Asymptotics for discrete time hedging errors under fractional Black–Scholes models

2019 ◽  
Vol 149 ◽  
pp. 160-170
Author(s):  
Wensheng Wang
Keyword(s):  
Discrete Time ◽  
Black Scholes ◽  
Discrete Time Hedging

Value-at-Risk and Expected Shortfall approaches for option premiums and the probability of default estimation based on ARMA models

2021 ◽  
Vol 57 (3) ◽  
pp. 126
Author(s):  
Nikolai Berzon
Keyword(s):  
Time Series ◽  
Option Pricing ◽  
Discrete Time ◽  
Value At Risk ◽  
Probability Of Default ◽  
Pricing Models ◽  
Arma Process ◽  
Underlying Asset ◽  
Black Scholes

The need to address the issue of risk management has given rise to a number of models for estimation the probability of default, as well as a special tool that allows to sell credit risk – a credit default swap (CDS). From the moment it appeared in 1994 until the crisis of 2008, that the CDS market was actively growing, and then sharply contracted. Currently, there is practically no CDS market in emerging economies (including Russia). This article is to improve the existing CDS valuation models by using discrete-time models that allow for more accurate assessment and forecasting of the selected asset dynamics, as well as new option pricing models that take into account the degree of risk acceptance by the option seller. This article is devoted to parametric discrete-time option pricing models that provide more accurate results than the traditional Black-Scholes continuous-time model. Improvement in the quality of assessment is achieved due to three factors: a more detailed consideration of the properties of the time series of the underlying asset (in particular, autocorrelation and heavy tails), the choice of the optimal number of parameters and the use of Value-at-Risk approach. As a result of the study, expressions were obtained for the premiums of European put and call options for a given level of risk under the assumption that the return on the underlying asset follows a stationary ARMA process with normal or Student's errors, as well as an expression for the credit spread under similar assumptions. The simplicity of the ARMA process underlying the model is a compromise between the complexity of model calibration and the quality of describing the dynamics of assets in the stock market. This approach allows to take into account both discreteness in asset pricing and take into account the current structure and the presence of interconnections for the time series of the asset under consideration (as opposed to the Black–Scholes model), which potentially allows better portfolio management in the stock market.

Option Hedging with Smooth Market Impact

2016 ◽  
Vol 02 (01) ◽  
pp. 1650002 ◽  
Author(s):  
Robert Almgren ◽  
Tianhui Michael Li
Keyword(s):  
Discrete Time ◽  
Market Impact ◽  
Large Collection ◽  
Public Markets ◽  
The Public ◽  
Option Hedging ◽  
A Value ◽  
First Case ◽  
Black Scholes ◽  
Optimal Hedge

We consider intraday hedging of an option position, for a large trader who experiences temporary and permanent market impact. We formulate the general model including overnight risk, and solve explicitly in two cases which we believe are representative. The first case is an option with approximately constant gamma: the optimal hedge trades smoothly towards the classical Black–Scholes delta, with trading intensity proportional to instantaneous mishedge and inversely proportional to illiquidity. The second case is an arbitrary non-linear option structure but with no permanent impact: the optimal hedge trades toward a value offset from the Black–Scholes delta. We estimate the effects produced on the public markets if a large collection of traders all hedge similar positions. We construct a stable hedge strategy with discrete time steps.

scholarly journals Hedging error estimate of the american put option problem in jump-diffusion processes

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2813-2824
Author(s):  
Sultan Hussain ◽  
Salman Zeb ◽  
Muhammad Saleem ◽  
Nasir Rehman
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Diffusion Processes ◽  
Asset Price ◽  
Independent Random Variables ◽  
American Put Option ◽  
Jump Diffusion Processes ◽  
Put Option ◽  
Discrete Time Hedging ◽  
American Put

We consider discrete time hedging error of the American put option in case of brusque fluctuations in the price of assets. Since continuous time hedging is not possible in practice so we consider discrete time hedging process. We show that if the proportions of jump sizes in the asset price are identically distributed independent random variables having finite moments then the value process of the discrete time hedging uniformly approximates the value process of the corresponding continuous-time hedging in the sense of L1 and L2-norms under the real world probability measure.

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