In this paper, we construct a numerical ‎method‎ to ‎the ‎solution ‎of‎ the Black-Scholes partial differential equation modeling the European option pricing problem with regard to a single asset‎. ‎W‎e use an explicit spline-difference scheme which is based on using a finite difference approximation for the temporal derivative and a cubic B-spline collocation for spatial derivatives. ‎The deriv...

In this paper, we present a stock model with Markov switching in the uncertainty markets, where the parameters of drift and volatility change according to the states of a Markov process. To price the option, we firstly establish a risk-neutral probability based on the uncertain measure given by Liu. Then a closed form of the European option pricing formula is obtained by applying the Laplace tr...

Differentiation matrices provide a compact and unified formulation for a variety of differential equation discretisation and timestepping algorithms. This paper illustrates their use for solving three differential equations of finance: the classic Black-Scholes equation (linear initial-boundary value problem), an American option pricing problem (linear complementarity problem), and an optimal m...

Reduced European call and put option formulas by risk-neutral valuation are given. It is shown that the European call and put options for log-uniformjump-diffusion models are worth more than that for the Black-Scholes (diffusion) model with the common parameters. Due to the complexity of the jump-diffusion models, obtaining a closed option pricing formula like that of Black-Scholes is not tract...

We propose a control variate method with multiple controls to effectively reduce variances of Monte Carlo simulations for pricing European options under multifactor stochastic volatility models. Based on an application of Ito’s formula, the option price is decomposed by its discounted payoff and stochastic integrals with the appearance of gradients of the unknown option price with respect to st...

OF THE DISSERTATION In this thesis, we propose the Markov tree option pricing model and subject it to large-scale empirical tests against market options and equity data to quantify its pricing and hedging performances. We begin by proposing a tree model that explicitly accounts for the dependence observed in the log-returns of underlying asset prices. The dynamics of the Markov tree model is ex...

In this paper, we present a reduced basis method for pricing European and American options based on the Black–Scholes and Heston models. To tackle each model numerically, we formulate the problem in terms of a time-dependent variational equality or inequality. We apply a suitable reduced basis approach for both types of options. The characteristic ingredients used in the method are a combined P...

In this paper, we combine modern portfolio theory and option pricing so that a trader taking position in European contract, the underlying assets, risk-free bond can constru...

*Correspondence: [email protected] 1School of Management, Hebei University, Baoding 071002, China 2School of Science, Hebei University of Engineering, Handan 056038, China Abstract Multi-asset options are created to accelerate investment among countries with the development of globalization and financial market integration. Considering the human uncertainty and the influence of sudden events...

This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk. As an example, and as our main motivation, we investigate option pricing under the class of q-optimal pricing measures. Using t...