American option pricing with imprecise risk neutral probabilities: from plain intervals to fuzzy sets

The aim f this paper is to price an American style option when there is uncertainty on the underlying asset volatility

OPTION VALUE CALCULATION AFFECTED COMPONENTS

As the subprime credit crisis has attracted attention to financial derivative instruments, more frequently arises questions about fairvalue calculations. Over the time, different models had been introduced. All of those models take into account factors affectingprices. Mostly, factors used in calculations on the same type of financial instruments are approximately the same. Therefore questionarises, which factor affects price more and which less, with no matter which model would be used for fair value calculations. One offinancial derivative instrument types is options. Options are agreements, which give to option buyer rights to buy or sell underlyingasset. While the seller or writer of option has obligation to buy or sell underlying asset. This research aims to explore the impact offactors on option fair value calculations and evaluate most important ones from those, which could be chosen by option buyer orseller. To reach the aim of research following tasks are developed: 1) review of fair value calculation models; 2) compare results ofusage of different models and changes in affecting factors; 3) highlight differences between option price affecting factors, modelsused in calculations and results provided. Research includes literature review and analysis of option pricing results. Option pricecalculations are based on historical option prices, using black-Scholes and Binomial option pricing models.KEYWORDS: options, fair value models, strike price, exercise date.DOI: http://dx.doi.org/10.15181/rfds.v11i3.61

A fuzzy real option approach for investment project valuation

[[abstract]]The main purpose of this paper is to propose a fuzzy approach for investment project valuation in uncertain environments from the aspect of real options. The traditional approaches to project valuation are based on discounted cash flows (DCF) analysis which provides measures like net present value (NPV) and internal rate of return (IRR). However, DCF-based approaches exhibit two major pitfalls. One is that DCF parameters such as cash flows cannot be estimated precisely in the uncertain decision making environments. The other one is that the values of managerial flexibilities in investment projects cannot be exactly revealed through DCF analysis. Both of them would entail improper results on strategic investment projects valuation. Therefore, this paper proposes a fuzzy binomial approach that can be used in project valuation under uncertainty. The proposed approach also reveals the value of flexibilities embedded in the project. Furthermore, this paper provides a method to compute the mean value of a project’s fuzzy expanded NPV that represents the entire value of project. Finally, we use the approach to practically evaluate a project.[[incitationindex]]SCI[[booktype]]紙

American Option Pricing of Future Contracts in an Effort to Investigate Trading Strategies; Evidence from North Sea Oil Exchange

In this paper, Black Scholes’s pricing model was developed to study American option on future contracts of Brent oil. The practical tests of the model show that market priced option contracts as future contracts less than what model did, which mostly represent option contracts with price rather than without price. Moreover, it suggests call option rather than put option. Using t hypothesis test, price differences were obtained, which can serve as a useful strategy for traders interested in arbitrage practice and risk hedging. This research introduces an optimal strategy (both for call and put option states and buy and sell of future contract ) for all options of buy and sell future contracts with and without price. In this research, six-month data of the end of 2015 about oil option and option of future contracts of North Sea oil for three different maturities were used

Fuzzy Optimization of Option Pricing Model and Its Application in Land Expropriation

Option pricing is irreversible, fuzzy, and flexible. The fuzzy measure which is used for real option pricing is a useful supplement to the traditional real option pricing method. Based on the review of the concepts of the mean and variance of trapezoidal fuzzy number and the combination with the Carlsson-Fuller model, the trapezoidal fuzzy variable can be used to represent the current price of land expropriation and the sale price of land on the option day. Fuzzy Black-Scholes option pricing model can be constructed under fuzzy environment and problems also can be solved and discussed through numerical examples

Nonparametric Predictive Inference for European Option Pricing based on the Binomial Tree Model

In finance, option pricing is one of the main topics. A basic model for option pricing is the Binomial Tree Model, proposed by Cox, Ross, and Rubinstein in 1979 (CRR). This model assumes that the underlying asset price follows a binomial distribution with a constant upward probability, the so-called risk-neutral probability. In this paper, we propose a novel method based on the binomial tree. Rather than using the risk-neutral probability, we apply Nonparametric Predictive Inference (NPI) to infer imprecise probabilities of movements, reflecting more uncertainty while learning from data. To study its performance, we price the same European options utilizing both the NPI method and the CRR model and compare the results in two different scenarios, firstly where the CRR assumptions are right, and secondly where the CRR model assumptions deviate from the real market. It turns out that our NPI method, as expected, cannot perform better than the CRR in the first scenario, but can do better in the second scenario

Generalised soft multi-mode real options model (fuzzy-stochastic approach)

Researchers and practitioners are dealing intensively with the real option valuation. One of the generalised types is reversible the multi-mode American real options. These options are solved mainly by applying the stochastic discrete binomial models. Uncertainty is a typical feature of valuation, and two basic types of representation are distinguished: risk (stochastic) and imprecision (fuzzy). The fuzzy-stochastic models indicate the generalised real options modelling containing both aspects. The objective of the paper is to develop and apply the generalised fuzzy-stochastic multi-mode real options model. This model is based on fuzzy numbers, the discrete binomial model, and the decomposition principle. Input data, particularly underlying cash-flows, are given by fuzzyrandom numbers; fuzzy numbers give terminal values, risk-free rate, switching cost. Furthermore, assumptions and computation procedures are also described. The proposed optimisation problem is used for the fuzzy multi-mode real option value calculation. Results are compared with sub-problems, crisp-stochastic multi-modes real options and partial fuzzy-stochastic multi-mode real options models. A stylised illustrative operational flexibility example of comparing the fuzzy-stochastic multi-mode real options models is presented and discussed. The model can serve to valuation, decision-making, generalised sensitivity analysis and control under a fuzzystochastic environment.Web of Science192art. no. 11638

American option pricing with imprecise risk-neutral probabilities

The aim of this paper is to price an American option in a multiperiod binomial model,when there is uncertainty on the volatility of the underlying asset.American option valuation is usually performed, under the risk-neutralvaluation paradigm, by using numerical procedures such as the binomialoption pricing model of Cox, Ross, Rubinstein (1979). A key input of themultiperiod binomial model is the volatility of the underlying asset,that is an unobservable parameter.As it is hard to give a precise estimate forthe volatility, in this paper we use a possibility distribution in order to modelthe uncertainty on the volatility. Possibility distributions are one of the mostpopular mathematical tools for modelling uncertainty. The standard risk-neutralvaluation paradigm requires the derivation of the risk-neutral probabilities, thatin a one period binomial model boils down to the solution of a linear system ofequations. As a consequence of the uncertainty in the volatility, we obtain apossibility distribution on the risk-neutral probabilities. Under these measures,we perform the risk-neutral valuation of the American option

Nonparametric predictive inference for option pricing based on the binomial tree model

Nonparametric Predictive Inference (NPI) is a frequentist statistical method based on only fewer assumptions, which has been developed for and applied to, several areas in statistics, reliability and finance. In this thesis, we introduce NPI for option pricing in discrete time models. NPI option pricing is applied to vanilla options and some types of exotic options. We first set up the NPI method for the European option pricing based on the binomial tree model. Rather than using the risk-neutral probability, we apply NPI to get the imprecise probabilities of underlying asset price movements, reflecting more uncertainty than the classic models with the constant probability while learning from data. As we assign imprecise probabilities to the option pricing procedure, surely, we get an interval expected option price with the upper and lower expected option prices as the boundaries, and we named the boundaries the minimum selling price and the maximum buying price. The put-call parity property of the classic model is also proved to be followed by the NPI boundary option prices. To study its performance, we price the same European options utilizing both the NPI method and the Cox, Ross, and Rubinstein binomial tree model (CRR) and compare the results in two different scenarios, first where the CRR assumptions are right, and second where the CRR model assumptions deviate from the real market. It turns out that our NPI method, as expected, cannot perform better than the CRR in the first scenario with small size historical data, but as enlarging the history data size, the NPI method's performance gets better. For the second scenario, the NPI method performs better than the CRR model. The American option pricing procedure is also presented from an imprecise statistical aspect. We propose a novel method based on the binomial tree. We prove through this method that it may be optimal for an American call option without dividends to be exercised early, and some influences of the stopping time toward option price prediction are investigated in some simulation examples. The conditions of the early exercise for both American call and put options are derived. The performance study of the NPI pricing method for American options is evaluated via simulation in the same two scenarios as the European options. Through the performance study, we conclude that the investor using the NPI method behaves more wisely in the second scenario than the investor using the CRR model, and faces to more profit and less loss than what it does in the first scenario. The NPI method can be applied to exotic options if the option payoffs are a monotone function of the number of upward movements in the binomial tree, like the digital option and the barrier option discussed in this thesis. Otherwise, either we can manipulate the binomial tree in order to assign the upper and lower probabilities, for instance, the look-back option with the float strike price, or a new probability mass is needed to be assigned to the payoff binomial tree according to the option definition which is attractive and challenging for future study