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In this study, we analyzed option pricing of rainfall derivatives based on stochastic daily rainfall model. We used Markov Chain Analogue Year model (MCAY) in order to describe occurrence process of daily rainfall. We have included the Analogue Year (AY) component in the Markov Chain (MC), which is a new component incorporated in this study and pricing rainfall derivatives. The inclusion of AY in the MC, provides excellent description of the occurrence process of daily rainfall. The amount of daily rainfall on wet days is obtained using Mixed Exponential distribution because it has the advantage of a better representation of extreme events. Combining the occurrence and amount model, we obtained Markov Chain Analogue Year Mixed Exponential model (MCAYMEM). Daily rainfall data from 2005 to 2017 were taken from Ethiopia National Meteorology Agency (ENMA) in order to assess the model performance. Based on the results of the daily rainfall models, we calculated an option price for different months. The price calculated using MCAYMEM gave an excellent result compared to the price calculated using MCMEM. This accuracy is mainly because of AY component included in the MC in the modeling of occurrence process., Applied mathematics; Rainfall; Call option; Weather derivative; Put option; Analogue Year; Option price

We prove in this paper a new integral relationship between the American put option early exercise boundary and its value function in the generalized Black–Scholes model. Based on this relationship we show that it is possible to construct the L 2 -approximation to the unknown early exercise boundary provided that we have at hand any uniform approximation of the American put option value function. Keywords: American put option, Early exercise boundary, Energy inequality for the difference of two convex functions

Climate adaptation requires large capital investments that could be provided not only by traditional sources like governments and banks, but also by derivatives markets. Such markets would allow two parties with different tolerances and expectations about climate risks to transact for their mutual benefit and, in so doing, finance climate adaptation. Here we calculate the price of a derivative called a European put option, based on future sea surface temperature (SST) in Tasmania, Australia, with an 18 C strike threshold. This price represents a quantifiable indicator of climate risk, and forms the basis for aquaculture industries exposed to the risk of higher SST to finance adaptation strategies through the sale of derivative contracts. Such contracts provide a real incentive to parties with different climate outlooks, or risk exposure to take a market assessment of climate change. Crown Copyright 2015 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

In this chapter we assume the perspective of environmentally conscious investors. We hold that the latter are interested in both the objective determinants and the subjective factors that determine if and to what extent owners of an externalities producing technology incorporate these externalities into their production functions. We use public value theory as concept that explains the normatively determined inclusion of externalities into a firm's production function as a result of changing consumer preferences and dynamic legislative environments. We propose that environmentally conscious investing can be compared to buying a put option on public value risks and describe the mechanics of this option.

The real options approach is an extension of financial options theory. The investment opportunities can be considered as corporate real options, which are integral for corporate resource allocation and planning. The value of flexibility of an investment project is basically a collection of real options, which can be valued with the techniques estimated for financial options. Basically, there are three main types of options associated with investment projects are: the option to postpone or delay, the option to expand, and the option to abandon. Option to expand and option to delay are valued as call option while option to abandon are valued as put option. Decision tree analysis can be used as a basic framework to determine the value of options embedded in the investment project. The operating synergies acquiring from a merger can be valued as a real option. Black Scholes model is also used for real option valuation.

[EN] American put option pricing under regime switching is modelled by a system of coupled partial differential equations. The proposed model combines better the reality of the market by incorporating the regime switching jointly with the emotional behaviour of traders using the rationality parameter approach recently introduced by Tågholt Gad and Lund Petersen to cope with possible irrational exercise policy. The classical rational exercise is recovered as a limit case of the rational parameter. The resulting nonlinear system of PDEs is solved by a weighted finite difference method, also known as θ-method. In order to avoid the need of an iterative method for nonlinear system, the term with rationality parameter and the coupling term are treated explicitly. Next, the resulting linear system is solved by Thomas algorithm. Stability conditions for the numerical scheme are studied by using von Neumann approach. Numerical examples illustrate the efficiency and accuracy of the proposed method., This work has been partially supported by the European Union in the FP7- PEOPLE-2012-ITN program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and the Ministerio de Economia y Competitividad Spanish grant MTM2013-41765-P. The fourth author has been partially supported by MINECO Spanish grant MTM2013-47800-C2-1-P and Xunta de Galicia grant CN2011/004.

A method based on Empirical Martingale Simulation (EMS) is presented to evaluate investments in wind energy. Risk-neutral prices are calculated, where electricity market prices are modeled using an ARIMA–GARCH method which shows conditional heteroskedasticity. The values of the put options are calculated a week ahead and it is observed that wind producers that invest in the options market can hedge against price risk and can also maximize their benefits. The use of Monte Carlo simulation with the EMS method in periods of high volatility is especially useful for investors facing price volatilities in order to improve their returns. The model is applied to the Colombian electricity market.

Publisher Summary This chapter focuses on the effects of short sale restrictions on bank's stock an option prices in the German bourse. The short selling restrictions introduced on September 22, 2008, by BaFin in Germany, affected the behavior of stock prices of 11 financial companies, whose naked short selling were specifically prohibited by BaFin. The analysis presented in the chapter involves these 11 banks and focuses on the period of July 5, 2007, to November 28, 2008, for which tick data of American call and put option trades and their underlying stock trades in Deutsche Borse were collected. The analysis did not find any evidence that short selling restrictions, introduced on September 22, 2008, affected bank stock prices or option prices in Germany. BaFin had introduced rules prohibiting naked short selling on financial stocks, while both the British and the American regulators (FSA and SEC) prohibited covered as well as naked short selling. Although prohibiting naked short selling may make the practice of short selling more costly, it is a less severe restriction than prohibiting covered short selling and as a result, the impact on market efficiency is minimal.

Publisher Summary Derivatives are instruments that derive their value from the value of another instrument or commodity. Derivatives are created by a contract that obliges the parties involved to take certain actions depending on the terms of the agreement they make. Although derivatives may include bespoke contracts, derivative exchanges have developed to trade standardized contracts that are fungible. Derivative instruments comprise of options, futures, forward contracts, and complex derivatives. Options are contracts that permit the owner (holder) to purchase (known as a call option) or sell (a put option) a security or some other financial instrument from or to the individual or firm that wrote the option contract (the option writer) in the event that the financial instrument known as the underlying reaches a certain value. A future is a derivative contract that promises to deliver a specified good, instrument, or cash during a set future month known as the delivery month under the terms set forth in the futures contract. Forwards may have many of the characteristics of a future such as standardized quantities and grades, but unlike most futures, most forward contracts are primarily commercial contracts that almost always result in physical delivery. Complex derivatives are structured or engineered contracts constructed for a holder, usually a financial firm or industrial corporation, by a writer, usually an investment or commercial bank. Each instrument is constructed to meet a very specific financial need.

With the present analysis the authors propose an approach for determining an adequate discount rate in environmental management problems, more specifically radioactive waste management. It is shown that the classical Black-Scholes pricing formula can be used for determining the adequate present funding to be set-aside for the future. The average funding is equal to the net present value (NPV) of the future costs, including technical-scenario uncertainties. For taking into account the financial uncertainties, the NPV is identified with the strike price of a European put option, and the asset value in the managed fund is identified with the current price. The risk-free rate is the expected return rate of the portfolio. The paper shows that the adequate present funding can be determined for given multi-generational risk levels and an asset allocation by fixing the discount rate and adding a premium to the NPV of future costs.

Publisher Summary This chapter briefly discusses the barriers that have historically promoted silo risk management contrary to the best interests of shareholders and regulators of national financial institutions. It introduces the concept of the Jarrow–Merton put option as the best single measure of total risk. The chapter highlights the contribution of asset and liability management to total risk, which is included in the value of this put option. Furthermore, it provides a review of outstanding options for Citigroup—a concrete market signal of the total risk of major financial services companies. The chapter also outlines an approach that allows going far beyond Black–Scholes using the Merton reduced form model of an option on the common stock of a company that can default. It uses this model to extract the term structure of default risk and business risk (that is, implied volatility) for Citigroup over a three-year horizon. It compares this term structure of default probabilities to a properly benchmarked reduced form model and confirms the finding of many others that the options market implied default.

Publisher Summary This chapter presents a discussion on option pricing. The chapter discusses the general properties of the stock options. Furthermore, the option pricing theory is presented in the chapter using two approaches: the method of the binomial trees and the classical Black-Scholes theory. In finance, derivatives are the instruments whose price depends on the value of another asset. In particular, the stock option is a derivative whose price depends on the underlying stock price. Derivatives have also been used for many other assets—including, but not limited to, commodities (e.g., cattle, lumber, and copper), Treasury bonds, and currencies. The chapter also discusses the forward and future contracts. In contrast to the forward and future contracts, “options” give an option holder the right to trade an underlying asset rather than the obligation to do this. In particular, the call option gives its holder the right to buy the underlying asset at a specific price (so-called exercise price or strike price) by a certain date (expiration date or maturity). The “put option” gives its holder the right to sell the underlying asset at a strike price by an expiration date. The chapter concludes with a discussion on the Black–Scholes theory and binomial trees.

Publisher Summary Option contracts have different characteristics than futures. The key difference is that the options are based on calls and puts. Options are traded on- and off-exchange and are usually referred to as traded options when based on a security or currency and as options on futures or futures options when the underlying is a futures contract. A call option gives the buyer the right, but not an obligation, to take delivery of the underlying, while a put option confers the right but not the obligation to make delivery of the underlying. As a result of this characteristic, the risk/reward profile is very different for the buyer and the seller. The buyer cannot lose more than they pay for securing the right. When making an opening purchase, the buyer of an option becomes known as a position holder and is said to be long of the market. When making an opening sale, the seller of an option is known as a position writer and is said to be short of the market. Option positions can be either long or short. In special circumstances, both long and short positions can be held gross. Long or short option positions are normally closed out at the time of transacting the opposite position.

Publisher Summary This chapter describes option which is a contract that can exist in several forms and can relate to a variety of assets including shares, foreign currencies, commodities, interest rate futures, stock indices, and treasury bill futures. The option granted is an option to sell. Option traders use combinations of different strategies to cover their risks and to enhance their return on the assets they own. A share option is a contract to purchase or sell a share at a stated price within a specified period of time. There are two types of option, namely, put and call options. A call option is an option to buy an asset at a stated price, called the exercise price, within a specified time period. A put option is the reverse of a call option—the buyer of a put option has the privilege of delivering the underlying asset to the seller, called the writer of the put option, at the stated price before the specified time period. Although options are considered as capital assets, they exist for a limited period only. They have expiration dates after which they are worthless. An option must be exercised on or before its expiration date, if there is to be an actual transfer of assets.