The aim of this paper is to research the establishment of Chinese rural drinking water market. On the basis of prior research on rural drinking water and water market, we study the market establishment and analyze trading elements, procedure and modes of Chinese rural drinking water market. Then we explore the operating mechanism of the market, such as the supply and demand mechanism, the price mechanism, the competition mechanism and the support mechanism. This study will conduce to the optimization of the allocation of water resources.

Abstract:
Based on empirical market data, a stochastic volatility model is proposed with volatility driven by fractional noise. The model is used to obtain a risk-neutrality option pricing formula and an option pricing equation.

Asset pricing under the certainty equivalent approach framework always raises the current value of the asset with the riskless rate first, followed immediately by risk adjustments. Clearly, this type of arrangement does not apply to assets that are expecting to lose values if it were to adhere to feasible economic reasoning. By using the put-call parity relationship and its underlying law of no arbitrage, the needed expected rates of return for the job of option pricing can thus be obtained. This study suggests a new model in old fashion, which can better satisfy the empirical criticism of the Black-Scholes option pricing model.

Abstract:
The study analyses some problems arising in stochastic volatility models by using Ito’s lemma and its applications to boundary Cauchy problem by giving the solution of vanilla option pricing models satisfying the partial differential equation obtained by assuming stochastic volatility in replication problems and risk neutral probability.

Abstract:
In this paper, we investigate recent developments in option pricing based on Black-Scholes processes, pure jump processes, jump diffusion process, and stochastic volatility processes. Results on Black-Scholes model with GARCH volatility (Gong, Thavaneswaran and Singh [1]) and Black-Scholes model with stochastic volatility (Gong, Thavaneswaran and Singh [2]) are studied. Also, recent results on option pricing for jump diffusion processes, partial differential equation (PDE) method together with FFT (fast Fourier transform) approximations of Pillay and O’ Hara [3] and a recently proposed method based on moments of truncated lognormal distribution (Thavaneswaran and Singh [4]) are also discussed in some detail.

Abstract:
How to price and hedge claims on nontraded assets are becoming increasingly important matters in option pricing theory today. The most common practice to deal with these issues is to use another similar or "closely related" asset or index which is traded, for hedging purposes. Implicitly, traders assume here that the higher the correlation between the traded and nontraded assets, the better the hedge is expected to perform. This raises the question as to how \textquoteleft{}closely related\textquoteright{} the assets really are. In this paper, the concept of twin assets is introduced, focusing the discussion precisely in what does it mean for two assets to be similar. Our findings point to the fact that, in order to have very similar assets, for example identical twins, high correlation measures are not enough. Specifically, two basic criteria of similarity are pointed out: i) the coefficient of variation of the assets and ii) the correlation between assets. From here, a method to measure the level of similarity between assets is proposed, and secondly, an option pricing model of twin assets is developed. The proposed model allows us to price an option of one nontraded asset using its twin asset, but this time knowing explicitly what levels of errors we are facing. Finally, some numerical illustrations show how twin assets behave depending upon their levels of similarities, and how their potential differences will traduce in MAPE (mean absolute percentage error) for the proposed option pricing model.

Abstract:
The literature on volatility modelling and option pricing is a large and diverse area due to its importance and applications. This paper provides a review of the most significant volatility models and option pricing methods, beginning with constant volatility models up to stochastic volatility. We also survey less commonly known models e.g. hybrid models. We explain various volatility types (e.g. realised and implied volatility) and discuss the empirical properties.

Abstract:
A statistical decision problem is hidden in the core of option pricing. A simple form for the price C of a European call option is obtained via the minimum Bayes risk, R_B, of a 2-parameter estimation problem, thus justifying calling C Bayes (B-)price. The result provides new insight in option pricing, among others obtaining C for some stock-price models using the underlying probability instead of the risk neutral probability and giving R_B an economic interpretation. When logarithmic stock prices follow Brownian motion, discrete normal mixture and hyperbolic Levy motion the obtained B-prices are "fair" prices. A new expression for the price of American call option is also obtained and statistical modeling of R_B can be used when pricing European and American call options.

Abstract:
In a model with no given probability measure, we consider asset pricing in the presence of frictions and other imperfections and characterize the property of coherent pricing, a notion related to (but much weaker than) the no arbitrage property. We show that prices are coherent if and only if the set of pricing measures is non empty, i.e. if pricing by expectation is possible. We then obtain a decomposition of coherent prices highlighting the role of bubbles. eventually we show that under very weak conditions the coherent pricing of options allows for a very clear representation from which it is possible, as in the original work of Breeden and Litzenberger, to extract the implied probability. Eventually we test this conclusion empirically via a new non parametric approach.

Abstract:
The problem of determining the European-style option price in the incomplete market has been examined within the framework of stochastic optimization. An analytic method based on the discrete dynamic programming equation (Bellman equation) has been developed that gives the general formalism for determining the option price and the optimal trading strategy (optimal control policy) that reduces total risk inherent in writing the option. The basic purpose of paper is to present an effective algorithm that can be used in practice. Keywords: option pricing, incomplete market, transaction costs, stochastic optimization, Bellman equation.