scholarly journals Pricing of Barrier Options on Underlying Assets with Jump-Diffusion Dynamics: A Mellin Transform Approach

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1271
Author(s):  
Marianito R. Rodrigo
Keyword(s):  
Mellin Transform ◽  
Jump Diffusion ◽  
Futures Contracts ◽  
Barrier Options ◽  
Barrier Option ◽  
Underlying Asset ◽  
Diffusion Dynamics ◽  
Path Dependent ◽  
The Right ◽  
Right To Buy

A barrier option is an exotic path-dependent option contract where the right to buy or sell is activated or extinguished when the underlying asset reaches a certain barrier price during the lifetime of the contract. In this article we use a Mellin transform approach to derive exact pricing formulas for barrier options with general payoffs and exponential barriers on underlying assets that have jump-diffusion dynamics. With the same approach we also price barrier options on underlying futures contracts.

scholarly journals PENENTUAN HARGA JUAL OPSI BARRIER TIPE EROPA DENGAN METODE ANTITHETIC VARIATE PADA SIMULASI MONTE CARLO

2018 ◽  
Vol 7 (2) ◽  
pp. 71
Author(s):  
LUH HENA TERECIA WISMAWAN PUTRI ◽  
KOMANG DHARMAWAN ◽  
I WAYAN SUMARJAYA
Keyword(s):  
Monte Carlo ◽  
Analytic Solution ◽  
Asset Price ◽  
Selling Price ◽  
Barrier Options ◽  
Option Prices ◽  
Barrier Option ◽  
Underlying Asset ◽  
Closing Price ◽  
Path Dependent

The purpose of this research is to compare the selling price of down and out barrier option when the prices are simulated by the Antithetic Variate Monte Carlo and the standar Monte Carlo. Barrier options are path dependent options and the payoff depend on whether the underlying asset price touched the barrier or not during the life of the option. In this research, we conducted simulations against the closing price of the shares of PT Adhi Karya using Standard Monte Carlo simulation and the Monte Carlo-Antithetic Variate simulation. After the simulation, we obtained that the option prices using Antithetic Variate produces a cheaper price than the standar one. We also found that the analytic solution has a smaller error on its confidence interval compare to the Monte Carlo Standar.

scholarly journals Penentuan Harga Opsi Barrier Menggunakan Metode Trinomial Kamrad-Ritchken Dengan Volatilitas Model Garch

2019 ◽  
Vol 10 (1) ◽  
pp. 83-92
Author(s):  
S Sulastri ◽  
Lienda Novieyanti ◽  
Sukono Sukono
Keyword(s):  
Stock Price ◽  
Option Price ◽  
Barrier Options ◽  
Barrier Option ◽  
Market Conditions ◽  
Strike Price ◽  
Daily Data ◽  
Closing Price ◽  
The Right ◽  
Initial Price

Abstract. This study aims to minimize the violation of the assumptions of determining price options by taking into account the actual market conditions in order to obtain the right price that will provide high profits for investors. The method used to determine the option price in this study is the Kamrad Ritchken trinomial with volatility values that will be modeled first using GARCH. The data used in this study is daily data (5 working days per week) from the closing price of the stock price of PT. Bank Rakyat Indonesia, Tbk (BBRI. Based on the results of the research, the best model is GARCH (1,1). For the call up barrier option, increase the strike price with the initial price and barrier which causes the option price to call up the barrier "in" and "out" decreases, on the contrary to the put barrier option, an increase in strike price with the initial price and a barrier that causes the put barrier option price to both put up-in and put up-out. initial and barrier which still causes the call down barrier option price both in and out decreases, on the contrary in the put down barrier option, increasing strike price with the initial price and barrier which causes the put down barrier option price to increase in and out.Keywords: Barrier Options, Trinomial, Kamrad Ritchken, Volatility, GARCH  Abstrak. Penelitian ini bertujuan untuk meminimalkan pelanggaran asumsi-asumsi penentuan harga opsi dengan memperhatikan kondisi pasar yang sebenarnya sehingga diperoleh harga yang tepat yang akan memberikan keuntungan tinggi bagi investor. Metode yang digunakan untuk menentukan harga opsi dalam penelitian ini adalah trinomial Kamrad Ritchken dengan nilai volatilitas yang akan dimodelkan terlebih dahulu dengan menggunakan GARCH. Data yang digunakan dalam penelitian ini adalah data harian (5 hari kerja per minggu) dari harga penutupan harga saham PT. Bank Rakyat Indonesia, Tbk (BBRI). Berdasarkan hasil penelitian diperoleh model yang paling baik adalah GARCH (1,1). Untuk opsi call up barrier, peningkatan strike price dengan harga awal dan barrier yang tetap menyebabkan harga opsi call up barrier baik "in" maupun "out" menurun, sebaliknya pada opsi put barrier, peningkatan strike price dengan harga awal dan barrier yang tetap menyebabkan harga opsi put barrier baik put up-in maupun put up-out meningkat. Sedangkan untuk opsi call barrier, peningkatan strike price dengan harga awal dan barrier yang tetap menyebabkan harga opsi call down barrier baik in maupun out menurun, sebaliknya pada opsi put down barrier, peningkatan strike price dengan harga awal dan barrier yang tetap menyebabkan harga opsi put down barrier baik in maupun out meningkat.Kata Kunci :  Opsi Barrier, Trinomial, Kamrad Ritchken, Volatilitas, GARCH

Brownian Excursions and Parisian Barrier Options

1997 ◽  
Vol 29 (1) ◽  
pp. 165-184 ◽  
Author(s):  
Marc Chesney ◽  
Monique Jeanblanc-Picqué ◽  
Marc Yor
Keyword(s):  
Essential Role ◽  
Fixed Number ◽  
Time Interval ◽  
Barrier Options ◽  
Barrier Option ◽  
Underlying Asset

In this paper we study a new kind of option, called hereinafter a Parisian barrier option. This option is the following variant of the so-called barrier option: a down-and-out barrier option becomes worthless as soon as a barrier is reached, whereas a down-and-out Parisian barrier option is lost by the owner if the underlying asset reaches a prespecified level and remains constantly below this level for a time interval longer than a fixed number, called the window. Properties of durations of Brownian excursions play an essential role. We also study another kind of option, called here a cumulative Parisian option, which becomes worthless if the total time spent below a certain level is too long.

A Mellin transform approach to barrier option pricing

2018 ◽  
Vol 31 (1) ◽  
pp. 49-67 ◽  
Author(s):  
Chiara Guardasoni ◽  
Marianito R Rodrigo ◽  
Simona Sanfelici
Keyword(s):  
Integral Representation ◽  
Option Pricing ◽  
Mellin Transform ◽  
Option Price ◽  
Representation Formula ◽  
Barrier Option ◽  
Knock Out ◽  
Path Dependent ◽  
Barrier Option Pricing ◽  
Numerical Approaches

Abstract A barrier option is an exotic path-dependent option contract that, depending on terms, automatically expires or can be exercised only if the underlying asset ever reaches a predetermined barrier price. Using a partial differential equation approach, we provide an integral representation of the barrier option price via the Mellin transform. In the case of knock-out barrier options, we obtain a decomposition of the barrier option price into the corresponding European option value minus a barrier premium. The integral representation formula can be expressed in terms of the solution to a system of coupled Volterra integral equations of the first kind. Moreover, we suggest some possible numerical approaches to the problem of barrier option pricing.

scholarly journals Volatility Timing: Pricing Barrier Options on DAX XETRA Index

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 722
Author(s):  
Carlos Esparcia ◽  
Elena Ibañez ◽  
Francisco Jareño
Keyword(s):  
Option Pricing ◽  
Term Structure ◽  
Barrier Options ◽  
Barrier Option ◽  
Probability Term ◽  
Volatility Timing ◽  
The Right ◽  
The Impact ◽  
Complementary Study ◽  
Frequency Current

This paper analyses the impact of different volatility structures on a range of traditional option pricing models for the valuation of call down and out style barrier options. The construction of a Risk-Neutral Probability Term Structure (RNPTS) is one of the main contributions of this research, which changes in parallel with regard to the Volatility Term Structure (VTS) in the main and traditional methods of option pricing. As a complementary study, we propose the valuation of options by assuming a constant or historical volatility. The study implements the GARCH (1,1) model with regard to the continuously compound returns of the DAX XETRA Index traded at daily frequency. Current methodology allows for obtaining accuracy forecasts of the realized market barrier option premiums. The paper highlights not only the importance of selecting the right model for option pricing, but also fitting the most accurate volatility structure.

BARRIER OPTIONS PRICING WITH JOINT DISTRIBUTION OF GAUSSIAN PROCESS AND ITS MAXIMUM

2017 ◽  
Vol 20 (06) ◽  
pp. 1750042
Author(s):  
PINGJIN DENG ◽  
XIUFANG LI
Keyword(s):  
Gaussian Process ◽  
Joint Distribution ◽  
Options Pricing ◽  
Time Interval ◽  
Exotic Options ◽  
Barrier Options ◽  
Rate Process ◽  
Barrier Option ◽  
Explicit Formulas ◽  
Underlying Asset

Barrier options are one of the most popular exotic options. In this contribution, we propose a performance barrier option, which is a type of barrier option defined with the [Formula: see text]th period logarithm return rate process on an underlying asset over the time interval [Formula: see text], [Formula: see text]. We show that the price of this performance barrier option is determined by the joint distribution of a Slepian process and its maximum. Furthermore, we derive a tractable formula for this joint distribution and obtain explicit formulas for the up-out-call performance option and up-out-put performance option.

scholarly journals A numerical scheme based on semi-static hedging strategy

2014 ◽  
Vol 20 (4) ◽  
Author(s):  
Yuri Imamura ◽  
Yuta Ishigaki ◽  
Toshiki Okumura
Keyword(s):  
Diffusion Process ◽  
Numerical Scheme ◽  
Numerical Results ◽  
Barrier Options ◽  
Barrier Option ◽  
Hedging Strategy ◽  
Underlying Process ◽  
Static Hedging ◽  
Black Scholes ◽  
Path Dependent

AbstractIn the present paper, we introduce a numerical scheme for the price of a barrier option when the price of the underlying follows a diffusion process. The numerical scheme is based on an extension of a static hedging formula of barrier options. To get the static hedging formula, the underlying process needs to have a symmetry. We introduce a way to “symmetrize” a given diffusion process. Then the pricing of a barrier option is reduced to that of plain options under the symmetrized process. To show how our symmetrization scheme works, we will present some numerical results of path-independent Euler–Maruyama approximation applied to our scheme, comparing them with the path-dependent Euler–Maruyama scheme when the model is of the type Black–Scholes, CEV, Heston, and (λ)-SABR, respectively. The results show the effectiveness of our scheme.

Alternative results for option pricing and implied volatility in jump-diffusion models using Mellin transforms

2016 ◽  
Vol 28 (5) ◽  
pp. 789-826 ◽  
Author(s):  
T. RAY LI ◽  
MARIANITO R. RODRIGO
Keyword(s):  
Differential Equation ◽  
Option Pricing ◽  
Implied Volatility ◽  
Asset Price ◽  
Jump Diffusion ◽  
Underlying Asset ◽  
Mellin Transforms ◽  
Diffusion Dynamics ◽  
Double Exponential ◽  
Jump Diffusion Model

In this article, we use Mellin transforms to derive alternative results for option pricing and implied volatility estimation when the underlying asset price is governed by jump-diffusion dynamics. The current well known results are restrictive since the jump is assumed to follow a predetermined distribution (e.g., lognormal or double exponential). However, the results we present are general since we do not specify a particular jump-diffusion model within the derivations. In particular, we construct and derive an exact solution to the option pricing problem in a general jump-diffusion framework via Mellin transforms. This approach of Mellin transforms is further extended to derive a Dupire-like partial integro-differential equation, which ultimately yields an implied volatility estimator for assets subjected to instantaneous jumps in the price. Numerical simulations are provided to show the accuracy of the estimator.

scholarly journals BARRIER OPTION PRICING BY BRANCHING PROCESSES

2009 ◽  
Vol 12 (07) ◽  
pp. 1055-1073 ◽  
Author(s):  
GEORGI K. MITOV ◽  
SVETLOZAR T. RACHEV ◽  
YOUNG SHIN KIM ◽  
FRANK J. FABOZZI
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Analytical Formula ◽  
Model Parameters ◽  
Barrier Options ◽  
Call Option ◽  
Barrier Option ◽  
Market Prices ◽  
Underlying Asset ◽  
Barrier Option Pricing

This paper examines the pricing of barrier options when the price of the underlying asset is modeled by a branching process in a random environment (BPRE). We derive an analytical formula for the price of an up-and-out call option, one form of a barrier option. Calibration of the model parameters is performed using market prices of standard call options. Our results show that the prices of barrier options that are priced with the BPRE model deviate significantly from those modeled assuming a lognormal process, despite the fact that for standard options, the corresponding differences between the two models are relatively small.

Brownian Excursions and Parisian Barrier Options

1997 ◽  
Vol 29 (01) ◽  
pp. 165-184 ◽  
Author(s):  
Marc Chesney ◽  
Monique Jeanblanc-Picqué ◽  
Marc Yor
Keyword(s):  
Essential Role ◽  
Fixed Number ◽  
Time Interval ◽  
Barrier Options ◽  
Barrier Option ◽  
Underlying Asset

In this paper we study a new kind of option, called hereinafter a Parisian barrier option. This option is the following variant of the so-called barrier option: a down-and-out barrier option becomes worthless as soon as a barrier is reached, whereas a down-and-out Parisian barrier option is lost by the owner if the underlying asset reaches a prespecified level and remains constantly below this level for a time interval longer than a fixed number, called the window. Properties of durations of Brownian excursions play an essential role. We also study another kind of option, called here a cumulative Parisian option, which becomes worthless if the total time spent below a certain level is too long.

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