A Full Asymptotic Series of European Call Option Prices in the SABR Model with Beta = 1

We develop two new pricing formulae for European options. The purpose of these formulae is to better understand the impact of each term of the model, as well as improve the speed of the calculations. We consider the SABR model (with β=1) of stochastic volatility, which we analyze by tools from Malliavin Calculus. We follow the approach of Alòs et al. (2006) who showed that under stochastic volatility framework, the option prices can be written as the sum of the classic Hull-White (1987) term and a correction due to correlation. We derive the Hull-White term, by using the conditional density of the average volatility, and write it as a two-dimensional integral. For the correction part, we use two different approaches. Both approaches rely on the pairing of the exponential formula developed by Jin, Peng, and Schellhorn (2016) with analytical calculations. The first approach, which we call “Dyson series on the return’s idiosyncratic noise” yields a complete series expansion but necessitates the calculation of a 7-dimensional integral. Two of these dimensions come from the use of Yor’s (1992) formula for the joint density of a Brownian motion and the time-integral of geometric Brownian motion. The second approach, which we call “Dyson series on the common noise” necessitates the calculation of only a one-dimensional integral, but the formula is more complex. This research consisted of both analytical derivations and numerical calculations. The latter show that our formulae are in general more exact, yet more time-consuming to calculate, than the first order expansion of Hagan et al. (2002).

On the Efficacy of Fourier Series Approximations for Pricing European Options

This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semi-closed form. The algorithms investigated here are the half-range Fourier cosine series, the half-range Fourier sine series and the full-range Fourier series. Their performance is assessed in simulation experiments in which an analytical solution is available and also for a simple affine model of stochastic volatility in which there is no closed-form solution. The results suggest that the half-range sine series approximation is the least effective of the three proposed algorithms. It is rather more difficult to distinguish between the performance of the half-range cosine series and the full-range Fourier series. However there are two clear differences. First, when the interval over which the density is approximated is relatively large, the full-range Fourier series is at least as good as the half-range Fourier cosine series, and outperforms the latter in pricing out-of-the-money call options, in particular with maturities of three months or less. Second, the computational time required by the half-range Fourier cosine series is uniformly longer than that required by the full-range Fourier series for an interval of fixed length. Taken together, these two conclusions make a case for pricing options using a full-range range Fourier series as opposed to a half-range Fourier cosine series if a large number of options are to be priced in as short a time as possible.

European Option Pricing under a Class of Fractional Market

In order to price European contingent claim in a class of fractional Black-Scholes market, where the prices of assets follow a Wick-Ito stochastic differential equation driven by the fractional Brownian motion and market coefficients are deterministic functions, the pricing formula of European call option was explicitly derived by the method of the stochastic calculus of tile fractional Brownian motion. A result about fractional Clark derivative was also obtained.

The application of option pricing theory to the evaluation of mining investment

A rational evaluation on an investment project forms the basis of a right investment decision making. The discounted cash flow (DCF for short) method is usually used as a traditional evaluation method for a project investment. However, as the mining investment is influenced by many uncertainties, DCF method cannot take into account these uncertainties and often underestimates the value of an investment project. Based on the option pricing theory of the modern financial assets, the characteristics of a real project investment are discussed, and the management option of mine managers and its pricing method are described.

A Boundary Element Formulation for the Pricing of Barrier Options

In this article, we derive a boundary element formulation for the pricing of barrier option. The price of a barrier option is modeled as the solution of Black-Scholes’ equation. Then the problem is transformed to a boundary value problem of heat equation with a moving boundary. The boundary integral representation and integral equation are derived. A boundary element method is designed to solve the integral equation. Special quadrature rules for the singular integral are used. A numerical example is also demonstrated. This boundary element formulation is correct.

Application of Binomial Option Pricing Model to the Appraisal of Knowledge Management Investment

This paper views knowledge management （KM） investment from the angle of real options, and demonstrates the utility of the real options approach to KM investment analysis. First, KM project has characteristics of uncertainty, irreversibility and choice of timing, which suggests that we can appraise KM investment by real options theory. Second, the paper analyses corresponding states of real options in KM and finance options. Then, this paper sheds light on the way to the application of binomial pricing method to KM investment model, which includes modeling and conducting KM options. Finally, different results are shown of using DCF method and binomial model of option evaluation via a case.

Pricing Catastrophe Options with Credit Risk in a Regime-Switching Model

In this paper,we consider the price of catastrophe options with credit risk in a regime-switching model.We assume that the macroeconomic states are described by a continuous-time Markov chain with a finite state space.By using the measure change technique,we derive the price expressions of catastrophe put options.Moreover,we conduct some numerical analysis to demonstrate how the parameters of the model affect the price of the catastrophe put option.

European option pricing model in a stochastic and fuzzy environment

The primary goal of this paper is to price European options in the Merton＇s frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes as fuzzy random variables and use the property of fuzzy set to deduce two different jump-diffusion models underlying principle of rational expectations equilibrium price. Unlike many conventional models, the European option price will now turn into a fuzzy number. One of the major advantages of this model is that it allows investors to choose a reasonable European option price under an acceptable belief degree. The empirical results will serve as useful feedback information for improvements on the proposed model.

Pricing of discrete barrier options based on an analytical method

The problem o f analytically pricing the discrete monitored European barrier options is studied under the assumption of the Black-Scholes market.First,using variable transformation,the mean vector and covariance matrix of multi-dimensional marginal distribution are given.Secondly,the analytica pricing formulas of the discrete monitored upknock-out European call option and the discrete monitored down-knock-out European put option a e obtained by using the conditional probability and the characteristics o f the multidimensional normal distribution.Finally,the effects of the discrete monitoring barriers on the prices of the barrier optionsare discussed and analyzed.The research results state that the price o f the discrete monitored up-knock-out European call option mcreases with the increase in the up barrier,a d the price o f the discrete monitored down-knock-out European put option decreases with the increase in the down barrier.

A New Binomial Tree Method for European Options under the Jump Diffusion Model

In this paper, the binomial tree method is introduced to price the European option under a class of jump-diffusion model. The purpose of the addressed problem is to find the parameters of the binomial tree and design the pricing formula for European option. Compared with the continuous situation, the proposed value equation of option under the new binomial tree model converges to Merton’s accurate analytical solution, and the established binomial tree method can be proved to work better than the traditional binomial tree. Finally, a numerical example is presented to illustrate the effectiveness of the proposed pricing methods.

A closed-form pricing formula for European options in an illiquid asset market

This article addresses the problem of pricing European options when the underlying asset is not perfectly liquid.A liquidity discounting factor as a function of market-wide liquidity governed by a mean-reverting stochastic process and the sensitivity of the underlying price to market-wide liquidity is firstly introduced,so that the impact of liquidity on the underlying asset can be captured by the option pricing model.The characteristic function is analytically worked out using the Feynman–Kac theorem and a closed-form pricing formula for European options is successfully derived thereafter.Through numerical experiments,the accuracy of the newly derived formula is verified,and the significance of incorporating liquidity risk into option pricing is demonstrated.

NEW METHOD TO OPTION PRICING FOR THE GENERAL BLACK-SCHOLES MODEL-AN ACTUARIAL APPROACH

Using physical probability measure of price process and the principle of fair premium, the results of Mogens Bladt and Hina Hviid Rydberg are generalized. In two cases of paying intermediate divisends and no intermediate dividends, the Black_Scholes model is generalized to the case where the risk_less asset (bond or bank account) earns a time_dependent interest rate and risk asset (stock) has time_dependent the continuously compounding expected rate of return, volatility. In these cases the accurate pricing formula and put_call parity of European option are obtained. The general approach of option pricing is given for the general Black_Scholes of the risk asset (stock) has the continuously compounding expected rate of return, volatility. The accurate pricing formula and put_call parity of European option on a stock whose price process is driven by general Ornstein_Uhlenback (O_U) process are given by actuarial approach.

Research on the Protection of Vulnerable Groups in Water Pollution Conflicts Based on Binomial Tree Pricing Model

The interests of vulnerable groups can’t be guaranteed due to their weaker capacity and the limited interests demand channels during the water pollution conflicts. The interest protection for the vulnerable people in the water pollution conflicts has attracted attentions of the international scholars. The paper tries to construct the market mechanism which can make the vulnerable people to involve in the emission trading. The vulnerable people can buy American put option in the emission trading market. When the price of the emission runs below the contract price, the vulnerable people can get the benefit through executing the option. When the price of the emission runs above the contract price, the vulnerable people can give up the right. The binomial tree option pricing model can help the vulnerable people to make a decision through the analysis of the worth of the American put option.

Black-Scholes Option Pricing Model Modified to Admit a Miniscule Drift Can Reproduce the Volatility Smile

This paper develops a closed-form solution to an extended Black-Scholes (EBS) pricing formula which admits an implied drift parameter alongside the standard implied volatility. The market volatility smiles for vanilla call options on the S&P 500 index are recreated fitting the best volatility-drift combination in this new EBS. Using a likelihood ratio test, the implied drift parameter is seen to be quite significant in explaining volatility smiles. The implied drift parameter is sufficiently small to be undetectable via historical pricing analysis, suggesting that drift is best considered as an implied parameter rather than a historically-fit one. An overview of option-pricing models is provided as background.

Valuation of European and American Options under Variance Gamma Process

Geometric Brownian Motion (GBM) is widely used to model the asset price dynamics. Option price models such as the Black-Sholes and the binomial tree models rely on the assumption that the underlying asset price dynamics follow the GBM. Modeling the asset price dynamics by using the GBM implies that the log return of assets at particular time is normally distributed. Many studies on real data in the markets showed that the GBM fails to capture the characteristic features of asset price dynamics that exhibit heavy tails and excess kurtosis. In our study, a class of Levy process, which is called a variance gamma (VG) process, performs much better than GBM model for modeling the dynamics of those stock indices. However, valuation of financial instruments, e.g. options, under the VG process has not been well developed. Here, we propose a new approach to the valuation of European option. It is based on the conditional distribution of the VG process. We also apply the path simulation model to value American options by assuming the underlying asset log return follow the VG process. Such a model is similar with that proposed by Tiley [1]. Simulation study shows that the proposed method performs well in term of the option price.

Early exercise European option and early termination American option pricing models

The maximum relative error between continuous-time American option pricing model and binomial tree model is very small. In order to improve the European and American options in trade course, the thesis tried to build early exercise European option and early termination American option pricing models. Firstly, the authors reviewed the characteristics of American option and European option, then there was compares between them. Base on continuous-time American option pricing model, this research analyzed the value of these options.

The SABR Model: Explicit Formulae of the Moments of the Forward Prices/Rates Variable and Series Expansions of the Transition Probability Density and of the Option Prices

The SABR stochastic volatility model with β-volatility β ? (0,1) and an absorbing barrier in zero imposed to the forward prices/rates stochastic process is studied. The presence of (possibly) nonzero correlation between the stochastic differentials that appear on the right hand side of the model equations is considered. A series expansion of the transition probability density function of the model in powers of the correlation coefficient of these stochastic differentials is presented. Explicit formulae for the first three terms of this expansion are derived. These formulae are integrals of known integrands. The zero-th order term of the expansion is a new integral formula containing only elementary functions of the transition probability density function of the SABR model when the correlation coefficient is zero. The expansion is deduced from the final value problem for the backward Kolmogorov equation satisfied by the transition probability density function. Each term of the expansion is defined as the solution of a final value problem for a partial differential equation. The integral formulae that give the solutions of these final value problems are based on the Hankel and on the Kontorovich-Lebedev transforms. From the series expansion of the probability density function we deduce the corresponding expansions of the European call and put option prices. Moreover we deduce closed form formulae for the moments of the forward prices/rates variable. The moment formulae obtained do not involve integrals or series expansions and are expressed using only elementary functions. The option pricing formulae are used to study synthetic and real data. In particular we study a time series (of real data) of futures prices of the EUR/USD currency's exchange rate and of the corresponding option prices. The website: http://www.econ.univpm.it/recchioni/finance/w18 contains material including animations, an interactive application and an app that helps the understanding of the paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website:http://www.econ.univpm.it/recchioni/finance.

Trinomial tree model of the real options approach used in mining investment price forecast and analysis

In order to effectively avoid the defects of a traditional discounted cash flow method, a trinomial tree pricing model of the real option is improved and used to forecast the investment price of mining. Taking Molybdenum ore as an example, a theoretical model for the hurdle price under the optimal investment timing is constructed. Based on the example data, the op- tion price model is simulated. By the model, mine investment price can be computed and forecast effectively. According to the characteristics of mine investment, cut-off grade, reserve estimation and mine life in different price also can be quantified. The result shows that it is reliable and practical to enhance the accuracy for mining investment decision.

Evaluation of call options

The European and American call options, for which the prices of their underlying asset follow compound Poisson process, are evaluated by a probability method. Formulas that can be used to evaluate the options are obtained, which include not only the elements of an option: the price of the call option, the exercise price and the expiration date, but also the riskless interest rate, nevertheless exclude the volatility of the underlying asset. In practice, the evaluated results obtained by these formulas can provide references of making strategic decision for an investor who buys the call option and a company who sells the call option.

How Load Aggregators Avoid Risks in Spot Electricity Market:In the Framework of Power Consumption Right Option Contracts

There is uncertainty in the electricity price of spot electricity market,which makes load aggregators undertake price risks for their agent users.In order to allow load aggregators to reduce the spot market price risk,scholars have proposed many solutions,such as improving the declaration decision-making model,signing power mutual insurance contracts,and adding energy storage and mobilizing demand-side resources to respond.In terms of demand side,calling flexible demand-side resources can be considered as a key solution.The user’s power consumption rights(PCRs)are core contents of the demand-side resources.However,there have been few studies on the pricing of PCR contracts and transaction decisions to solve the problem of price forecast deviation and to manage the uncertainty of spot market prices.In addition,in traditional PCR contracts,PCRs are mostly priced using a single price mechanism,that is,the power user is compensated for part of the electricity that was interrupted or reduced in power supply.However,some power users might engage in speculative behaviours under this mechanism.Further,for load aggregators,their price risk avoidance ability has not substantially improved.As a financial derivative,options can solve the above problems.In this article,firstly,the option method is used to build an option pricing optimization model for power consumption right contracts that can calculate the optimal option premium and strike price of option contracts of power consumption rights.Secondly,from the perspective of power users and load aggregators,a simulation model of power consumption right transaction decision-making is constructed.The results of calculation examples show that(1)Under the model in this article,the pricing of option contracts for power consumption rights with better risk aversion capabilities than traditional compensation contracts can be obtained.(2)The decision to sell or purchase the power consumption rights will converge at respective highvalue periods,and option contracts will expedite the process.(3)Option contracts can significantly reduce the loss caused by the uncertainty of spot electricity prices for load aggregators without reducing users’willingness to sell power consumption rights.