LOCALLY RISK-NEUTRAL VALUATION OF OPTIONS IN GARCH MODELS BASED ON VARIANCE-GAMMA PROCESS

2012 ◽  
Vol 15 (02) ◽  
pp. 1250015 ◽  
Author(s):  
LIE-JANE KAO
Keyword(s):  
Option Pricing ◽  
Sufficient Conditions ◽  
Garch Model ◽  
Gamma Process ◽  
Type Model ◽  
Underlying Asset ◽  
Variance Gamma ◽  
Variance Gamma Process ◽  
Black Scholes ◽  
Risk Neutral

This study develops a GARCH-type model, i.e., the variance-gamma GARCH (VG GARCH) model, based on the two major strands of option pricing literature. The first strand of the literature uses the variance-gamma process, a time-changed Brownian motion, to model the underlying asset price process such that the possible skewness and excess kurtosis on the distributions of asset returns are considered. The second strand of the literature considers the propagation of the previously arrived news by including the feedback and leverage effects on price movement volatility in a GARCH framework. The proposed VG GARCH model is shown to obey a locally risk-neutral valuation relationship (LRNVR) under the sufficient conditions postulated by Duan (1995). This new model provides a unified framework for estimating the historical and risk-neutral distributions, and thus facilitates option pricing calibration using historical underlying asset prices. An empirical study is performed comparing the proposed VG GARCH model with four competing pricing models: benchmark Black–Scholes, ad hoc Black–Scholes, normal NGARCH, and stochastic volatility VG. The performance of the VG GARCH model versus these four competing models is then demonstrated.

The Variance Gamma Process and Option Pricing

1998 ◽  
Vol 2 (1) ◽  
pp. 79-105 ◽  
Author(s):  
Dilip B. Madan ◽  
Peter P. Carr ◽  
Eric C. Chang
Keyword(s):  
Option Pricing ◽  
Gamma Process ◽  
Variance Gamma ◽  
Variance Gamma Process

OPTION PRICING IN THE VARIANCE-GAMMA MODEL UNDER THE DRIFT JUMP

2018 ◽  
Vol 21 (04) ◽  
pp. 1850018 ◽  
Author(s):  
ROMAN V. IVANOV
Keyword(s):  
Option Pricing ◽  
Drift Rate ◽  
Research Direction ◽  
Gamma Process ◽  
Gamma Model ◽  
Hyperbolic Functions ◽  
Variance Gamma ◽  
Variance Gamma Process ◽  
Variance Gamma Model ◽  
Analytical Expressions

This paper continues elements of the research direction of the work of Madan et al. [(1998) The variance gamma process and option pricing, European Finance Review 2, 79–105] and gives analytical expressions for the prices of digital and European call options in the variance-gamma model under the assumption that the linear drift rate of stock log-returns can suddenly jump downwards. The time of the jump is taken to be exponentially distributed. The formulas obtained require the computation of some generalized hyperbolic functions.

scholarly journals Additive logistic processes in option pricing

2021 ◽  
Author(s):  
Peter Carr ◽  
Lorenzo Torricelli
Keyword(s):  
Option Pricing ◽  
Pricing Models ◽  
Underlying Asset ◽  
Formula One ◽  
Additive Type ◽  
Black Scholes ◽  
Minimum Price ◽  
Risk Neutral ◽  
Risk Neutral Probabilities ◽  
Security Price

AbstractIn option pricing, it is customary to first specify a stochastic underlying model and then extract valuation equations from it. However, it is possible to reverse this paradigm: starting from an arbitrage-free option valuation formula, one could derive a family of risk-neutral probabilities and a corresponding risk-neutral underlying asset process. In this paper, we start from two simple arbitrage-free valuation equations, inspired by the log-sum-exponential function and an $\ell ^{p}$ ℓ p vector norm. Such expressions lead respectively to logistic and Dagum (or “log-skew-logistic”) risk-neutral distributions for the underlying security price. We proceed to exhibit supporting martingale processes of additive type for underlying securities having as time marginals two such distributions. By construction, these processes produce closed-form valuation equations which are even simpler than those of the Bachelier and Samuelson–Black–Scholes models. Additive logistic processes provide parsimonious and simple option pricing models capturing various important stylised facts at the minimum price of a single market observable input.

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.

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