The Black–Scholes formula which was introduced by three economists, Black et al. (1973) has been widely used to calculate the theoretical price of the European call option. In 1979, Cox, Ross and ...Rubinstein (Cox et al., 1979) gave the binomial formula which is a tool to find the price of European option and showed that this formula converges to the Black–Scholes formula as the number of periods (n) converges to infinity. In 1988, Boyle investigated another formula that is used to find the price of European option, that is the trinomial formula. In 2015, Puspita et al. gave examples to show that the trinomial formula is closed to the Black–Scholes formula. After that, Ratibenyakool and Neammanee (2020) gave the rigorous proof of this convergence. In this paper, we show that the rate of convergence is of order 1n.
We derive new formulas for the price of the European call and put options in the Black-Scholes model, under the form of uniformly convergent series generalizing previously known approximations; these ...series are obtained by means of tools from multidimensional complex analysis. We also provide precise boundaries for the convergence speed and apply the results to the calculation of hedge parameters (Greeks).
In this paper, we consider non-linear transformations of classical telegraph process. The main results consist of deriving a general partial differential Equation (PDE) for the probability density ...(pdf) of the transformed telegraph process, and then presenting the limiting PDE under Kac's conditions, which may be interpreted as the equation for a diffusion process on a circle. This general case includes, for example, classical cases, such as limiting diffusion and geometric Brownian motion under some specifications of non-linear transformations (i.e., linear, exponential, etc.). We also give three applications of non-linear transformed telegraph process in finance: (1) application of classical telegraph process in the case of balance, (2) application of classical telegraph process in the case of dis-balance, and (3) application of asymmetric telegraph process. For these three cases, we present European call and put option prices. The novelty of the paper consists of new results for non-linear transformed classical telegraph process, new models for stock prices based on transformed telegraph process, and new applications of these models to option pricing.
The article concerns the generalised Cox‑Ross‑Rubinstein (CRR) option pricing model with new formulas for changes in upper and lower stock prices. The formula for option pricing in this model, which ...is the Black‑Scholes type formula, and its asymptotics are presented. The aim of the paper is to analyse limiting cases of the obtained asymptotics using probability theory and later data from the Warsaw Stock Exchange. Empirical analyses of option pricing in the generalised CRR model confirm the calculated limits.
The Black–Scholes formula for a European option price, which resulted in the 1997 Nobel Prize in Economic Sciences, is known to be the unique solution of the boundary-value problem consisting of the ...Black–Scholes partial differential equation and the terminal condition defined by the European call option. This has been one of the most popular tools of finance in theory as well as in practice. Here we present infinitely many solutions of the boundary value problem, involving Hermite polynomials. This indicates that the Black–Scholes boundary-value problem violates the law of one price, which is one of the fundamental concepts in economics.
•We present infinitely many solutions of the Black–Scholes boundary problem.•The Black–Scholes option valuation formula is included as a special solution.•The solutions consist of many independent functions, involving Hermite polynomials.•The Black–Scholes boundary-value problem violates the law of one price.
The installation of household-scale renewable energy (RE) assets including the likes of solar home systems, micro-wind turbines, pico-hydro systems, biomass space heaters and improved cook-stoves, ...offers householders various benefits. These include the possibility of working longer hours, enhancing the efficiency of production processes, improving the quality of life, and of gaining greater control over their immediate environment. In several settings, artisans pursuing the same vocation work from homes located in clusters. Consequent to procuring and deploying the RE asset, the community of individual investors bestows upon itself the option to derive incremental money incomes. This is subject to each member’s access to working-capital credit and raw material,skill levels and levels of effort, productivity, and more. This paper argues that householders assess the option to derive incremental incomes and go on to makethe investment decision in RE micro-infrastructure based on the estimated value of such options. The model so developed is applied to a community of silk weavers in southern India to estimate the premiumsthatinvestors pay to opt into derivingincremental incomes. This study could estimate that by installing a Solar Home System, a weaver could derive an economic benefit of 17.36% and an intangible benefit of 82.64% of the amount invested into the asset.
This paper presents a Bayesian approach to bandwidth selection for multivariate kernel regression. A Monte Carlo study shows that under the average squared error criterion, the Bayesian bandwidth ...selector is comparable to the cross-validation method and clearly outperforms the bootstrapping and rule-of-thumb bandwidth selectors. The Bayesian bandwidth selector is applied to a multivariate kernel regression model that is often used to estimate the state-price density of Arrow–Debreu securities with the S&P 500 index options data and the DAX index options data. The proposed Bayesian bandwidth selector represents a data-driven solution to the problem of choosing bandwidths for the multivariate kernel regression involved in the nonparametric estimation of the state-price density pioneered by Aït-Sahalia and Lo Aït-Sahalia, Y., Lo, A.W., 1998. Nonparametric estimation of state-price densities implicit in financial asset prices. The Journal of Finance, 53, 499, 547.
Defense in depth is a pillar of nuclear power plant design and it also plays an important factor for the reliability of fuel supply. While most safety systems of a classical light water reactor ...usually remain unchanged for many years of operation the first and second fission product barrier - the fuel and the cladding - are regularly exchanged by loading fresh fuel assemblies into the reactor core. We review our experience of achieving high reliability in fuel supply by means of real options. The required activities for option creation are explained in detail. Real option valuation is known to be more difficult than financial option valuation and we review the valuation metrics which we found most useful. Due to the incompleteness and illiquidity of the considered option market and due to the barriers of realizing arbitrage, lower and upper bounds are easier to determine than exact option values. Notwithstanding these constraints using concepts based on volatility and stochastic processes in our experience is good practice to avoid naked bets on future power plant availability and avoids the pitfalls of static forecasts.
In this paper we show how the results of Bernstein (1943) and recent results of Zubkov and Serov (2012) on the normal approximation to the binomial distribution lead to an alternative derivation of ...the Black–Scholes formula from a binomial option pricing model.
We prove a Black–Scholes type formula when the geometric Brownian motion originates from approximations by multinomial distributions. It is shown that the variance appearing in the Black–Scholes ...formula for option pricing can be structured according to occurrences of different types of events at each time instance using a local limit theorem for multinomial distributions in Richter (1956). The general approach has first been developed in Kan (2005).
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