Pricing bounds for discrete arithmetic Asian options under Lévy models

Analytical bounds for Asian options are almost exclusively available in the Black-Scholes framework. In this paper we derive bounds for the price of a discretely monitored arithmetic Asian option when the underlying asset follows an arbitrary Lévy process. Explicit formulas are given for Kou’s model, Merton’s model, the normal inverse Gaussian model, the CGMY model and the variance gamma model. The results are compared with the comonotonic upper bound, existing numerical results, Monte carlo simulations and in the case of the variance gamma model with an existing lower bound. The method outlined here provides lower and upper bounds that are quick to evaluate, and more accurate than existing bounds.     

Option Pricing in Subdiffusive Bachelier Model

The earliest model of stock prices based on Brownian diffusion is the Bachelier model. In this paper we propose an extension of the Bachelier model, which reflects the subdiffusive nature of the underlying asset dynamics. The subdiffusive property is manifested by the random (infinitely divisible) periods of time, during which the asset price does not change. We introduce a subdiffusive arithmetic Brownian motion as a model of stock prices with such characteristics. The structure of this process agrees with two-stage scenario underlying the anomalous diffusion mechanism, in which trapping random events are superimposed on the Langevin dynamics. We find the corresponding fractional Fokker-Planck equation governing the probability density function of the introduced process. We construct the corresponding martingale measure and show that the model is incomplete. We derive the formulas for European put and call option prices. We describe explicit algorithms and present some Monte-Carlo simulations for the particular cases of α-stable and tempered α-stable distributions of waiting times.     

The predictive performance of a path-dependent exotic-option credit risk model in the emerging market

Most empirical research of the path-dependent, exotic-option credit risk model focuses on developed markets. Taking Taiwan as an example, this study investigates the bankruptcy prediction performance of the path-dependent, barrier option model in the emerging market. We adopt Duan’s (1994) [11], (2000) [12] transformed-data maximum likelihood estimation (MLE) method to directly estimate the unobserved model parameters, and compare the predictive ability of the barrier option model to the commonly adopted credit risk model, Merton’s model. Our empirical findings show that the barrier option model is more powerful than Merton’s model in predicting bankruptcy in the emerging market. Moreover, we find that the barrier option model predicts bankruptcy much better for highly-leveraged firms. Finally, our findings indicate that the prediction accuracy of the credit risk model can be improved by higher asset liquidity and greater financial transparency.     

Long-memory volatility in derivative hedging

The aim of this work is to take into account the effects of long memory in volatility on derivative hedging. This idea is an extension of the work by Fedotov and Tan [Stochastic long memory process in option pricing, Int. J. Theor. Appl. Finance 8 (2005) 381–392] where they incorporate long-memory stochastic volatility in option pricing and derive pricing bands for option values. The starting point is the stochastic Black–Scholes hedging strategy which involves volatility with a long-range dependence. The stochastic hedging strategy is the sum of its deterministic term that is classical Black–Scholes hedging strategy with a constant volatility and a random deviation term which describes the risk arising from the random volatility. Using the fact that stock price and volatility fluctuate on different time scales, we derive an asymptotic equation for this deviation in terms of the Green's function and the fractional Brownian motion. The solution to this equation allows us to find hedging confidence intervals.     

A quantum model of option pricing: When Black-Scholes meets Schrödinger and its semi-classical limit

The Black-Scholes equation can be interpreted from the point of view of quantum mechanics, as the imaginary time Schrödinger equation of a free particle. When deviations of this state of equilibrium are considered, as a product of some market imperfection, such as: Transaction cost, asymmetric information issues, short-term volatility, extreme discontinuities, or serial correlations; the classical non-arbitrage assumption of the Black-Scholes model is violated, implying a non-risk-free portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the Black-Scholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new Black-Scholes-Schrödinger model based on the endogenous arbitrage option pricing formulation introduced by Contreras et al. (2010) [2]. Hence, we derive a more general quantum model of option pricing, that incorporates arbitrage as an external time dependent force, which has an associated potential related to the random dynamic of the underlying asset price. This new resultant model can be interpreted as a Schrödinger equation in imaginary time for a particle of mass 1/σ² with a wave function in an external field force generated by the arbitrage potential. As pointed out above, this new model can be seen as a more general formulation, where the perfect market equilibrium state postulated by the Black-Scholes model represent a particular case. Finally, since the Schrödinger equation is in place, we can apply semiclassical methods, of common use in theoretical physics, to find an approximate analytical solution of the Black-Scholes equation in the presence of market imperfections, as it is the case of an arbitrage bubble. Here, as a numerical illustration of the potential of this Schrödinger equation analogy, the semiclassical approximation is performed for different arbitrage bubble forms (step, linear and parabolic) and compare with the exact solution of our general quantum model of option pricing.     

Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing

Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.     

Ground state energy of N Frenkel excitons

By using the composite many-body theory for Frenkel excitons we have recently developed, we here derive the ground state energy of N Frenkel excitons in the Born approximation through the Hamiltonian mean value in a state made of N identical Q = 0 excitons. While this quantity reads as a density expansion in the case of Wannier excitons, due to many-body effects induced by fermion exchanges between N composite particles, we show that the Hamiltonian mean value for N Frenkel excitons only contains a first order term in density, just as for elementary bosons. Such a simple result comes from a subtle balance, difficult to guess a priori, between fermion exchanges for two or more Frenkel excitons appearing in Coulomb term and the ones appearing in the N exciton normalization factor – the cancellation being exact within terms in 1/N_s where N_s is the number of atomic sites in the sample. This result could make us naively believe that, due to the tight binding approximation on which Frenkel excitons are based, these excitons are just bare elementary bosons while their composite nature definitely appears at various stages in the precise calculation of the Hamiltonian mean value.     

The application of fractional derivatives in stochastic models driven by fractional Brownian motion

In this paper, in order to establish connection between fractional derivative and fractional Brownian motion (FBM), we first prove the validity of the fractional Taylor formula proposed by Guy Jumarie. Then, by using the properties of this Taylor formula, we derive a fractional Itô formula for H∈[1/2,1), which coincides in form with the one proposed by Duncan for some special cases, whose formula is based on the Wick Product. Lastly, we apply this fractional Itô formula to the option pricing problem when the underlying of the option contract is supposed to be driven by a geometric fractional Brownian motion. The case that the drift, volatility and risk-free interest rate are all dependent on t is also discussed.     

A quantile-based Time at Risk: A new approach for assessing risk in financial markets

In this paper, we provide a new measure for evaluation of risk in financial markets. This measure is based on the return interval of critical events in financial markets or other investment situations. Our main goal was to devise a model like Value at Risk (VaR). As VaR, for a given financial asset, probability level and time horizon, gives a critical value such that the likelihood of loss on the asset over the time horizon exceeds this value is equal to the given probability level, our concept of Time at Risk (TaR), using a probability distribution function of return intervals, provides a critical time such that the probability that the return interval of a critical event exceeds this time equals the given probability level. As an empirical application, we applied our model to data from the Tehran Stock Exchange Price Index (TEPIX) as a financial asset (market portfolio) and reported the results.     

Beyond Black-Scholes: semimartingales and Lévy processes for option pricing

We consider the problem of option pricing when the underlying asset follows a general semimartingale process. After reviewing the foundations of arbitrage pricing theory for semimartingales and the link with Lévy processes, we introduce a general method to price options in this framework based on Fourier and Wavelet analysis. Received 4 September 2000     

A study about the existence of the leverage effect in stochastic volatility models

The empirical relationship between the return of an asset and the volatility of the asset has been well documented in the financial literature. Named the leverage effect or sometimes risk-premium effect, it is observed in real data that, when the return of the asset decreases, the volatility increases and vice versa.Consequently, it is important to demonstrate that any formulated model for the asset price is capable of generating this effect observed in practice. Furthermore, we need to understand the conditions on the parameters present in the model that guarantee the apparition of the leverage effect.In this paper we analyze two general specifications of stochastic volatility models and their capability of generating the perceived leverage effect. We derive conditions for the apparition of leverage effect in both of these stochastic volatility models. We exemplify using stochastic volatility models used in practice and we explicitly state the conditions for the existence of the leverage effect in these examples.     

A maximum (non-extensive) entropy approach to equity options bid–ask spread

The cross-section of options bid–ask spreads with their strikes are modelled by maximising the Kaniadakis entropy. A theoretical model results with the bid–ask spread depending explicitly on the implied volatility; the probability of expiring at-the-money and an asymmetric information parameter (κ$\kappa$). Considering AIG as a test case for the period between January 2006 and October 2008, we find that information flows uniquely from the trading activity in the underlying asset to its derivatives. Suggesting that κ$\kappa$ is possibly an option implied measure of the current state of trading liquidity in the underlying asset.     

Non-monotonic swelling of a macroion due to correlation-induced charge inversion

It is known that a large, charged body immersed in a solution of multivalent counterions may undergo charge inversion as the counterions adsorb to its surface. We use the theory of charge inversion to examine the case of a deformable, porous macroion which may adsorb multivalent ions into its bulk to form a three-dimensional strongly-correlated liquid. This adsorption may lead to non-monotonic changes in the size of the macroion as multivalent ions are added to the solution. The macroion first shrinks as its bare charge is screened and then reswells as the adsorbed ions invert the sign of the net charge. We derive a value for the outward pressure experienced by such a macroion as a function of the ion concentration in solution. We find that for small deviations in the concentration of multivalent ions away from the neutral point (where the net charge of the body is zero), the swollen size grows parabolically with the logarithm of the ratio of multivalent ion concentration to the concentration at the neutral point.     

The (φ 4)3 + 1 theory with infinitesimal bare coupling constants

We study the (φ ⁴)_{3 + 1} theory by means of a variational method improved with a BCS-type vacuum state. We examine the theory with both negative and positive infinitesimal bare coupling constants, where the theory has been suggested to exist nontrivially and stably in the infinite ultraviolet cutoff limit. When the cutoff is sent to infinity, we find the instability of the vacuum energy at the end point value of the variational parameter in the case of the negative bare coupling constant. For the positive bare coupling constant, we can renormalize the vacuum energy without using the extremal condition with respect to the variational mass parameter. We do not find an instability for the whole range of parameters including the end point. We still have a possibility that the theory with this bare coupling constant is nontrivial and stable.     

Meson-Cloud Effects in the Electromagnetic Nucleon Structure

We study how the electromagnetic structure of the nucleon is influenced by a pion cloud. To this aim we make use of a constituent-quark model with instantaneous confinement and a pion that couples directly to the quarks. To derive the invariant 1-photon-exchange electron-nucleon scattering amplitude we employ a Poincaré-invariant coupled-channel formulation which is based on the point-form of relativistic quantum mechanics. We argue that the electromagnetic nucleon current extracted from this amplitude can be reexpressed in terms of pure hadronic degrees of freedom with the quark substructure of the pion and the nucleon being encoded in electromagnetic and strong vertex form factors. These are form factors of bare particles, i.e. eigenstates of the pure confinement problem. First numerical results for (bare) photon-nucleon and pion-nucleon form factors, which are the basic ingredients of the further calculation, are given for a simple 3-quark wave function of the nucleon.     

J-plane structure of the “cylinder”; slope and intercept of the bare pomeron

We derive and investigate an integral equation for the “cylinder” with full t-dependence. In contrast to the pure pole planar model, the “cylinder” contains, in addition to the bare pomeron pole, also a Reggeon-Reggeon cut. The strongly correlated slope and intercept of the bare pomeron have the correct observed values. We also briefly comment on the concept of “asymptotic planarity”.     

Optimized quantum random-walk search algorithm for multi-solution search

This study investigates the multi-solution search of the optimized quantum random-walk search algorithm on the hypercube. Through generalizing the abstract search algorithm which is a general tool for analyzing the search on the graph to the multi-solution case, it can be applied to analyze the multi-solution case of quantum random-walk search on the graph directly. Thus, the computational complexity of the optimized quantum random-walk search algorithm for the multi-solution search is obtained. Through numerical simulations and analysis, we obtain a critical value of the proportion of solutions q. For a given q, we derive the relationship between the success rate of the algorithm and the number of iterations when q is no longer than the critical value.     

Point-Form Hamiltonian Dynamics and Applications

This short review summarizes recent developments and results in connection with point-form dynamics of relativistic quantum systems. We discuss a Poincaré invariant multichannel formalism which describes particle production and annihilation via vertex interactions that are derived from field theoretical interaction densities. We sketch how this rather general formalism can be used to derive electromagnetic form factors of confined quark?Cantiquark systems. As a further application it is explained how the chiral constituent quark model leads to hadronic states that can be considered as bare hadrons dressed by meson loops. Within this approach hadron resonances acquire a finite (non-perturbative) decay width. We will also discuss the point-form dynamics of quantum fields. After recalling basic facts of the free-field case we will address some quantum field theoretical problems for which canonical quantization on a space?Ctime hyperboloid could be advantageous.     

Pricing on electricity market based on coupled-continuous-time-random-walk concept

In this paper we propose a model of electricity market based on the forward rate dynamics described by a diffusion with jumps as a generalization of the classical diffusion approach. We consider jump components resulting from a coupled continuous-time random walk (CTRW) with jump lengths proportional to the corresponding inter-jump time intervals. In the framework of the model we derive a formula for the EURO-price of a standard European call option, showing applicability of CTRW processes for pricing of financial instruments. The result, obtained by an advance theory of semimartingales, is an essential extension of the pricing formula derived in the classical diffusion model of the forward rate dynamics. It indicates an influence of both, the continuous and the jump parts of the forward rate process on the option price.