ANALYTICAL STUDY AND GENERALISATION OF SELECTED STOCK OPTION VALUATION MODELS

In this work, the classical Black-Scholes model for stock option valuation on the basis of some stochastic dynamics was considered. As a result, a stock option val- uation model with a non-�xed constant drift coe�cient was derived. The classical Black-Scholes model was generalised via the application of the Constant Elasticity of Variance Model (CEVM) with regard to two cases: case one was without a dividend yield parameter while case two was with a dividend yield parameter. In both cases, the volatility of the stock price was shown to be a non-constant power function of the underlying stock price and the elasticity parameter unlike the constant volatility assumption of the classical Black-Scholes model. The It^o's theorem was applied to the associated Stochastic Di�erential Equations (SDEs) for conversion to Partial Dif- ferential Equations (PDEs), while two approximate-analytical methods: the Modi�ed Di�erential Transformation Method (MDTM) and the He's Polynomials Technique (HPT) were applied to the Black-Scholes model for stock option valuation; in both cases the integer and time-fractional orders were considered, and the results obtained proved the latter as an extension of the former. In addition, a nonlinear option pric- ing model was obtained when the constant volatility assumption of the classical linear Black-Scholes option pricing model was relaxed through the inclusion of transaction cost (Bakstein and Howison model). Thereafter, this nonlinear option pricing model was extended to a time-fractional ordered form, and its approximate-analytical solu- tions were obtained via the proposed solution technique. For e�ciency and reliability of the method, two cases with �ve examples were considered: Case 1 with two ex- amples for time-integer order, and Case 2 with three examples for time-fractional order, and the results obtained show that the time-fractional order form generalises the time-integer order form. Thus, the Black-Scholes and the Bakstein and Howison models for stock option valuation were generalised and extended to time-fractional order, and analytical solutions of these generalised models were provided

Modelling Crop Insurance Based on Weather Index Using The Homotopy Analysis for American Put Option

The crop insurance in Indonesia (AUTP) is much focused on the area impacted by flood, drought, and pest attack. The complication of the procedure to claim the loss must follow several conditions. The different approaches in the insurance sector, using weather index can be taken into consideration to produce a variety of insurance products. This insurance product used the American put option with the primary asset is the rainfall and the cumulative rainfall to exercise the claim, considering the optimal execution limit. The homotopic analysis is used to determine the valuation of the American put option, which also becomes the insurance premium. The case study is focused on areas experiencing a drought so that insurance claims can be exercise when the rainfall index value is below a predetermined limit. Considering the normality of the rainfall data, the calculation of insurance premium was done for the first growing season. The insurance premium is varies based on the optimal execution limit, while the calculation of profit is based on the optimum limit exercise and the minimum rainfall for the growing season, and its different depended on insurance claim acceptance limits

Analytical Solutions of the Black–Scholes Pricing Model for European Option Valuation via a Projected Differential Transformation Method

In this paper, a proposed computational method referred to as Projected Differential Transformation Method (PDTM) resulting from the modification of the classical Differential Transformation Method (DTM) is applied, for the first time, to the Black–Scholes Equation for European Option Valuation. The results obtained converge faster to their associated exact solution form; these easily computed results represent the analytical values of the associated European call options, and the same algorithm can be followed for European put options. It is shown that PDTM is more efficient, reliable and better than the classical DTM and other semi-analytical methods since less computational work is involved. Hence, it is strongly recommended for both linear and nonlinear stochastic differential equations (SDEs) encountered in financial mathematics

Do Heterogeneous Beliefs Matter for Asset Pricing?

We study how heterogeneous beliefs affect returns and examine whether heterogeneous beliefs are a priced factor in traditional asset pricing models. To accomplish this task, we suggest new empirical measures based on the disagreement among analysts about expected (short-term and long-term) earnings are good proxies. Having established that heterogeneity of beliefs matters for asset pricing we turn our attention to estimating a structural model in which we use the forecasts of financial analysts to proxy for the beliefs of agents. Finally, we investigate if the amount of heterogeneity in analysts' forecasts can help explain asset pricing puzzlesHeterogeneous Beliefs, Asset pricing

A quasi-closed-form solution for the valuation of american put option

This study develops a quasi-closed-form solution for the valuation of an American put option and the critical price of the underlying asset. This is an important area of research both because of a large number of transactions for American put options on dierent underlying assets (stocks, currencies, commodities, etc.) and because this type of evaluation plays a role in determining the value of other financial assets such as mortgages, convertible bonds or life insurance policies. The procedure used is commonly known as the method of lines, which is considered to be a formulation in which time is discrete rather than continuous. To improve the quality of the results obtained, the Richardson extrapolation is applied, which allows the convergence of the outputs to be accelerated to values close to reality. The model developed in this paper derives an explicit formula of the finite-maturity American put option. The results obtained, besides allowing us to quickly determine the option value and the critical price, enable the graphical representation—in two and three dimensions—of the option value as a function of the other components of the modelinfo:eu-repo/semantics/publishedVersio

Mortgage valuation: a quasi-closed form solution

The main objective of this study consists in developing a quasi-analytical solution for the valuation of commercial mortgages. We consider the existence of a single source of risk - the risk of defaulting on a mortgage - and therefore, the existence of a single state variable - the value of the mortgaged property. The value of the mortgage corresponds to the present value of the future payments on the loan, minus the value of the embedded American default option. The major difficulty in designing such a model consists in calculating the value of this option, since for that purpose it is necessary to determine the lowest property price below which it must be immediately exercised, i.e. the critical value of the property

A quasi-closed-form solution for the valuation of American put options

This study develops a quasi-closed-form solution for the valuation of an American put option and the critical price of the underlying asset. This is an important area of research both because of a large number of transactions for American put options on different underlying assets (stocks, currencies, commodities, etc.) and because this type of evaluation plays a role in determining the value of other financial assets such as mortgages, convertible bonds or life insurance policies. The procedure used is commonly known as the method of lines, which is considered to be a formulation in which time is discrete rather than continuous. To improve the quality of the results obtained, the Richardson extrapolation is applied, which allows the convergence of the outputs to be accelerated to values close to reality. The model developed in this paper derives an explicit formula of the finite-maturity American put option. The results obtained, besides allowing us to quickly determine the option value and the critical price, enable the graphical representation—in two and three dimensions—of the option value as a function of the other components of the model.info:eu-repo/semantics/publishedVersio

Mortgage valuation: a quasi-closed-form solution

The substantial growth observed in commercial mortgage contracts during the last two decades justifies a greater academic effort in order to develop adequate valuation models. In the vast majority of cases, the literature in this field has presented only numeric solutions. In order to obtain these numeric solutions, highly complex calculation techniques are required, which make the process overly slow and expensive. The main objective of this work is to make a contribution in a hitherto underexplored area of financial research: the development of a contingent claims commercial mortgage valuation model with a closed-form solution. The model developed in the present paper constitutes one of the first attempts to identify closed-form solutions for commercial mortgage valuation. It is also a valid alternative to models proposed up to now in the specific field of commercial mortgage valuationinfo:eu-repo/semantics/publishedVersio

Option Pricing with Expansion Methods: New Approaches to Advanced Stochastic Volatility Models and American Options

In the thesis, we aim to develop a new framework for pricing advanced options quickly and accurately. Specifically, we price European options under stochastic volatility models and American options under the Black – Scholes model using expansion methods, which are widely used in physical sciences.Efficient and accurate pricing of option contracts has long been the central problem of mathematical finance. Apart from the classic Black – Scholes model, which has a closedform solution, there is no universally accepted method for pricing of options. The Monte Carlo simulation is general but slow, while finite-difference and numerical integration (Fourier transform) methods are comparably accurate but are not always applicable to exotic options and non-affine models. Furthermore, although those popular numerical methods can provide decent estimates of option prices, their discrete nature makes it difficult for them to achieve efficiency and accuracy simultaneously. While considerable amount of research has been devoted to these methods, we believe that the potential of expansion methods are underestimated.The expansion methodology, which is widely used in mathematics and physics, divides an unknown quantity into an infinite and converging series whose neighbouring terms are related by algebraic or differential equations. Therefore, starting from the known leading terms, we can work out the iteration equations and obtain any number of terms in the series, as long as the iteration equations are exact and explicitly solvable. In the context of finance, the option price can be expanded as an infinite series of analytical functions, which are related by the pricing partial differential equation (PDE). Besides the flexibility to deal with different models and options, the biggest advantage of expansion methods is that, users can evaluate the formulae derived by the author by plugging in parameter values, which greatly reduces computational intensity.First, we show that European options under stochastic volatility models can be expanded with various pairs of parameters in the volatility process, such as initial volatility, speed of mean-reversion, volatility of volatility and long-term volatility. The methods use powers of parameters as basis functions, and work with small parameter values. To achieve better performance, a modified version of expansion with initial volatility and volatility of volatility is proposed to reduce the pricing error when the parameters are large. The new method uses bounded basis functions, rather than the unbounded power series, and the numerical results confirm that the promotion from unbounded to bounded greatly improves the ability of expansion methods to approximate option prices. Moreover, symmetry considerations are also helpful for expansion methods. When the scale invariance is broken, we are equipped with one more degree of freedom to fine-tune the convergence, which is not proven or guaranteed.Then, we show that the non-linear problem of American options under the Black – Scholes model can be solved as a series of special functions that we defined earlier. Such special functions remain in the same family under many operations, making explicit expression of the option prices possible. We formally demonstrate two of the many ways of expansion, which work numerically except in the case of low volatility and high interest rate. Thus, an improved version, is proposed. It treats the Black – Scholes model as an advanced model with an additional operator. The improved method is able to deal with reasonable values of volatility, interest rate, moneyness and maturity. Finally, we outline the possibility of combining advanced models with advanced option types. American options can be treated similarly under many popular models, as long as the extra operators preserve the closedness of the special functions.The main contribution of the thesis is the demonstration that expansion methods can be used efficiently with non-affine stochastic volatility models and American options, which have no closed-form solutions. Additionally, explicit formulae, instead of formal relations in terms of integrals, are derived and available for reproduction