European Option Pricing Model Research Articles

An accurate European option pricing model under Fractional Stable Process based on Feynman Path Integral

In this paper, we propose to use the Fractional Stable Process (FSP) for option pricing. The FSP is one of the few candidates to directly model a number of desired empirical properties of asset price risk neutral dynamics. However, pricing the vanilla European option under FSP is difficult and problematic. In the paper, built upon the developed Feynman Path Integral inspired techniques, we present a novel computational model for option pricing, i.e. the Fractional Stable Process Path Integral (FSPPI) model under a general fractional stable distribution that tackles this problem. Numerical and empirical experiments show that the proposed pricing model provides a correction of the Black–Scholes pricing error — overpricing long term options, underpricing short term options; overpricing out-of-the-money options, underpricing in-the-money options without any additional structures such as stochastic volatility and a jump process.

Symmetry Analysis of Options Pricing with Transactions Costs Driven by Fractional Brownian Noises

We provide a closed-form solution for the European and Asian option pricing models when the source of randomness is a fractional Brownian motion as opposed to the geometric Brownian motion. In addition to the source of randomness, transaction costs are considered to be non-negligible. For the case of the European option, proportional transaction costs hide in the volatility and do not change the form of the model. The construction of the solution is based on the symmetries of the model. The model for Asian options has an additional parameter that makes the volatility time-dependent, which complicates the solution process. However, we are still able to obtain solutions using Lie symmetry methods.

European option pricing model in a stochastic and fuzzy environment

The primary goal of this paper is to price European options in the Merton’s framework with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes as fuzzy random variables and use the property of fuzzy set to deduce two different jump-diffusion models underlying principle of rational expectations equilibrium price. Unlike many conventional models, the European option price will now turn into a fuzzy number. One of the major advantages of this model is that it allows investors to choose a reasonable European option price under an acceptable belief degree. The empirical results will serve as useful feedback information for improvements on the proposed model.

Rational Krylov methods in exponential integrators for European option pricing

The aim of this paper is to analyze efficient numerical methods for time integration of European option pricing models. When spatial discretization is adopted, the resulting problem consists of an ordinary differential equation that can be approximated by means of exponential Runge–Kutta integrators, where the matrix-valued functions are computed by the so-called shift-and-invert Krylov method. To our knowledge, the use of this numerical approach is innovative in the framework of option pricing, and it reveals to be very attractive and efficient to solve the problem at hand. In this respect, we propose some a posteriori estimates for the error in the shift-and-invert approximation of the core-functions arising in exponential integrators. The effectiveness of these error bounds is tested on several examples of interest. They can be adopted as a convenient stopping criterion for implementing the exponential Runge–Kutta algorithm in order to perform time integration.

Applying the Model Order Reduction method to a European option pricing model

This paper presents a European option pricing model by applying the Model-Order-Reduction (MOR) method. A European option pricing theorem based on Black–Scholes' equation is implemented by the Finite-Difference Method (FDM). However, the numerical models generated by the FDM could be simplified through the MOR technique, which is based on the concept of an Arnoldi-based Model-Order Reduction algorithm. In terms of computational cost, the MOR models are at least 2 orders of magnitude faster than the original FDM models with a negligible compromise in accuracy.

A new closed-form solution as an extension of the Black–Scholes formula allowing smile curve plotting

In this paper, as a generalization of the Black–Scholes (BS) model, we elaborate a new closed-form solution for a uni-dimensional European option pricing model called the J-model. This closed-form solution is based on a new stochastic process, called the J-process, which is an extension of the Wiener process satisfying the martingale property. The J-process is based on a new statistical law called the J-law, which is an extension of the normal law. The J-law relies on four parameters in its general form. It has interesting asymmetry and tail properties, allowing it to fit the reality of financial markets with good accuracy, which is not the case for the normal law. Despite the use of one state variable, we find results similar to those of Heston dealing with the bi-dimensional stochastic volatility problem for pricing European calls. Inverting the BS formula, we plot the smile curve related to this closed-form solution. The J-model can also serve to determine the implied volatility by inverting the J-formula and can be used to price other kinds of options such as American options.

Early exercise European option and early termination American option pricing models

Early exercise European option and early termination American option pricing models

European Put-Call Parity and the Early Exercise Premium for American Currency Options

The European put-call parity condition is used to estimate the early exercise premium for American currency options traded on the Philadelphia Stock Exchange. Using a sample of 331 pairs of call and put options with the same exercise price and time to expiration, evidence is provided for early exercise premiums that average 5.03% for put options and 4.60% for call options. The premiums for both call and put options are strongly related to the interest rate differential and time to expiration. These results have implications for the use of European option pricing models in the valuation of American options.

Pricing and Inference with Mixtures of Conditionally Normal Processes

The main objective of this paper is to exploit important properties of the mixtures of normal distributions and of conditionally normal processes in order to propose flexible and tractable discrete-time option pricing models, based on the exponential-affine stochastic discount factor (SDF) modeling principle. These models deal simultaneously and coherently with the historical and risk-neutral worlds, they can be static or dynamic and, moreover, they can be parametric or semi-parametric. In the static setting our approach provide explicit historical and risk-neutral stock returns distributions and closed-form formulas for European option prices of any residual maturity. These pricing formulas turn out to be convex combinations of Black and Scholes formulas. The previous approach is extended to the dynamic case in several ways. First, by assuming that a Mixed Normal distribution is valid for the standardized error terms. Second, by considering mixtures of conditionally Gaussian processes and, third, by adopting a semi-parametric approach which estimates the distribution of standardized error terms by mixtures of normals based on a kernel method. Some numerical exercise shows the ability of our European option pricing models to replicate implied volatility smiles, volatility skews and volatility surfaces coherently with observations.

TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL

We present two approaches of the stochastic interest rate European option pricing model. One is a bond numeraire ap- proach which is applicable to a nonzero value asset. In this ap- proach, we assume log-normality of returns of the asset normalized by a bond whose maturity is the same as the expiration date of an option instead that of an asset itself. Another one is the expec- tation hypothesis approach for value zero asset which has futures- style margining. Bond numeraire approach allows us to calculate volatilities implied in options even though stochastic interest rate is considered.

Put-Call Parity and the Early Exercise Premium for Currency Options

Put-call parity is used to study the early exercise premium for currency options traded on the Philadelphia Stock Exchange. Using 564 pairs of call and put options evidence is provided that the early exercise premia is on average 5.71% for put options and 6.88% for call options. The premiums for both call and put options are strongly related to time to maturity and the interest rate differential. These results are important when using a European option pricing model for the valuation of American options.

Numerical analysis on binomial tree methods for a jump-diffusion model

This paper studies the numerical approximation for an European option pricing model with jump-diffusion. Equivalence of the Binomial tree method and an explicit difference scheme is discussed. The optimal error estimation of the Binomial tree approximation is also given. Another explicit difference scheme is constructed, which has higher accuracy than the Binomial tree method. Numerical results coincide with the theoretical results.

MSM Estimators of European Options on Assets with Jumps

This paper shows that, under some regularity conditions, the method of simulated moments estimator of European option pricing models developed by Bossaerts and Hillion (1993) can be extended to the case where the prices of the underlying asset follow Lévy processes, which allow for jumps, with no losses on their asymptotic properties, still allowing for the joint test of the model.

An upwind numerical approach for an American and European option pricing model

The numerical solution of several mathematical models arising in financial economics for the valuation of both European and American call options on different types of assets is considered. All the models are based on the Black-Scholes partial differential equation. In the case of European options a numerical upwind scheme for solving the boundary backward parabolic partial differential equation problem is presented. When treating with American options an additional inequality constraint leads to a discretized linear complementarity problem. In each case, the numerical approximation of option values is computed by means of optimization algorithms. In particular, Uzawa's method allows to compute the optimal exercise boundary which corresponds to the classical concept of moving boundary in continuum mechanics.

The Intertemporal Relationship Between the Currency Spot Market and the Currency Option Market

This study examines the lead/lag relationship between currency option and currency spot markets for the Deutsche mark and the Japanese yen. Using intraday currency option transactions data for the year 1989 and applying a European type currency option pricing model, pair data series of the implied and the observed exchange rates are compiled. Causality tests are then employed to test the causal relation between the observed and the implied exchange rate changes. The results indicate that the currency spot market leads the currency option market by about ninety minutes.