural’ methods to a large class of hard optimisation problems. The author derives a new algorithm of the elastic-net type based on statistical mechanics. It has some of the ‘positive’ ingredients of the elastic-net method, yet it does not have an intrinsic problem (discussed in this paper) of the original algorithm.
Some time ago Durbin and Willshaw proposed an interesting parallel algorithm (the “elastic net”) ... more Some time ago Durbin and Willshaw proposed an interesting parallel algorithm (the “elastic net”) for approximately solving some geometric optimization problems, such as the Traveling Salesman Problem. Recently it has been shown that their algorithm is related to neural networks of Hopfield and Tank, and that they both can be understood as the semiclassical approximation to statistical mechanics of related physical models. The main point of the elastic net algorithm is seen to be in the way one deals with the constraints when evaluating the effective cost function (free energy in the thermodynamic analogy), and not in its geometric foundation emphasized originally by Durbin and Willshaw. As a consequence, the elastic net algorithm is a special case of the more general physically based computations and can be generalized to a large class of nongeometric problems. In this paper we further elaborate on this observation, and generalize the elastic net to the quadratic assignment problem. We work out in detail its special case, the graph matching problem, because it is an important problem with many applications in computational vision and neural modeling. Simulation results on random graphs, and on structured (hand-designed) graphs of moderate size (20-100 nodes) are discussed.
We describe the use of neural networks for optimization and inference associated with a variety o... more We describe the use of neural networks for optimization and inference associated with a variety of complex systems. We show how a string formalism can be used for parallel computer decomposition, message routing and sequential optimizing compilers. We extend these ideas to a general treatment of spatial assessment and distributed artificial intelligence.
We show that the asymptotic value of the magnification exponent of the one-dimensional elastic ne... more We show that the asymptotic value of the magnification exponent of the one-dimensional elastic net algorithm is equal to 1, indicating optimal information conserving properties of the algorithm.
An analogue-computer structure consisting of operational amplifiers with no more than three input... more An analogue-computer structure consisting of operational amplifiers with no more than three inputs and multipliers is described for generating arbitrary orthogonal polynomials and some orthogonal functions.
In this note we describe briefly our HJM(H&W) model for DI Future Options which models the underl... more In this note we describe briefly our HJM(H&W) model for DI Future Options which models the underlying overnight CDI rate (represented as an equivalent continuously compounded short rate) as mean-reverting normal process embedded into the general no-arbitrage HJM framework. We present the final pricing formula and sketch its derivation using standard no-arbitrage arguments and some idealizations. I. NOTATION Notation. Lets start with establishing some notation. Today's date expressed as an offset in years from start of some "asof date" Expiry of the option expressed as time offset in years Futures expiration time last reserve date before futures expiration date as time (close) expressed as time offset in years Number of reserve days between option expiry date (including) and future expiration date (excluded) Market price of a DI futures contract at time t Closing price of the DI future on day DI Future's rate. Futures rate and not price is the standard market quote for DI Future DI Futures Option strike is typically expressed as futures rate quote DI Futures Option strike expressed as futures price quote Call price per one "dollar" of the underlying DI future notional Put price per one "dollar" of the underlying DI future notional Price of discount bond maturing at Volatility of the discount bond maturing at Volatility of continuously compounded forward rate for accrual period [ at time II. DIF OPTION UNDERLYING-DI FUTURE At the highest level the DI Futures Option underlying is the DI Futures contract and the payoff of the option depends on the difference between the future's price at the option's expiry and an effective strike expressed as futures price. A call option payoff of is given by the familiar canonical Call payoff expression: (1) where is DI futures price at option's expiry time, , and is a strike expressed as futures price. Because DI future is traded based on the futures rate quote and not the price quote, the strike of DIF option is contractually specified as futures rate strike, , the relationship between the two being:
In this note we describe the HJM(LLM) model for pricing Mid-Curve Money Market Future Options. Th... more In this note we describe the HJM(LLM) model for pricing Mid-Curve Money Market Future Options. The model is based on assuming a lognormal process for the relevant forward money market (“LIBOR”) rates and imbedding it into the general no-arbitrage HJM interest rate modeling framework (understood in the most general sense as a manifestly self-consistent framework for formulating interest rate models in terms of any type of interest rate, including the observable market rates [2, 3, 4]). The relevant convexity adjustment factor is calculated by leveraging on the known functional form for the discount bond price volatility in any no-arbitrage lognormal “LIBOR” rate model, and approximating it with a leading order deterministic approximation. This makes it possible to derive an analytic expression for the HJM(LLM) model Discount Propagator using the path integral techniques and leads to the explicit analytic pricing formulas for the Mid-Curve futures options.
We consider the general nonstationary passive systems modeled by linearRCG (G = gyrator) netw... more We consider the general nonstationary passive systems modeled by linearRCG (G = gyrator) networks. Such networks in general contain conventional topological degeneracies (i.e., capacitor-only loops and/or inductor-only cutsets) as well as the topological degeneracies due to gyrator positioning in the network. Our central result is the explict demonstration that the existence of topological degeneracies does not impose any obstruction to the existence of the explicit state model, which we derive under no restriction on the topology of the network. We also discuss the conditions for the existence of the state model, its unique solutions, and the continuity of the state vector. The nature of the degeneracies inherent in the formulation is highlighted and it is shown that a gyrator-only subnetwork is accountable for algebraic degeneracies. Since the nature and the existence of topological degeneracies does not have anything to do with whether the element characteristics are linear or nonlinear, passive or active, our results are easy to extend to a large class of nonlinear and/or activeRCG networks.
Petar Simic was supported by a Fellowship from the Rockefeller University.
The Rockefeller University - Dissertation Abstracts International
This thesis consists of two parts. In the first, we develop a correct canonical formulation of lo... more This thesis consists of two parts. In the first, we develop a correct canonical formulation of local field theory in the colored monopole sector of realistic, spontaneously broken non-Abelian theories with fermions. Then, we study the charge properties and excitations of the ground state in the soliton (monopole) sector of ordinary Grand Unified Theories (GUTs) as well as in GUTs in which the Peccei -Quinn symmetry is used to solve the strong CP problem. In the second part, we study the problem of constructing the effective low energy theory of the light mesons in QCD starting from first principles. We show that the partition functional of QCD in the limit of large number of colors (N) and extremely low energies can be written in terms of (massive) quarks and (light) meson degrees of freedom. Integrating out the quarks, we derive a somewhat extended version of the Skyrme model; it contains the anomaly term (Wess-Zumino term) as well as non-topological terms, the coefficients of which depend on a classical scalar meson background. Soliton stability is discussed and an interpretation of our results as the link between QCD and the topological soliton-bag model is proposed.
A systematic application of the collective coordinate method to quantization of spontaneously bro... more A systematic application of the collective coordinate method to quantization of spontaneously broken gauge theories with fermions, such as GUTs, in the monopole sector is presented. Usual and unusual charge properties of the ground state GUT monopole induced by CP violation, and in particular by the fact that thetaQCD != thetaEM, are studied. In the presence of axions, semiclassically the ground state of GUT monopole is an actual realization of a spherical color-capacitor with the capacitance controlled by the magnitude of the Peccei-Quinn symmetry breaking scale. The "vacuum" around it is characterized by strongly enhanced (strong) CP violation and extends considerably far outside the monopole core. The effect of light fermions is studied in some detail. The possibility that even in the presence of axions (i.e. thetaQCD=0) there might be a non-vanishing and, in general, order-one color electric charge on the monopole ground state is pointed out.
A framework for extracting the low-energy dynamics of pseudoscalar mesons from QCD at large N is ... more A framework for extracting the low-energy dynamics of pseudoscalar mesons from QCD at large N is developed and, to leading order in the decoupling of heavy mesons, the pure pseudoscalar theory is calculated truncated to four derivatives. The soliton is found not to be manifestly stabilized by the four derivative terms. Under a plausible assumption about the classical solution for
In the limit of extremely low energies QCD describes essentially the interactions between the app... more In the limit of extremely low energies QCD describes essentially the interactions between the approximately massless pseudoscalars. They are the bound states of the ''chiral'' quarks. Equivalently they are collective Goldstone modes of dynamically broken chiral symmetry, and interact in a nonlinear way with the ''constituent'' quarks. These two pictures are related by going into the ''constituent gauge,'' which we define in this paper as a QCD analog of the unitary gauge in theories with Higgs scalars. We develop a framework for extracting the low-energy dynamics of pions directly from QCD in the limit of a large number of colors, and under some additional assumptions we calculate the pure pion theory truncated to four derivatives. The model obtained is a somewhat extended Skyrme model, and contains the anomaly term (Wess-Zumino term) as well as the nontopological terms, the coefficients of which depend on a classical scalar-meson-field background. Stability of the soliton is discussed. We show that in the limit in which symmetry breaking is turned off, the coefficients in front of the pseudoscalar interactions vanish. Under a plausible assumption about the behavior of a scalar-meson background, we interpret this as a natural realization of the space cutoff entering the topological soliton bag model.
The onset of dynamical chaos is studied numerically in (2+1)-dimensional non-Abelian field theory... more The onset of dynamical chaos is studied numerically in (2+1)-dimensional non-Abelian field theory with the Chern-Simons topological term. In the limit of strong fields, slowly varying in space (spatially homogeneous fields), this theory is an analog to a system of three charged particles moving in a plane in an orthogonal magnetic field and under the influence of a quartic potential. The ‘‘phase transition’’ (order chaos) is observed within a narrow energy range. The threshold of the transition depends on the sign of the angular momentum of the field reflecting parity violation in the underlying field theory. The transition region is investigated in some detail and the hyperfine structure of order-chaos-order-... transitions is observed suggesting the necessity of probabilistic description.
There is an interesting connection between two, recently popular, methods for finding good approx... more There is an interesting connection between two, recently popular, methods for finding good approximate solutions to hard optimisation problems, the ‘neural’ approach of Hopfield and Tank and the elastic-net method of Durbin and Willshaw. They both have an underlying statistical mechanics foundation and can be derived as the leading approximation to the thermodynamic free energy of related physical models. The apparent difference in the form of the two algorithms comes from different handling of constraints when evaluating the thermodynamic partition function. If all the constraints are enforced ‘softly’, the ‘mean-field’ approximation to the thermodynamic free energy is just the neural network Lyapunov function. If, on the other hand, half of the constraints are enforced ‘strongly’, the leading approximation to the thermodynamic free energy is the elastic-net Lyapunov function. Our results have interesting implications for the general problem of mapping optimisation problems to ‘neural’ and ‘elastic’ networks, and suggest a natural and systematic way to generalise the elastic net and ‘neural’ methods to a large class of hard optimisation problems. The author derives a new algorithm of the elastic-net type based on statistical mechanics. It has some of the ‘positive’ ingredients of the elastic-net method, yet it does not have an intrinsic problem (discussed in this paper) of the original algorithm.
Some time ago Durbin and Willshaw proposed an interesting parallel algorithm (the “elastic net”) ... more Some time ago Durbin and Willshaw proposed an interesting parallel algorithm (the “elastic net”) for approximately solving some geometric optimization problems, such as the Traveling Salesman Problem. Recently it has been shown that their algorithm is related to neural networks of Hopfield and Tank, and that they both can be understood as the semiclassical approximation to statistical mechanics of related physical models. The main point of the elastic net algorithm is seen to be in the way one deals with the constraints when evaluating the effective cost function (free energy in the thermodynamic analogy), and not in its geometric foundation emphasized originally by Durbin and Willshaw. As a consequence, the elastic net algorithm is a special case of the more general physically based computations and can be generalized to a large class of nongeometric problems. In this paper we further elaborate on this observation, and generalize the elastic net to the quadratic assignment problem. We work out in detail its special case, the graph matching problem, because it is an important problem with many applications in computational vision and neural modeling. Simulation results on random graphs, and on structured (hand-designed) graphs of moderate size (20-100 nodes) are discussed.
We describe the use of neural networks for optimization and inference associated with a variety o... more We describe the use of neural networks for optimization and inference associated with a variety of complex systems. We show how a string formalism can be used for parallel computer decomposition, message routing and sequential optimizing compilers. We extend these ideas to a general treatment of spatial assessment and distributed artificial intelligence.
We show that the asymptotic value of the magnification exponent of the one-dimensional elastic ne... more We show that the asymptotic value of the magnification exponent of the one-dimensional elastic net algorithm is equal to 1, indicating optimal information conserving properties of the algorithm.
An analogue-computer structure consisting of operational amplifiers with no more than three input... more An analogue-computer structure consisting of operational amplifiers with no more than three inputs and multipliers is described for generating arbitrary orthogonal polynomials and some orthogonal functions.
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Papers by Petar D Simic
Petar Simic was supported by a Fellowship from the Rockefeller University.
Low-energy meson action from QCD: Extended Skyrme model - ResearchGate. Available from: http://www.researchgate.net/publication/13332669_Low-energy_meson_action_from_QCD_Extended_Skyrme_model [accessed Jun 4, 2015].
Talks by Petar D Simic
Petar Simic was supported by a Fellowship from the Rockefeller University.
Low-energy meson action from QCD: Extended Skyrme model - ResearchGate. Available from: http://www.researchgate.net/publication/13332669_Low-energy_meson_action_from_QCD_Extended_Skyrme_model [accessed Jun 4, 2015].