This section details the mathematical develop- ment of the Willow tree. To model Brownian motion efficiently, the tree implements a discrete approximation to the normal distribution at each time point. Normal distributions are rele- vant to **pricing** derivatives since they play an integral role in diffusion processes, which, in turn, are the building blocks of the large major- ity of **derivative** **pricing** models. The basic struc- ture of the tree is constructed using this principle as a guideline, with the transitions linking successive time points together coming later.

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In this paper, we have a firm whose value process is dynamically controlled by an individual manager. That manager holds restricted shares in the firm as well as an employee stock option position. However, he is subject to dismissal for poor performance if the firm’s value declines to a lower boundary. The manager also has external wealth that he can dynamically allocate between a riskless asset and an index fund which is correlated with firm value. He chooses the optimal joint control positions at the firm and for his outside wealth to maximize his expected utility. We examine how his restricted share and option positions influence that optimal control behavior. Typically, he alters firm risk through time depending on firm value relative to the exercise price of his option and distance to the lower boundary. This makes standard **derivative** **pricing** models inappropriate; however once we have identified his optimal dynamic behavior, we can value **derivative** securities on that controlled process from a market perspective using a risk-neutral **pricing** procedure.

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The most popular of the one-factor equilibrium models is the Cox, Ingersoll, and Ross model (denoted CIR model). Since it uses a square root process in the volatility term, the model does not allow negative interest rates. This also implies that as interest rates increase, the standard deviation of changes in the interest rate will increase as well. The CIR model also incorporates mean-reversion like the Vasicek model (Hull, 2003). Of these three models, the CIR model has received the most attention in the term structure modeling literature. However, the Vasicek model is the one that was extended by Hull and White to develop the Hull-White model. Section 1.3 contains more detailed explanations of the Hull-White model, including the formulae behind it. Two-factor equilibrium models receive only a brief discussion in Hull (2003) and in Hunt and Kennedy (2004). Hull (2003) mentions two models, Brennan-Schwartz and Longstaff-Schwartz, that both have a second factor that allows for more flexibility in modeling the term structure. Hunt and Kennedy (2004) note that two-factor models are rarely used in **derivative** **pricing** primarily because of the difficulty in implementa- tion.

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Financial markets are known to be far from deterministic but stochastic and hence time dependent correlation tends to suit the markets. We price for European Options by us- ing three dimensional assets under stochastic correlation. The **pricing** equations under constant correlation and stochastic correlation are derived numerically by using finite difference method called the Crank Nicolson method. We compare the **pricing** equa- tions when the correlation is stochastic and constant by using real data from emerging financial markets, that is, exchange rates data for Kenya as the domestic currency and South Africa as the foreign currency. **Pricing** equation for the European option with stochastic correlation performed better than that with constant correlation.

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DOI: 10.4236/jmf.2019.94035 692 Journal of Mathematical Finance manner. [3] runs a comparison of different machine learning methods on a monthly frequency in US equity market. [4] extends the analysis to fixed income markets. In [5], the authors recover the stochastic discount factor, project it onto the asset span and obtain the optimal weights considering the no-arbitrage con- straints. In all the references, the authors use a brute-force supervised learning approach to predict the asset returns as a regression problem. In this paper, we factor in the clustering techniques to compute conditional expected asset returns, first proposed in a **derivative** **pricing** setting in [6] to evaluate the conditional expectation at each point of time, expressed as function of risk factors. The me- thod utilizes non-supervised learning techniques, such as k-means clustering, to partition the factor space and in each of the sub-spaces, a simple functional form is used to approximate the non-linear relationship between the future asset re- turns and current values of underlying risk factors.

For application we extend the canonical polynomial representation analy- sis to a Wold decomposition of call option prices. There, we show how a GARCH(1,1) model can be used to construct an empirical **pricing** kernel for call option by a signal extraction procedure for unobservable **pricing** kernel. To the best of our knowledge that procedure is new. However, (Chernov, 2003, pp. 332-333) also assumed an unobservable **pricing** kernel but used a two stage estimation proce- dure that involves first stage estimation of parameters from a continuous time as- set **pricing** model. At the second stage, he used an equivalent martingale measure, that includes parameters from the first stage asset **pricing** model, together with the asset(s) payoff to construct the **pricing** kernel. He then used a **derivative** **pricing** relation that includes parameters of the underlying asset **pricing** model ion order to derive “independent” equations. Whereupon, “simultaneous equations” are solved to infer the **pricing** kernel in second stage estimation. Our “two stage” procedure is distinguished because we calibrate second stage residuals from a discrete risk **pricing** model for the underlying asset under consideration, after a first stage Wold decomposition of a call option on the asset.

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This thesis consists of a collection of studies investigating various aspects of the interplay between the markets for **derivative** securities and their respective underlying assets in the presence of market imperfections. The classic theory of **derivative** **pricing** and hedging hinges on three rather unrealistic assumptions regarding the m arket for the underlying asset. Markets are assumed to be perfectly elastic, complete and frictionless. This thesis studies some effects of relaxing one or more of these assumptions. Chapter 1 provides an introduction to the thesis, details the structure of what follows, and gives a selective review of the relevant literature. Chapter 2 focuses on the effects th at the implementation of hedging strategies has on equilibrium asset prices when markets are imperfectly elastic. The results show th a t the feedback effect caused by such hedging strategies generates excess volatility of equilibrium asset prices, thus violating the very assumptions from which these strategies are derived. However, it is shown th at hedging is nonetheless possible, albeit at a slightly higher price. In Chapter 3, a model is developed which describes equilibrium asset prices when m arket participants use technical trading rules. The results confirm th at technical trading leads to the emergence of speculative price “bubbles”. However, it is shown th at although technical trading rules are irrational ex-ante, they turn out to be profitable ex-post. In Chapter 4, a general framework is developed in which th e optim al trading behaviour of a large, informed trader can be studied in

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As the major barrier of studying weather **derivative** modeling in China is due to the absence of real market data, one can only focus on the accuracy of temperature modeling rather than on the **derivative** **pricing** in order to examine the performances of different models. In this chapter, we utilize the DAT data described in the preceding Chapter to conduct empirical analyses on the stochastic seasonal variation (SSV) model along with three established empirical temperature and **pricing** models, i.e. the Alaton model [1], the continuous auto-regressive (CAR) model [2], and the Spline model [3]. We then compare heuristically the four temperature models, in terms of simulation results, residual normality, auto-correlation function (ACF), Akaike Information Criterion (AIC) and error measures, with the hope to find out the most suitable model for modeling and **pricing** temperature-based derivatives in China. Meanwhile, we look into the option prices generated by different models. The results show that the SSV model dominates the other three models by providing a more precise fitting of the temperature process. Further, the Spline model [3] displays inconsistencies when it is applied to Chinese temperature data. This model has the smallest relative errors, but the worst result for the normality of residuals.

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Above four examples have demonstrated that these laws can be seemly applied to different fields in both natural and social science. Although there are very little in common among the Rational Choice Theory in economics, genetic mutations in biology, uncertainty in historic facts, and **derivative** **pricing** theory in finance, these physics laws of social science can provide coherent answers to fundamental questions in different fields and powerful guidance to improve existing theories. These new laws have removed the invisible wall, which is artificially separating social science from natural science. Therefore, for the first time, we have established a shared physics foundation for all fields in social and natural science.

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Previously measurement of the **derivative** has been delayed until these HF oscillations have decayed, meaning that all PWM vectors (under which a **derivative** measurement is required) have to be applied for an amount of time (the minimum pulse width) sufficient to allow the high frequency oscillations to decay and a **derivative** measurement to be made [1-4]. This often involves PWM vector extension (when vectors are less than the minimum pulse width) and subsequent compensation [5], the side effects of which include current distortion, audible noise, torque ripple vibration and heating. By estimating the **derivative** before the HF oscillations have decayed, the minimum pulse width and its associated side effects can be reduced. Additionally many methods use specialist current **derivative** sensors. These represent an additional cost since they are not found in standard industrial drives.

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Introduction: There are appreciable concerns among European health authorities with growing expenditure on cancer medicines and issues of sustainability. The enhanced use of low cost generics could help. Aims: Consequently, there is a need to comprehensively document current and future arrangements regarding the **pricing** of generic cancer medicines across Europe, and whether these are indication specific, as well as how this translates into actual prices to provide future direction. Methodology: Mixed method approach with qualitative research among senior health authority personnel and their advisers. Quantitative research via health authority databases to ascertain current prices for oral cancer medicines that had lost their patent and the influence of population size and economics on prices. Results: 25 European countries participated. Currently we see (a) variable approaches to the **pricing** of generic cancer medicines, which will continue; (b) no concerns with substitution for oral generic cancer medicines; (c) substantial price reductions versus originators for generic capecitabine (up to -93.1%), generic imatinib (up to -97.8%) and generic temozolomide (up to -80.7%). Prices for oncology medicines are not indication specific, and are not affected by population size although influenced by **pricing** approaches. There have also been price increases for some non- patented cancer medicines following manufacturer changes although now stabilising. Conclusion: The considerable price reductions seen for some generics means health authorities should further encourage the use of generic oncology medicines when they become available to fund increased volumes and new valued cancer medicines. Countries are also starting to address price increases for generics following changes in the manufacturer

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Cloud computing has been coined as an umbrella term to describe a category of sophisticated on-demand computing services initially offered by commercial providers, such as Amazon, Google, and Microsoft. It denotes a model on which a computing infrastructure is viewed as a “cloud,” from which businesses and individuals access applications from anywhere in the world on demand [1]. In Cloud computing, a provider leases its computing resources in the form of virtual machine instances to users, and a price is charged for the period for which they are used [2]. When the resources are requested by users, they are provided as per Service Level Agreement (SLA) by the CP. An SLA is a contract established between CP and its users, that defines the terms and conditions of the services to be provided by the CP. This agreement includes the **pricing** mechanism of the resources offered by CP to its users.

Electricity is used to illuminate and condition our homes, to power our businesses and factories, and to operate the appliances and devices that enhance our quality of life. Retail electricity markets generally offer flat **pricing** or block **pricing**. Prices remain unchanged irrespective of demand in the first case, while in the second, the per unit rate of electricity either increases or decreases with increasing slabs of electricity consumption [4].

**Derivative** spectrophotometry involves the transformation of absorption spectra into first, second or higher order **derivative** spectra. In **derivative** spectroscopy, the ability to detect and to measure minor spectral features is considerably enhanced. It can be used in quantitative analysis to measure the concentration of an analyte whose peak is obscured by a larger overlapping peak. It is useful in eliminating matrix interference in the assay of many medicinal substances. **Derivative** spectrum is done by wavelength modulation with dual wavelength photometers and microprocessor controlled digital photometer.

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Electricity is used to illuminate and condition our homes, to power our businesses and factories, and to operate the appliances and devices that enhance our quality of life. Retail electricity markets generally offer flat **pricing** or block **pricing**. Prices remain unchanged irrespective of demand in the first case, while in the second, the per unit rate of electricity either increases or decreases with increasing slabs of electricity consumption [4].

to pay. This paper does not examine the specific **pricing** models in particular niche, but wants to give a new angle to value the information goods the demand side decides it. I compare the online fixed **pricing** scheme with online dynamic **pricing** - online auctions and bargaining, to identify the preferences of users when trading information online. The online auction changes the parameters of the traditional auction, and allows the seller and buyer more degrees of freedom (Kumar, 1999; Beam, 1997). Transaction cost is significantly reduced with the development of information technology. Fraud is a problem in online auction since bidders and sellers do not meet each other. Goods auctioning in Internet range from the intangible to the unwieldy. Niche/collectibles, overstocks, commodities, perishables, software, and information, are all actively auctioned in Internet. The delay, security, and easy collaboration aspects of the Internet will cause online auctions on the Internet to be different than the traditional auctions. Information is an experienced good each time it is consumed. I don't know whether today's Wall Street Journal is worth 75 cents until I have read it. Its intrinsic value varies across people. People tend to buy it at higher price if they value it more. When supplier is difficult to determine the price of the information goods, why not use auctions? Let bidding side to determine the price. Hypothesis la: People prefer online dynamic **pricing** to online fixed **pricing** when trading information goods.

values have been determined to establish SAR of the series. The synthesized antihistamine drug loratadine (2) and its amide derivatives (3-5) showed noteworthy cytotoxic activity. Among the synthesized compounds, diethylamine **derivative** (3) showed least cytotoxic activity, whereas ethylenediamine **derivative** (4) and propylenediamine **derivative** (5) showed more cytotoxic activity than their precursor drug loratadine (2) with the propylenediamine **derivative** being the most toxic of all. From this investigation we concluded that increased presence of electronegative atoms, such as Oxygen (O), Nitrogen (N) etc. in a particular compound enhances its chance of being biologically more active.

assumes rational and egoistic individuals and predicts that in a one-shot anonymous interaction a utility maximizing buyer pays no more than the minimum price. However, in applied PWYW situations this is not always the case. Empirical evidence shows that many buyers often pay more than the minimum price. These findings are in line with theories of other regarding preferences. Furthermore, even if buyers pay more than a minimum price, this **pricing** mechanism is bound to fail if payments are too low to cover the production costs. In this case the application of PWYW **pricing** can have negative effects on a corporate profit even if people do not behave selfishly.

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consists of the strategies and tactics used to identify, create and maintain satisfying relationships with customers that result in value for both. Marketing includes anticipating demand, which requires a firm to do customer research on a regular bases so that it develops and introduces products that are desired by consumers, Management of demand which consists of stimulation, facilitation, and regulation of tasks; and satisfaction of demand which involves actual performance, safety, availability of options, after sale service and other factors (Stanton, 1990). In marketing, there are combinations of activities, which start before the creation of a product and don’t end until customers are satisfied. Therefore, product planning, **pricing**, distribution and promotion are the main activities performed in marketing (Kotler, 1996). Ethiop

Price fairness is important amongst construction and engineering consultants as a perceived lack of it engenders unwillingness to pay amongst clients. This can create contractual disputes that negatively impact upon a consultant’s ability to generate sufficient revenue to ensure business continuity and survival. With this in mind, this research aims to analyze the **pricing** measurement forces needed to attain **pricing** fairness within a Ghanaian construction cost consultant (CC) practice. Specific objectives were to identify the key variables responsible for CC services price fairness and establish any interrelationships between them.

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