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.

Pricing Discrete Barrier Options Under the Jump-Diffusion Model with Stochastic Volatility and Stochastic Intensity

This paper considers the problem of numerically evaluating discrete barrier option prices when the underlying asset follows the jump-diffusion model with stochas-tic volatility and stochastic intensity.We derive the three-dimensional characteristic function of the log-asset price,the volatility and the jump intensity.We also provide the approximate formula of the discrete barrier option prices by the three-dimensional Fourier cosine series expansion(3D-COS)method.Numerical results show that the 3D-COS method is rather correct,fast and competent for pricing the discrete barrier options.

The valuation of barrier options under a threshold rough Heston model

In this paper,we propose a novel model for pricing double barrier options,where the asset price is modeled as a threshold geometric Brownian motion time changed by an integrated activity rate process,which is driven by the convolution of a fractional kernel with the CIR process.The new model both captures the leverage effect and produces rough paths for the volatility process.The model also nests the threshold diffusion,Heston and rough Heston models.We can derive analytical formulas for the double barrier option prices based on the eigenfunction expansion method.We also implement the model and numerically investigate the sensitivities of option prices with respect to the parameters of the model.

PRICING BARRIER OPTIONS UNDER STOCHASTIC VOLATILITY FRAMEWORK

Abstract Option pricing problem plays an extremely important role in quantitative finance. In com- plete market, Black-Scholes-Merton theory has been central to the development of financial engineering as both discipline and profession. However, in incomplete market, there are not any replicating port- folios for those options, and thus, the market traders cannot apply the law of one price for obtaining a unique solution. Fortunately, the authors can get a fair price via local-equilibrium principle. In this paper, the authors apply the stochastic control theory to price the exotic option-barrier options, and analyze the relationship between the price and the current positions. The authors get the explicit expression for the market price of the risk. The position effect plays a significant role in option pricing, because it can tell the trader how many and which direction to trade with the market in order to reach the local equilibrium with the market.

Pricing Continuously Monitored Barrier Options under the SABR Model:A Closed‐Form Approximation

The stochastic alpha beta rho(SABR)model introduced by Hagan et al.(2002)is widely used in both fixed income and the foreign exchange(FX)markets.Continuously monitored barrier option contracts are among the most popular derivative contracts in the FX markets.In this paper,we develop closed-form formulas to approximate various types of barrier option prices(down-and-out/in,up-and-out/in)under the SABR model.We first derive an approximate formula for the survival density.The barrier option price is the one-dimensional integral of its payoff function and the survival density,which can be easily implemented and quickly evaluated.The approximation error of the survival density is also analyzed.To the best of our knowledge,it is the first time that analytical(approximate)formulas for the survival density and the barrier option prices for the SABR model are derived.Numerical experiments demonstrate the validity and efficiency of these formulas.

Toxic Currency Options in Poland as a Consequence of the 2008 Financial Crisis

The paper is on toxic foreign exchange options problem which occurred in Poland just prior to and after the outbreak of the recent crisis. Especially Polish enterprises were severely stricken by transactions on fx and interest rate derivatives contracted with their banks. Poland was the only EU country which did not precipitate into recession during the financial crisis beginning in 2008. However, the toxic fx and interest rate derivatives transmitted the shockwaves from global financial markets into Poland. Huge dimensions of losses resulted in conflicts between banks and their customers, who claimed just being cheated by the financial institutions. The article deeply researches into reasons for such developments on Polish fx over-the-counter derivatives market. As a case study, an authentic strategy has been presented. The contract was concluded between the construction company and one of the biggest commercial banks in Poland. Because the case study may be representative for many other cases, the analysis includes exact pricing of option strategy and therefore reveals inequality of the contract. The consequences of non-implementing the MiFID directive in the context of derivatives offering to non-financial customers were also touched in the paper.

The Utilization of Exotic Options in the Formation of Structured Products

This paper deals with the issue of investment certificate formation in the financial market. Investment certificate is a type of structured products, the value of which is derived from the value of an underlying asset. The underlying asset is usually a share in a company, a basket of shares, or an entire index, etc.. It can be stated that for every estimated development of an asset （growth, fall, and stagnation） or for every attitude to risks （conservative or aggressive investors）, there is a suitable kind of certificate. The main objective is to perform an analysis of the structured product--Austria/Germany Bond 3 and its guarantee certificate construction using digital-barrier options. The authors have found an alternative opportunity to the purchase of this certificate, i.e., investment in a bank deposit, together with a purchase of cash or nothing down and four-knock-out call options and a sale of cash or nothing down and four-knock-out put options. The authors prove that the alternative investment has the same profit profile as the certificate. The authors made this analysis with the objective to contribute to the intellectualization of investors.

AMERICAN CONTINUOUS-INSTALLMENT OPTIONS OF BARRIER TYPE

This paper analyzes and values an American barrier option with continuous payment plan written on a dividend paying asset under the classical Black-Scholes model.The integral representation of the initial premium along with the delta hedge parameter for an American continuous-installment down-and-out call option are obtained by using the decomposition technique.This offers a system of nonlinear integral equations for determining the optimal exercise and stopping boundaries,which can be utilized to approximate the option price and delta hedge parameter.The implementation is based on discretizing the quadrature formula in the system of equations and using the Newton-Raphson method to compute the two optimal boundaries at each time points.Numerical results are provided to illustrate the computational accuracy and the effects on the initial premium and optimal boundaries with respect to barrier.

AN INTEGRO-DIFFERENTIAL PARABOLIC VARIATIONAL INEQUALITY ARISING FROM THE VALUATION OF DOUBLE BARRIER AMERICAN OPTION

This paper studies the nonlinear variational inequality with integro-differential term arising from valuation of American style double barrier option. First, the authors use the penalty method to transform the variational inequality into a nonlinear parabolic initial boundary problem(i.e., penalty problem). Second, the existence and uniqueness of solution to the penalty problem are proved by using the Scheafer fixed point theory. Third, the authors prove the existence of variational inequality' solution by showing the fact that the penalized PDE converges to the variational inequality. The uniqueness of solution to the variational inequality is also proved by contradiction.

American Barrier Option Pricing Formulas for Currency Model in Uncertain Environment

Option pricing problem is one of the central issue in the theory of modern finance.Uncertain currency model has been put forward under the foundation of uncertainty theory as a tool to portray the foreign exchange rate in uncertain finance market.This paper uses uncertain differential equation involved by Liu process to dispose of the foreign exchange rate.Then an American barrier option of currency model in uncertain environment is investigated.Most important of all,the authors deduce the formulas to price four types of American barrier options for this currency model in uncertain environment by rigorous derivation.

RBSDEs with optional barriers:monotone approximation

In this short note we consider reflected backward stochastic differential equations(RBSDEs)with a Lipschitz driver and barrier processes that are optional and right lower semicontinuous.In this case,the barrier is represented as a nondecreasing limit of right continuous with left limit(RCLL)barriers.We combine some well-known existence results for RCLL barriers with comparison arguments for the control process to construct solutions.Finally,we highlight the connection of these RBSDEs with standard RCLL BSDEs.

ON PRICING MODEL OF THE RESET OPTION WITH N PREDETERMINED LEVELS

Motivated by the reset option with n predetermined dates analyzed by W.Cheng, we consider a kind of reset option with uncertain dates by introducing N pie-specifiedbarrier levels. We claim this reset option consists of some standard knock-in and knock-out barrieroptions. The closed-form pricing formula is derived by means of a PDE's approach.