美国灾害保险期货期权的保险精算定价

考虑连续情形、几何平均保险期货价格的基础上研究欧式看涨保险期货期权的定价,运用保险精算定价的方法,最终给出了连续情形、几何平均欧式看涨保险期货期权的定价.     

A non-Gaussian Ornstein–Uhlenbeck model for pricing wind power futures

The recent introduction of wind power futures written on the German wind power production index has brought with it new interesting challenges in terms of modelling and pricing. Some particularities of this product are the strong seasonal component embedded in the underlying, the fact that the wind index is bounded from both above and below and also that the futures are settled against a synthetically generated spot index. Here, we consider the non-Gaussian Ornstein–Uhlenbeck type processes proposed by Barndorff-Nielsen and Shephard in the context of modelling the wind power production index. We discuss the properties of the model and estimation of the model parameters. Further, the model allows for an analytical formula for pricing wind power futures. We provide an empirical study, where the model is calibrated to 37 years of German wind power production index that is synthetically generated assuming a constant level of installed capacity. Also, based on 1 year of observed prices for wind power futures with different delivery periods, we study the market price of risk. Generally, we find a negative risk premium whose magnitude decreases as the length of the delivery period increases. To further demonstrate the benefits of our proposed model, we address the pricing of European options written on wind power futures, which can be achieved through Fourier techniques.     

A Non‐Gaussian Ornstein–Uhlenbeck Process for Electricity Spot Price Modeling and Derivatives Pricing

A mean‐reverting model is proposed for the spot price dynamics of electricity which includes seasonality of the prices and spikes. The dynamics is a sum of non‐Gaussian Ornstein–Uhlenbeck processes with jump processes giving the normal variations and spike behaviour of the prices. The amplitude and frequency of jumps may be seasonally dependent. The proposed dynamics ensures that spot prices are positive, and that the dynamics is simple enough to allow for analytical pricing of electricity forward and futures contracts. Electricity forward and futures contracts have the distinctive feature of delivery over a period rather than at a fixed point in time, which leads to quite complicated expressions when using the more traditional multiplicative models for spot price dynamics. In a simulation example it is demonstrated that the model seems to be sufficiently flexible to capture the observed dynamics of electricity spot prices. The pricing of European call and put options written on electricity forward contracts is also discussed.     

Stochastic models for bidding strategies on oligopoly electricity market

In this paper we consider the forward/futures contracts and Asian-type call options for power delivery as important components of the bidding strategies of the players’ profits on the electricity market. We show how these derivatives can affect their profit. We use linear asymmetric supply function equilibrium (SFE) and Cournot models to develop firms’ optimal bidding strategies by including forward/futures contracts and Asian-type options. We extend the methodology proposed by Niu et al. (IEEE Trans Power Syst 20(4):1859–1867, 2005), where only forward contracts for power delivery were considered in the SFE model.     

The European options hedge perfectly in a Poisson-Gaussian stock market model

It is shown that n + 1 European call options written on a stock S with different strike prices (or the stock and n calls) are non-redundant assets in a model for the stock driven by a Brownian motion and n independent Poisson processes. That extends the result obtained for n = 1 by Pham and implies that the proposed model can price and perfectly hedge any integrable derivative on S.     

沪铜期货及其期权的量化定价实证研究

作为最大的铜消费市场,铜在我国占据着最重要地位。随着2018年9月沪铜期货期权正式登陆上海期货交易所,我国铜交易产品进一步与国际接轨。虽然拥有世界领先的铜交易市场,但我国学术领域尚缺乏采用量化方法对沪铜期货及其期权的深入实证研究。在大宗商品量化领域,Schwartz二因子模型^[1]被看作是量化期货定价的基准模型,本文通过实证分析发现经典二因子模型在中国市场应用时会产生“水土不服”,因此本文采用波动参数的思想,修正了Schwartz二因子模型在中国沪铜市场的“水土不服”。另外,根据早期文献对认购期货期权的量化定价模型^[2,3],本文明确了认沽期货期权的量化定价公式,并通过对沪铜期货期权定价模型的实证分析,发现目前沪铜期货期权存在套利空间。     

Valuation of futures options with initial margin requirements and daily price limit

The paper presents a valuation model of futures options trading at exchanges with initial margin requirements and daily price limit, and this result gives an academic guidance to design trading rules at exchanges. Unlike the leading work of Black, certain trading rules are considered so as to be more fit for practical futures markets. The paper prices futures options with initial margin requirements and daily price limit by duplicating them with the help of the theory of backward stochastic differential equations （BSDEs, for short）. Furthermore, an explicit expression of the price Of the call （or the put） futures option is given and also is shown to be the unique solution of the associated nonlinear partial differential equation.     

A systematic approach to pricing and hedging international derivatives with interest rate risk: analysis of international derivatives under stochastic interest rates

This paper deals with the valuation and the hedging of non-path-dependent European options on one or several underlying assets in a model of an international economy allowing for both, interest rate risk and exchange rate risk. Using martingale theory and, in particular, the change of numeraire technique we provide a unified and easily applicable approach to pricing and hedging exchange options on stocks, bonds, futures, interest rates and exchange rates. We also cover the pricing and hedging of compound exchange options.     

A simple model of deferred callability in defaultable debt

Banks and other financial institutions issue hybrid capital as part of their risk capital. Hybrid capital has no maturity, but, similarly to most corporate debt, includes an embedded issuer’s call option. To obtain acceptance as risk capital, the first possible exercise date of the embedded call is contractually deferred by several years, generating a protection period. We value the call feature as a European option on perpetual defaultable debt. We do this by first modifying the underlying asset process to incorporate a time-dependent bankruptcy level before the expiration of the embedded option. We identify a call option on debt as a fixed number of put options on a modified asset, which is lognormally distributed, as opposed to the market value of debt. To include the possibility of default before the expiration of the option we apply barrier options results. The formulas are quite general and may be used for valuing both embedded and third-party options. All formulas are developed in the seminal and standard Black–Scholes–Merton model and, thus, standard analytical tools such as ‘the greeks’, are immediately available.     

Sustainable energy futures: Methodological challenges in combining scenarios and participatory multi-criteria analysis

This paper analyses the combined use of scenario building and participatory multi-criteria analysis (PMCA) in the context of renewable energy from a methodological point of view. Scenarios have been applied increasingly in decision-making about long-term consequences by projecting different possible pathways into the future. Scenario analysis accounts for a higher degree of complexity inherent in systems than the study of individual projects or technologies. MCA is a widely used appraisal method, which assesses options on the basis of a multi-dimensional criteria framework and calculates rankings of options. In our study, five renewable energy scenarios for Austria for 2020 were appraised against 17 sustainability criteria. A similar process was undertaken on the local level, where four renewable energy scenarios were developed and evaluated against 15 criteria. On both levels, the scenario development consisted of two stages: first an exploratory stage with stakeholder engagement and second a modelling stage with forecasting-type scenarios. Thus, the scenarios consist of a narrative part (storyline) and a modeled quantitative part. The preferences of national and local energy stakeholders were included in the form of criteria weights derived from interviews and participatory group processes, respectively. Especially in the case of renewable energy promotion in Austria, the paper systematically analyses the potentials and limitations of the methodology (1) for capturing the complexity of decision-making about the long-term consequences of changes in socio-economic and biophysical systems and (2) for appraising energy futures. The paper concludes that assessing scenarios with PMCA is resource intense, but this methodology captures successfully the context of technology deployment and allows decision-making based on a robust and democratic process, which addresses uncertainties, acknowledges multiple legitimate perspectives and encourages social learning.     

Ornstein–Uhlenbeck processes time changed with additive subordinators and their applications in commodity derivative models

We characterize Ornstein–Uhlenbeck processes time changed with additive subordinators as time-inhomogeneous Markov semimartingales, based on which a new class of commodity derivative models is developed. Our models are tractable for pricing European, Bermudan and American futures options. Calibration examples show that they can be better alternatives than those developed in Li and Linetsky (2012)  [6]. Our method can be applied to many other processes popular in various areas besides finance to develop time-inhomogeneous Markov processes with desirable features and tractability.     

Black-Scholes期权定价公式推广

在Black-Scholes期权定价模型的基础上,进一步考虑标的资产受多个跳跃源影响的情况,用含有多维Poisson过程的Ito-Skorohod随机微分方程描述标的资产价格的动态运动,应用等价鞅测度变换方法导出一般形式的欧式期权定价公式,并讨论了利率,波动率不是常数情况下的拓广形式.     

A partial introduction to financial asset pricing theory

We present an introduction to mathematical Finance Theory for mathematicians. The approach is to start with an abstract setting and then introduce hypotheses as needed to develop the theory. We present the basics of European call and put options, and we show the connection between American put options and backwards stochastic differential equations.     

Pricing Parisian down-and-in options

In this paper, we price American-style Parisian down-and-in call options under the Black–Scholes framework. Usually, pricing an American-style option is much more difficult than pricing its European-style counterpart because of the appearance of the optimal exercise boundary in the former. Fortunately, the optimal exercise boundary associated with an American-style Parisian knock-in option only appears implicitly in its pricing partial differential equation (PDE) systems, instead of explicitly as in the case of an American-style Parisian knock-out option. We also recognize that the “moving window” technique developed by Zhu and Chen (2013) for pricing European-style Parisian up-and-out call options can be adopted to price American-style Parisian knock-in options as well. In particular, we obtain a simple analytical solution for American-style Parisian down-and-in call options and our new formula is written in terms of four double integrals, which can be easily computed numerically.     

企业债券的欧式期权的定价公式

本文研究了带有信用风险的企业债券的欧式衍生资产的定价方法,建立风险债券与无风险债券期权价格的相互关系。     

Option pricing with mean reversion and stochastic volatility

Many underlying assets of option contracts, such as currencies, commodities, energy, temperature and even some stocks, exhibit both mean reversion and stochastic volatility. This paper investigates the valuation of options when the underlying asset follows a mean-reverting lognormal process with stochastic volatility. A closed-form solution is derived for European options by means of Fourier transform. The proposed model allows the option pricing formula to capture both the term structure of futures prices and the market implied volatility smile within a unified framework. A bivariate trinomial lattice approach is introduced to value path-dependent options with the proposed model. Numerical examples using European options, American options and barrier options demonstrate the use of the model and the quality of the numerical scheme.     

Viscosity Solutions of Integro-Differential Equations and Passport Options in a Jump-Diffusion Model

We study the viscosity solutions of integro-differential Hamilton–Jacobi–Bellman equations of degenerate parabolic type. These equations are from the pricing problem for the European passport options in a jump-diffusion model. The passport option is a call option on a trading account. We discuss the mathematical model for pricing problem. We prove the comparison principle, uniqueness and convexity preserving for the viscosity solutions of related pricing equations.     

Real (investment) options with multiple sources of rare events

We value real (investment) options when the underlying asset follows a mixed jump-diffusion process involving various types (sources) of rare events (jumps). These jumps are assumed independent of each other, with each type having a log-normally distributed jump size and a random (Poisson-distributed) arrival time. They may represent uncertainties about the arrival and impact (on the underlying investment) of new information concerning technological innovation, competition, political risk, regulatory effects and other sources. An analytic solution is presented for European claims (call or put options) with multiple sources of jumps. A discrete-time (Markov-chain) methodology (implemented within a finite-difference scheme) is proposed for the valuation of American as well as European options. The approach is also applicable to financial options with multiple types of rare events. The approach is illustrated through valuing complex real options with compound features involving interactions between optimal investment and subsequent operating decisions. Specifically, we value a growth option and an extension option.     

Numerical simulation of a Finite Moment Log Stable model for a European call option

Compared to the classical Black-Scholes model for pricing options, the Finite Moment Log Stable (FMLS) model can more accurately capture the dynamics of the stock prices including large movements or jumps over small time steps. In this paper, the FMLS model is written as a fractional partial differential equation and we will present a new numerical scheme for solving this model. We construct an implicit numerical scheme with second order accuracy for the FMLS and consider the stability and convergence of the scheme. In order to reduce the storage space and computational cost, we use a fast bi-conjugate gradient stabilized method (FBi-CGSTAB) to solve the discrete scheme. A numerical example is presented to show the efficiency of the numerical method and to demonstrate the order of convergence of the implicit numerical scheme. Finally, as an application, we use the above numerical technique to price a European call option. Furthermore, by comparing the FMLS model with the classical B-S model, the characteristics of the FMLS model are also analyzed.     

Valuation of employee stock options using the exercise multiple approach and life tables

Employee stock options (ESOs) are common in performance-based employee remuneration. Financial reporting standards such as IFRS2 and AASB2 require public corporations to report on the cost of providing ESOs, and mandate the incorporation of voluntary and involuntary early exercise. In this paper we extend the exercise multiple approach of Hull and White (2004) and decompose the attrition unadjusted voluntary exercise ESO into a gap call option and two partial-time barrier options. We use exit probabilities obtained from empirically determined multiple decrement or life tables to model involuntary early exercise or forfeiture. We provide a new analytic valuation formula which expresses the ESO value in terms of a portfolio of exotic European bivariate power options and which correctly accounts for both voluntary exercise and employee attrition. Recent approaches seek to model employee attrition using a constant hazard rate. Our approach uses an empirically driven actuarial method for incorporating employee attrition in the valuation.