OPTIMAL TRADE EXECUTION UNDER GEOMETRIC BROWNIAN MOTION IN THE ALMGREN AND CHRISS FRAMEWORK

2011 ◽  
Vol 14 (03) ◽  
pp. 353-368 ◽  
Author(s):  
JIM GATHERAL ◽  
ALEXANDER SCHIED
Keyword(s):  
Brownian Motion ◽  
Stock Price ◽  
Closed Form Solution ◽  
Form Solution ◽  
Geometric Brownian Motion ◽  
Price Process ◽  
Risk Criterion ◽  
Execution Strategy ◽  
Optimal Trade Execution ◽  
Alternative Choice

With an alternative choice of risk criterion, we solve the HJB equation explicitly to find a closed-form solution for the optimal trade execution strategy in the Almgren–Chriss framework assuming the underlying unaffected stock price process is geometric Brownian motion.

scholarly journals LOCAL RISK-MINIMIZATION WITH MULTIPLE ASSETS UNDER ILLIQUIDITY WITH APPLICATIONS IN ENERGY MARKETS

2018 ◽  
Vol 21 (04) ◽  
pp. 1850028 ◽  
Author(s):  
PANAGIOTIS CHRISTODOULOU ◽  
NILS DETERING ◽  
THILO MEYER-BRANDIS
Keyword(s):  
Stock Price ◽  
Closed Form Solution ◽  
Energy Markets ◽  
Form Solution ◽  
Supply Curve ◽  
Risk Minimization ◽  
Curve Model ◽  
Risk Criterion ◽  
Multiple Assets ◽  
Local Risk

We propose a hedging approach for general contingent claims when liquidity is a concern and trading is subject to transaction cost. Multiple assets with different liquidity levels are available for hedging. Our risk criterion targets a tradeoff between minimizing the risk against fluctuations in the stock price and incurring low liquidity costs. We work in an arbitrage-free setting assuming a supply curve for each asset. In discrete time, we prove the existence of a locally risk-minimizing strategy under mild conditions on the price process. Under stochastic and time-dependent liquidity risk we give a closed-form solution for an optimal strategy in the case of a linear supply curve model. Finally we show how our hedging method can be applied in energy markets where futures with different maturities are available for trading. The futures closest to their delivery period are usually the most liquid but depending on the contingent claim not necessarily optimal in terms of hedging. In a simulation study, we investigate this tradeoff and compare the resulting hedge strategies with the classical ones.

scholarly journals The Geometric Brownian Motion of Indosat Telecommunications Daily Stock Price During the Covid-19 Pandemic in Indonesia

2021 ◽  
Vol 2084 (1) ◽  
pp. 012012
Author(s):  
Tiara Shofi Edriani ◽  
Udjianna Sekteria Pasaribu ◽  
Yuli Sri Afrianti ◽  
Ni Nyoman Wahyu Astute
Keyword(s):  
Brownian Motion ◽  
Stock Price ◽  
Service Providers ◽  
Main Idea ◽  
Geometric Brownian Motion ◽  
Government Policies ◽  
Percentage Error ◽  
Price Movement ◽  
Data Movement ◽  
Total Data

Abstract One of the major telecommunication and network service providers in Indonesia is PT Indosat Tbk. During the coronavirus (COVID-19) pandemic, the daily stock price of that company was influenced by government policies. This study addresses stock data movement from February 5, 2020 to February 5, 2021, resulted in 243 data, using the Geometric Brownian motion (GBM). The stochastic process realization of this stock price fluctuates and increases exponentially, especially in the 40 latest data. Because of this situation, the realization is transformed into log 10 and calculated its return. As a result, weak stationary in variance is obtained. Furthermore, only data from December 7, 2020 to February 5, 2021 fulfill the GBM assumption of stock price return, as R t 1 * , t 1 * = 1 , 2 , 3 , … , 40 . The main idea of this study is adding datum one by one as much as 10% – 15% of the total data R t 1 * , starting from December 4, 2020 backwards. Following this procedure, and based on the 3% < p-value < 10%, the study shows that its datum can be included in R t 1 * , so t 1 * = − 4. − 3 , − 2 , … , 40 and form five other data groups, R t 2 * , … , R t 6 * . Considering Mean Absolute Percentage Error (MAPE) and amount of data from each group, R t 6 * is selected for modelling. Thus, GBM succeeded in representing the stock price movement of the second most popular Indonesian telecommunication company during COVID-19 pandemic.

scholarly journals An Interval of No-Arbitrage Prices in Financial Markets with Volatility Uncertainty

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Hanlei Hu ◽  
Zheng Yin ◽  
Weipeng Yuan
Keyword(s):  
Brownian Motion ◽  
Financial Markets ◽  
Stock Price ◽  
Contingent Claims ◽  
Price Dynamics ◽  
Complex Situation ◽  
No Arbitrage ◽  
Price Process ◽  
Arbitrage Opportunities

In financial markets with volatility uncertainty, we assume that their risks are caused by uncertain volatilities and their assets are effectively allocated in the risk-free asset and a risky stock, whose price process is supposed to follow a geometric G-Brownian motion rather than a classical Brownian motion. The concept of arbitrage is used to deal with this complex situation and we consider stock price dynamics with no-arbitrage opportunities. For general European contingent claims, we deduce the interval of no-arbitrage price and the clear results are derived in the Markovian case.

Time-changed generalized mixed fractional Brownian motion and application to arithmetic average Asian option pricing

2017 ◽  
Vol 6 (3) ◽  
pp. 85
Author(s):  
ömer önalan
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stock Price ◽  
Long Range Dependence ◽  
Price Process ◽  
Inverse Gamma ◽  
Proposed Model ◽  
Mixed Fractional Brownian Motion ◽  
Time Periods

In this paper we present a novel model to analyze the behavior of random asset price process under the assumption that the stock price pro-cess is governed by time-changed generalized mixed fractional Brownian motion with an inverse gamma subordinator. This model is con-structed by introducing random time changes into generalized mixed fractional Brownian motion process. In practice it has been observed that many different time series have long-range dependence property and constant time periods. Fractional Brownian motion provides a very general model for long-term dependent and anomalous diffusion regimes. Motivated by this facts in this paper we investigated the long-range dependence structure and trapping events (periods of prices stay motionless) of CSCO stock price return series. The constant time periods phenomena are modeled using an inverse gamma process as a subordinator. Proposed model include the jump behavior of price process because the gamma process is a pure jump Levy process and hence the subordinated process also has jumps so our model can be capture the random variations in volatility. To show the effectiveness of proposed model, we applied the model to calculate the price of an average arithmetic Asian call option that is written on Cisco stock. In this empirical study first the statistical properties of real financial time series is investigated and then the estimated model parameters from an observed data. The results of empirical study which is performed based on the real data indicated that the results of our model are more accuracy than the results based on traditional models.

scholarly journals Robust Volatility Estimation and Analysis of the Leverage Effect

2021 ◽  
Author(s):  
◽  
John Randal
Keyword(s):  
Option Pricing ◽  
Stock Price ◽  
Closed Form Solution ◽  
Estimation Procedure ◽  
Form Solution ◽  
Careful Study ◽  
Financial Problem ◽  
Volatility Estimation ◽  
Real Price ◽  
Price Relationships

<p>Using volatility estimation as the underlying commonality this thesis traverses the statistical problem of robust estimation of scale, through to the financial problem of valuing call options over stock. We use a large simulation study of robust scale estimators to benchmark a nonparametric volatility estimation procedure, which not only uses techniques which are particularly suited to observed financial returns, but also addresses the problem of bias in any robust volatility estimation procedure. Existing option pricing models are discussed with careful study of the assumed volatility and elasticity of volatility with respect to stock price relationships for each of these models. An option pricing formula is derived which extends existing methods, and provides a closed form solution which can be readily computed. Preliminary analysis of real price data suggests this model is able to explain observed leverage phenomena.</p>

scholarly journals Robust Volatility Estimation and Analysis of the Leverage Effect

2021 ◽  
Author(s):  
◽  
John Randal
Keyword(s):  
Option Pricing ◽  
Stock Price ◽  
Closed Form Solution ◽  
Estimation Procedure ◽  
Form Solution ◽  
Careful Study ◽  
Financial Problem ◽  
Volatility Estimation ◽  
Real Price ◽  
Price Relationships

<p>Using volatility estimation as the underlying commonality this thesis traverses the statistical problem of robust estimation of scale, through to the financial problem of valuing call options over stock. We use a large simulation study of robust scale estimators to benchmark a nonparametric volatility estimation procedure, which not only uses techniques which are particularly suited to observed financial returns, but also addresses the problem of bias in any robust volatility estimation procedure. Existing option pricing models are discussed with careful study of the assumed volatility and elasticity of volatility with respect to stock price relationships for each of these models. An option pricing formula is derived which extends existing methods, and provides a closed form solution which can be readily computed. Preliminary analysis of real price data suggests this model is able to explain observed leverage phenomena.</p>

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