PRICING TWO-ASSET BARRIER OPTIONS UNDER STOCHASTIC CORRELATION VIA PERTURBATION

2015 ◽  
Vol 18 (03) ◽  
pp. 1550018 ◽  
Author(s):  
MARCOS ESCOBAR ◽  
BARBARA GÖTZ ◽  
DANIELA NEYKOVA ◽  
RUDI ZAGST
Keyword(s):  
Perturbation Theory ◽  
Stochastic Volatility ◽  
Correlation Structure ◽  
Barrier Options ◽  
Correlation Effects ◽  
Stochastic Correlation ◽  
Black Scholes ◽  
Leading Term ◽  
Path Dependent ◽  
Correction Terms

The correlation structure is crucial when pricing multi-asset products, in particular barrier options. In this work, we price two-asset path-dependent derivatives by means of perturbation theory in the context of a bi-dimensional asset model with stochastic correlation and volatilities. To our best knowledge, this is the first attempt at pricing barriers with stochastic correlation. It turns out that the leading term of the approximation corresponds to a constant covariance Black–Scholes type price with correction terms adjusting for stochastic volatility and stochastic correlation effects. The practicability of the presented method is illustrated by some numerical implementations.

SIMPLIFIED HEDGE FOR PATH-DEPENDENT DERIVATIVES

2016 ◽  
Vol 19 (07) ◽  
pp. 1650045 ◽  
Author(s):  
CAROLE BERNARD ◽  
JUNSEN TANG
Keyword(s):  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Asian Option ◽  
Distributional Properties ◽  
Lookback Option ◽  
Volatility Model ◽  
Black Scholes ◽  
Delta Hedging ◽  
Path Dependent

Path-dependent derivatives are typically difficult to hedge. Traditional dynamic delta hedging does not perform well because of the difficulty to evaluate the Greeks and the high cost of constantly rebalancing. We propose to price and hedge path-dependent derivatives by constructing simplified alternatives that preserve certain distributional properties of their terminal payoffs, and that can be hedged by semi-static replication. The method is illustrated by a geometric Asian option and by a lookback option in the Black–Scholes setting, for which explicit forms of the simplified alternatives exist. Extensions to a Lévy market and to a Heston stochastic volatility model are discussed as well.

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.

scholarly journals CEV MODEL WITH STOCHASTIC VOLATILITY

2019 ◽  
Vol 6 (3-4) ◽  
pp. 22-28
Author(s):  
IVAN BURTNYAK ◽  
ANNA MALYTSKA
Keyword(s):  
Perturbation Theory ◽  
Boundary Value Problems ◽  
Stochastic Volatility ◽  
Boundary Value ◽  
Singular Perturbation Theory ◽  
Barrier Options ◽  
Regular Perturbation ◽  
Explicit Formulas ◽  
Wide Range ◽  
Adjoint Operators

This paper develops a systematic method for calculating approximate prices for a wide range of securities implying the tools of spectral analysis, singular and regular perturbation theory. Price options depend on stochastic volatility, which may be multiscale, in the sense that it may be driven by one fast-varying and one slow-varying factor. The found the approximate price of two-barrier options with multifactor volatility as a schedule for own functions. The theorem of estimation of accuracy of approximation of option prices is established. Explicit formulas have been found for finding the value of derivatives based on the development of eigenfunctions and eigenvalues of self-adjoint operators using boundary-value problems for singular and regular perturbations. This article develops a general method of obtaining a guide price for a broad class of securities. A general theory of derivative valuation of options generated by diffusion processes is developed. The algorithm of calculating the approximate price is given. The accuracy of the estimates is established. The theory developed is applied to a diffusion operator, which is decomposed by eigenfunctions and eigenvalues. The purpose of the article is to develop an algorithm for finding the approximate price of two-barrier options and to find explicit formulas for finding the value of derivatives based on the development of self-functions and eigenvalues of self-adjoint operators using boundary-value problems for singular and regular perturbations. Price finding is reduced to the problem solving of eigenvalues and eigenfunctions of a certain equation. The main advantage of our pricing methodology is that, by combining methods in spectral theory, regular perturbation theory, and singular perturbation theory, we reduce everything to equations to find eigenfunctions and eigenvalues.

scholarly journals Pricing Parisian Option under a Stochastic Volatility Model

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Min-Ku Lee ◽  
Kyu-Hwan Jang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Barrier Options ◽  
Barrier Option ◽  
Partial Differential ◽  
Model Based ◽  
Volatility Model ◽  
Black Scholes

We study the pricing of a Parisian option under a stochastic volatility model. Based on the manipulation problem that barrier options might create near barriers, the Parisian option has been designed as an extended barrier option. A stochastic volatility correction to the Black-Scholes price of the Parisian option is obtained in a partial differential equation form and the solution is characterized numerically.

scholarly journals Quanto Pricing beyond Black–Scholes

2021 ◽  
Vol 14 (3) ◽  
pp. 136
Author(s):  
Holger Fink ◽  
Stefan Mittnik
Keyword(s):  
Option Pricing ◽  
Goodness Of Fit ◽  
Calibration Procedure ◽  
Barrier Options ◽  
Pricing Models ◽  
Modeling Framework ◽  
Double Barrier ◽  
Parameter Stability ◽  
Comprehensive Data ◽  
Black Scholes

Since their introduction, quanto options have steadily gained popularity. Matching Black–Scholes-type pricing models and, more recently, a fat-tailed, normal tempered stable variant have been established. The objective here is to empirically assess the adequacy of quanto-option pricing models. The validation of quanto-pricing models has been a challenge so far, due to the lack of comprehensive data records of exchange-traded quanto transactions. To overcome this, we make use of exchange-traded structured products. After deriving prices for composite options in the existing modeling framework, we propose a new calibration procedure, carry out extensive analyses of parameter stability and assess the goodness of fit for plain vanilla and exotic double-barrier options.

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