Abstract:
We consider an economic model with a deterministic money market account and a finite set of basic economic risks. The real-world prices of the risks are represented by continuous time stochastic processes satisfying a stochastic differential equation of diffusion type. For the simple class of log-normally distributed instantaneous rates of return, we construct an explicit state-price deflator. Since this includes the Black-Scholes and the Vasicek (Ornstein-Uhlenbeck) return models, the considered deflator is called Black-Scholes- Vasicek deflator. Besides a new elementary proof of the Black-Scholes and Margrabe option pricing formulas a validation of these in a multiple risk economy is achieved.

Abstract:
The Black-Scholes options formula is the breakthrough in valuating options prices. However, the formula is heavily based on several assumptions that are not realistic in practice. The extensions of the assumptions are needed to make options pricing model more realistic. This paper has reviewed the relaxation of the formula to European options on shares with the focus on its analytical solutions. The assumptions that are relaxed are non-dividends assumption, constant interest rate, constant volatility, and continuous time. Abstract in Bahasa Indonesia : Rumus opsi saham Black-Scholes merupakan terobosan dalam penentuan nilai suatu wahana keuangan derivatif opsi saham. Namun demikian, rumus ini didasari beberapa asumsi yang dalam praktiknya tidak realistis. Pengembangan asumsi tersebut diperlukan agar model penilaian harga opsi saham lebih realistis. Tulisan ini membahas relaksasi asumsi dalam rumus Black-Scholes terhadap opsi Eropa pada saham yang berfokus pada solusi analitis. Relaksasi asumsi yang dibahas merupakan asumsi tanpa dividen, suku bunga konstan, volatilitas tetap, dan waktu yang kontinu. Kata kunci: Opsi, penilaian opsi saham, solusi analitis, rumus Black-Scholes.

Abstract:
We consider a risk-neutral stock-price model where the volatility and the return processes are assumed to be dependent. The market is complete and arbitrage-free. Using a linear regression approach, explicit functions of risk-neutral density functions of stock return functions are obtained and closed form solutions of the corresponding Black-Scholes-type option pricing results are derived. Implied volatility skewness properties are illustrated.

Abstract:
In this paper, we extend the Johnson, Pawlukiwicz, and Mehta [1] skewness-adjusted binomial model to the pricing of futures options and examine in some detail the asymptotic properties of the skewness model as it applies to futures and spot options. The resulting skewness-adjusted futures options model shows that for a large number of subperiods, the price of futures options depends not only on the volatility and mean but also on the risk-free rate, asset-yield, and other carrying-cost parameters when skewness exists.

Abstract:
This paper develops a closed-form solution to an extended Black-Scholes (EBS) pricing formula which admits an implied drift parameter alongside the standard implied volatility. The market volatility smiles for vanilla call options on the S&P 500 index are recreated fitting the best volatility-drift combination in this new EBS. Using a likelihood ratio test, the implied drift parameter is seen to be quite significant in explaining volatility smiles. The implied drift parameter is sufficiently small to be undetectable via historical pricing analysis, suggesting that drift is best considered as an implied parameter rather than a historically-fit one. An overview of option-pricing models is provided as background.

Abstract:
In this paper, we consider the Black-Scholes (BS) equation for option pricing with constant volatility. Here, we construct the first-order Darboux transformation and the real valued condition of transformed potential for BS corresponding equation. In that case we also obtain the transformed of potential and wave function. Finally, we discuss the factorization method and investigate the supersymmetry aspect of such corresponding equation. Also we show that the first order equation is satisfied by commutative algebra.

Abstract:
this study aims to test the existence of near unit roots and local persistence in several important variables of economic models (product market, ccapm and black-scholes' formula). it is argued that the rejection of the unit root hypothesis will not necessarily imply in accepting a stationary and ergodic behavior for the time series. in order to do that, the near unit root model developed by phillips, moon and xiao (2001) was selected and an estimation strategy was used. such strategy is described as follows: a) the df-gls test, suggested by elliott, rothenberg and stock (1996); b) optimal selection of lags used by ng and perron (2001); c) the non parametric correction for terms of perturbation non i.i.d., from the kernel smoothing. the empirical results show, for some series, a characterization of the dgp from the local persistence.

Abstract:
In the framework of path integral the evolution operator kernel for the Merton-Garman Hamiltonian is constructed. Based on this kernel option formula is obtained, which generalizes the well-known Black-Scholes result. Possible approximation numerical schemes for path integral calculations are proposed.

The aim of this paper is to show how options with transaction costs under fractional, mixed Brownian-fractional, and subdiffusive fractional Black-Scholes models can be efficiently computed by using the barycentric Jacobi spectral method. The reliability of the barycentric Jacobi spectral method for space (asset) direction discretization is demonstrated by solving partial differential equations (PDEs) arising from pricing European options with transaction costs under these models. The discretization of these PDEs in time relies on the implicit Runge-Kutta Radau IIA method. We conducted various numerical experiments and compared our numerical results with existing analytical solutions. It was found that the proposed method is efficient, highly accurate and reliable, and is an alternative to some existing numerical methodsfor pricing financial options.