Efficient option pricing with path integral

An efficient computational algorithm to price financial derivatives is presented. It is based on a path integral formulation of the pricing problem. It is shown how the path integral approach can be worked out in order to obtain fast and accurate predictions for the value of a large class of options, including those with path-dependent and early exercise features. As examples, the application of the method to European and American options in the Black-Scholes model is illustrated. The results of the algorithm are compared with those obtained with the standard procedures known in the literature and found to be in good agreement. Received 31 December 2001     

Option pricing and perfect hedging on correlated stocks

We develop a theory for option pricing with perfect hedging in an inefficient market model where the underlying price variations are autocorrelated over a time τ0. This is accomplished by assuming that the underlying noise in the system is derived by an Ornstein-Uhlenbeck, rather than from a Wiener process. With a modified portfolio consisting in calls, secondary calls and bonds we achieve a riskless strategy which results in a closed and exact expression for the European call price which is always lower than Black-Scholes price. We obtain the same price and a modified delta hedging if we start from an effective one-dimensional market model. We compare these strategies and study the sensitivity of the call price to several parameters where the correlation effects are also observed.     

Market memory and fat tail consequences in option pricing on the expOU stochastic volatility model

The expOU stochastic volatility model is capable of reproducing fairly well most important statistical properties of financial markets daily data. Among them, the presence of multiple time scales in the volatility autocorrelation is perhaps the most relevant which makes appear fat tails in the return distributions. This paper wants to go further on with the expOU model we have studied in Ref. [J. Masoliver, J. Perelló, Quant. Finance 6 (2006) 423] by exploring an aspect of practical interest. Having as a benchmark the parameters estimated from the Dow Jones daily data, we want to compute the price for the European option. This is actually done by Monte Carlo, running a large number of simulations. Our main interest is to “see” the effects of a long-range market memory from our expOU model in its subsequent European call option. We pay attention to the effects of the existence of a broad range of time scales in the volatility. We find that a richer set of time scales brings the price of the option higher. This appears in clear contrast to the presence of memory in the price itself which makes the price of the option cheaper.     

Supersymmetry in option pricing

We use supersymmetry to find the isospectral partners of Black-Scholes Hamiltonian without a potential and with a double knock out barrier potential. The pricing kernels for these Hamiltonians have also been obtained.     

Parametric option pricing: A divide-and-conquer approach

Non-parametric option pricing models, such as artificial neural networks, are often found to outperform their parametric counterparts in empirical option pricing exercises. In this context, non-parametric models are viewed as more flexible and amenable to adaptive learning. However, the main drawback of non-parametric approaches is their lack of stability, which is detrimental to out-of-sample performance. This is the key reason why one may prefer a parsimonious parametric model. This paper proposes a parametric Takagi-Sugeno-Kang (TSK) fuzzy rule-based option pricing model that requires only a small number of rules to describe highly complex non-linear functions. The findings for this data-driven approach indicate that the TSK model presents a robust option pricing tool that is superior to an array of well-known parametric models from the literature. In addition, its predictive performance is consistently no worse than that of a non-parametric feedforward neural network model.     

Dynamic option pricing with endogenous stochastic arbitrage

Only few efforts have been made in order to relax one of the key assumptions of the Black-Scholes model: the no-arbitrage assumption. This is despite the fact that arbitrage processes usually exist in the real world, even though they tend to be short-lived. The purpose of this paper is to develop an option pricing model with endogenous stochastic arbitrage, capable of modelling in a general fashion any future and underlying asset that deviate itself from its market equilibrium. Thus, this investigation calibrates empirically the arbitrage on the futures on the S&P 500 index using transaction data from September 1997 to June 2009, from here a specific type of arbitrage called “arbitrage bubble”, based on a t-step function, is identified and hence used in our model. The theoretical results obtained for Binary and European call options, for this kind of arbitrage, show that an investment strategy that takes advantage of the identified arbitrage possibility can be defined, whenever it is possible to anticipate in relative terms the amplitude and timespan of the process. Finally, the new trajectory of the stock price is analytically estimated for a specific case of arbitrage and some numerical illustrations are developed. We find that the consequences of a finite and small endogenous arbitrage not only change the trajectory of the asset price during the period when it started, but also after the arbitrage bubble has already gone. In this context, our model will allow us to calibrate the B-S model to that new trajectory even when the arbitrage already started.     

Stochastic model for market stocks with floors

We present a model to describe the stochastic evolution of stocks that show a strong resistance at some level and generalize to this situation the evolution based upon geometric Brownian motion. If volatility and drift are related in a certain way we show that our model can be integrated in an exact way. The related problem of how to prize general securities that pay dividends at a continuous rate and earn a terminal payoff at maturity T is solved via the martingale probability approach.     

Thermoalgebras and path integral

Using a representation for Lie groups closely associated with thermal problems, we derive the algebraic rules of the real-time formalism for thermal quantum field theories, the so-called thermo-field dynamics (TFD), including the tilde conjugation rules for interacting fields. These thermo-group representations provide a unified view of different approaches for finite-temperature quantum fields in terms of a symmetry group. On these grounds, a path integral formalism is constructed, using Bogoliubov transformations, for bosons, fermions and non-abelian gauge fields. The generalization of the results for quantum fields in topology is addressed.     

Comment on “Time-changed geometric fractional Brownian motion and option pricing with transaction costs” by Hui Gu et al.

The purpose of this comment is to point out the inappropriate assumption of “3αH>1$3\alpha H>1$” and two problems in the proof of “Theorem 3.1” in section 3 of the paper “Time-changed geometric fractional Brownian motion and option pricing with transaction costs” by Hui Gu et al. [H. Gu, J.R. Liang, Y. X. Zhang, Time-changed geometric fractional Brownian motion and option pricing with transaction costs, Physica A 391 (2012) 3971–3977]. Then we show the two problems will be solved under our new assumption.     

On the path integral representation of stochastic processes

We derive the path integral representation of the conditional probability for a Markovian process starting from the master equation. Existing derivations require both the variable and the transition probability to be extensive. We show that this requirement may be relaxed if Langer's formula for the transition probability is used. We prove that different path integral representations appearing in the literature are in fact equivalent and correspond to various choices of an arbitrary parameter.This work was supported in part by the National Science Foundation and the Office of Naval Research (Project SQUID).     

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.     

Spin in the worldline path integral in 2 + 1 dimensions

We present a constructive derivation of a worldline path integral for the effective action and the propagator of a Dirac field in 2 + 1 dimensions, in terms of spacetime and SU(2) paths. After studying some general properties of this representation, we show that the auxiliary gauge-group variable can be integrated, deriving a worldline action depending only on x(τ), the spacetime paths. We then show that the functional integral automatically imposes the constraint , while there is a spin action, which agrees with the one one should expect for a spin- field.     

Credit risk—A structural model with jumps and correlations

We set up a structural model to study credit risk for a portfolio containing several or many credit contracts. The model is based on a jump-diffusion process for the risk factors, i.e. for the company assets. We also include correlations between the companies. We discuss that models of this type have much in common with other problems in statistical physics and in the theory of complex systems. We study a simplified version of our model analytically. Furthermore, we perform extensive numerical simulations for the full model. The observables are the loss distribution of the credit portfolio, its moments and other quantities derived thereof. We compile detailed information about the parameter dependence of these observables. In the course of setting up and analyzing our model, we also give a review of credit risk modeling for a physics audience.     

Scaling and long-range dependence in option pricing III: A fractional version of the Merton model with transaction costs

A model for option pricing of fractional version of the Merton model with ‘Hurst exponent’ H being in [1/2,1) is established with transaction costs. In particular, for H(1/2,1) the minimal price C_min(t,S_t) of an option under transaction costs is obtained, which displays that the timestep δt and the ‘Hurst exponent’ H play an important role in option pricing with transaction costs.     

Scaling and long-range dependence in option pricing I: Pricing European option with transaction costs under the fractional Black–Scholes model

This paper deals with the problem of discrete time option pricing by the fractional Black–Scholes model with transaction costs. By a mean self-financing delta-hedging argument in a discrete time setting, a European call option pricing formula is obtained. The minimal price of an option under transaction costs is obtained as timestep , which can be used as the actual price of an option. In fact, is an adjustment to the volatility in the Black–Scholes formula by using the modified volatility to replace the volatility σ, where is the Hurst exponent, and k is a proportional transaction cost parameter. In addition, we also show that timestep and long-range dependence have a significant impact on option pricing.     

Scaling and long-range dependence in option pricing II: Pricing European option with transaction costs under the mixed Brownian–fractional Brownian model

This paper deals with the problem of discrete-time option pricing by the mixed Brownian–fractional Brownian model with transaction costs. By a mean-self-financing delta hedging argument in a discrete-time setting, a European call option pricing formula is obtained. In particular, the minimal pricing c_min(t,s_t) of an option under transaction costs is obtained, which shows that timestep δt and Hurst exponent H play an important role in option pricing with transaction costs. In addition, we also show that there exists fundamental difference between the continuous-time trade and discrete-time trade and that continuous-time trade assumption will result in underestimating the value of a European call option.     

Stochastic relaxational dynamics applied to finance: Towards non-equilibrium option pricing theory

Non-equilibrium phenomena occur not only in the physical world, but also in finance. In this work, stochastic relaxational dynamics (together with path integrals) is applied to option pricing theory. Equilibrium in financial markets is defined as the absence of arbitrage, i.e. profits “for nothing”. A recently proposed model (by Ilinski et al.) considers fluctuations around this equilibrium state by introducing a relaxational dynamics with random noise for intermediate deviations called “virtual” arbitrage returns. In this work, the model is incorporated within a martingale pricing method for derivatives on securities (e.g. stocks) in incomplete markets using a mapping to option pricing theory with stochastic interest rates. The arbitrage return is considered as a component of a fictitious short-term interest rate in a virtual world. The influence of intermediate arbitrage returns on the price of derivatives in the real world can be recovered by performing an average over the (non-observable) arbitrage return at the time of pricing. Using a famous result by Merton and with some help from the path integral method, exact pricing formulas for European call and put options under the influence of virtual arbitrage returns (or intermediate deviations from economic equilibrium) are derived where only the final integration over initial arbitrage returns needs to be performed numerically. This result, which has not been given previously and is at variance with results stated by Ilinski et al., is complemented by a discussion of the hedging strategy associated to a derivative, which replicates the final payoff but turns out to be not self-financing in the real world, but self-financing when summed over the derivative's remaining life time. Numerical examples are given which underline the fact that an additional positive risk premium (with respect to the Black-Scholes values) is found reflecting extra hedging costs due to intermediate deviations from economic equilibrium. Received 16 June 1999 and Received in final form 26 September 1999     

A contribution to the systematics of stochastic volatility models

We systematically compare several classes of stochastic volatility models of stock market fluctuations. We show that the long-time return distribution is either Gaussian or develops a power-law tail, while the short-time return distribution has generically a stretched-exponential form, but can also assume an algebraic decay, in the family of models which we call “GARCH” type. The intermediate regime is found in the exponential Ornstein-Uhlenbeck process. We also calculate the decay of the autocorrelation function of volatility.