STOCHASTIC VOLATILITY

2002 ◽  
Vol 05 (05) ◽  
pp. 515-530 ◽  
Author(s):  
SOTIRIOS SABANIS
Keyword(s):  
Brownian Motion ◽  
Stochastic Volatility ◽  
Series Representation ◽  
Random Variable ◽  
Call Option ◽  
Underlying Asset ◽  
European Call Option ◽  
Practical Convergence ◽  
Security Price ◽  
Simulation Results

Hull and White [1] have priced a European call option for the case in which the volatility of the underlying asset is a lognormally distributed random variable. They have obtained their formula under the assumption of uncorrelated innovations in security price and volatility. Although the option pricing formula has a power series representation, the question of convergence has been left unanswered. This paper presents an iterative method for calculating all the higher order moments of volatility necessary for the process of proving convergence theoretically. Moreover, simulation results are given that show the practical convergence of the series. These results have been obtained by using a displaced geometric Brownian motion as a volatility process.

Option pricing under the fractional stochastic volatility model

2021 ◽  
Vol 63 ◽  
pp. 123-142
Author(s):  
Yuecai Han ◽  
Zheng Li ◽  
Chunyang Liu
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Stochastic Volatility ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
European Call Option ◽  
Volatility Model ◽  
The Impact

We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented. doi:10.1017/S1446181121000225

scholarly journals Series representation of the pricing formula for the European option driven by space-time fractional diffusion

2018 ◽  
Vol 21 (4) ◽  
pp. 981-1004 ◽  
Author(s):  
Jean-Philippe Aguilar ◽  
Cyril Coste ◽  
Jan Korbel
Keyword(s):  
Asset Price ◽  
Series Representation ◽  
Option Price ◽  
Double Series ◽  
Fractional Diffusion ◽  
Space Time ◽  
Esscher Transform ◽  
European Option ◽  
Underlying Asset ◽  
European Call Option

Abstract In this paper, we show that the price of an European call option, whose underlying asset price is driven by the space-time fractional diffusion, can be expressed in terms of rapidly convergent double-series. This series formula is obtained from the Mellin-Barnes representation of the option price with help of residue summation in ℂ2. We also derive the series representation for the associated risk-neutral factors, obtained by Esscher transform of the space-time fractional Green functions.

scholarly journals A Stochastic Volatility Alternative to SABR

2008 ◽  
Vol 45 (04) ◽  
pp. 1071-1085
Author(s):  
L. C. G. Rogers ◽  
L. A. M. Veraart
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Closed Form ◽  
Stochastic Volatility ◽  
Option Prices ◽  
Stochastic Volatility Models ◽  
Underlying Asset ◽  
Volatility Models ◽  
Sabr Model ◽  
Modified Dynamics

We present two new stochastic volatility models in which option prices for European plain-vanilla options have closed-form expressions. The models are motivated by the well-known SABR model, but use modified dynamics of the underlying asset. The asset process is modelled as a product of functions of two independent stochastic processes: a Cox-Ingersoll-Ross process and a geometric Brownian motion. An application of the models to options written on foreign currencies is studied.

OPTION PRICING UNDER THE FRACTIONAL STOCHASTIC VOLATILITY MODEL

2021 ◽  
pp. 1-20
Author(s):  
Y. HAN ◽  
Z. LI ◽  
C. LIU
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Stochastic Volatility ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
European Call Option ◽  
Volatility Model ◽  
The Impact

Abstract We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented.

scholarly journals Successive Approximation of SFDEs with Finite Delay Driven byG-Brownian Motion

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Litan Yan ◽  
Qinghua Zhang
Keyword(s):  
Brownian Motion ◽  
Existence And Uniqueness ◽  
Functional Differential Equations ◽  
Call Option ◽  
Stochastic Functional Differential Equations ◽  
European Call Option ◽  
Finite Delay ◽  
Carathéodory Conditions ◽  
Functional Differential ◽  
Stochastic Functional

We consider the stochastic functional differential equations with finite delay driven byG-Brownian motion. Under the global Carathéodory conditions we prove the existence and uniqueness, and as an application, we price the European call option when the underlying asset's price follows such an equation.

PRICING FORMULA FOR EXCHANGE OPTION BASED ON STOCHASTIC DELAY DIFFERENTIAL EQUATION WITH JUMPS

2020 ◽  
pp. 1-16
Author(s):  
Kyong-Hui Kim ◽  
Jong-Kuk Kim ◽  
Ho-Bom Jo
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Asset Price ◽  
Call Option ◽  
Stochastic Delay Differential Equations ◽  
Underlying Asset ◽  
Stochastic Delay ◽  
European Call Option ◽  
Delay Differential ◽  
Exchange Option

This paper deals with pricing formulae for a European call option and an exchange option in the case where underlying asset price processes are represented by stochastic delay differential equations with jumps (hereafter “SDDEJ”). We introduce a new model in which Poisson jumps are added in stochastic delay differential equations to capture behaviors of an underlying asset process more precisely. We derive explicit pricing formulae for the European call option and the exchange option by proving a Lemma on the conditional expectation. Finally, we show that our “SDDEJ” model is meaningful through some numerical experiments and discussions.

scholarly journals Estimasi Harga Multi-State European Call Option Menggunakan Model Binomial

CAUCHY ◽  
2011 ◽  
Vol 1 (4) ◽  
pp. 182
Author(s):  
Mila Kurniawaty, Endah Rokhmati ◽  
Endah Rokhmati
Keyword(s):  
Mean Square Error ◽  
Mean Square ◽  
Call Option ◽  
Expiration Date ◽  
Strike Price ◽  
Underlying Asset ◽  
Put Option ◽  
European Call Option ◽  
Black Scholes

Option merupakan kontrak yang memberikan hak kepada pemiliknya untuk membeli (call option) atau menjual (put option) sejumlah aset dasar tertentu (underlying asset) dengan harga tertentu (strike price) dalam jangka waktu tertentu (sebelum atau saat expiration date). Perkembangan option belakangan ini memunculkan banyak model pricing untuk mengestimasi harga option, salah satu model yang digunakan adalah formula Black-Scholes. Multi-state option merupakan sebuah option yang payoff-nya didasarkan pada dua atau lebih aset dasar. Ada beberapa metode yang dapat digunakan dalam mengestimasi harga call option, salah satunya masyarakat finance sering menggunakan model binomial untuk estimasi berbagai model option yang lebih luas seperti multi-state call option. Selanjutnya, dari hasil estimasi call option dengan model binomial didapatkan formula terbaik berdasarkan penghitungan eror dengan mean square error. Dari penghitungan eror didapatkan eror rata-rata dari masing-masing formula pada model binomial. Hasil eror rata-rata menunjukkan bahwa estimasi menggunakan formula 5 titik lebih baik dari pada estimasi menggunakan formula 4 titik.

scholarly journals Proactive Hedging European Call Option Pricing with Linear Position Strategy

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Meng Li ◽  
Xuefeng Wang ◽  
Fangfang Sun
Keyword(s):  
Asset Price ◽  
Stock Option ◽  
Call Option ◽  
European Option ◽  
Price Movement ◽  
Underlying Asset ◽  
European Call Option ◽  
Exotic Option ◽  
Risk Of Exposure ◽  
Geometric Fractional Brownian Motion

Proactive hedging option is an exotic European stock option designed for hedgers. Such option requires option holders to buy in (or sell out) the underlying asset (stock) and allows them to adjust the holdings of the underlying asset per its price changes within an option period. The proactive hedging option is an attractive choice for hedgers because its price is lower than that of classical options and because it completely hedges the risk of exposure for option holders. In this study, the underlying asset price movement is assumed to follow geometric fractional Brownian motion. The pricing formula for proactive hedging call options is derived with a linear position strategy by applying the risk-neutral evaluation principle. We use simulations to confirm that the price of this exotic option is always no more than that of the classical European option under the same parameters.

scholarly journals Pricing Collar Options with Stochastic Volatility

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Pengshi Li ◽  
Jianhui Yang
Keyword(s):  
Brownian Motion ◽  
Numerical Experiment ◽  
Stochastic Volatility ◽  
Asset Price ◽  
Analytical Approximation ◽  
Stochastic Volatility Model ◽  
Approximation Formula ◽  
Underlying Asset ◽  
Volatility Model ◽  
Mean Reverting Process

This paper studies collar options in a stochastic volatility economy. The underlying asset price is assumed to follow a continuous geometric Brownian motion with stochastic volatility driven by a mean-reverting process. The method of asymptotic analysis is employed to solve the PDE in the stochastic volatility model. An analytical approximation formula for the price of the collar option is derived. A numerical experiment is presented to demonstrate the results.

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