CONSTANT ELASTICITY OF VARIANCE IN RANDOM TIME: A NEW STOCHASTIC VOLATILITY MODEL WITH PATH DEPENDENCE AND LEVERAGE EFFECT

2007 ◽  
Vol 10 (06) ◽  
pp. 915-937 ◽  
Author(s):  
DMITRY OSTROVSKY
Keyword(s):  
Term Structure ◽  
Path Dependence ◽  
Implied Volatility ◽  
Sufficient Conditions ◽  
Stochastic Volatility Model ◽  
Random Time ◽  
Cev Model ◽  
Constant Elasticity ◽  
No Arbitrage ◽  
Volatility Model

An arbitrage-free CEV economy driven by Brownian motion in independent, continuous random time is introduced. European options are priced by the no-arbitrage principle as conditional averages of their classical CEV values over the CEV-modified random time to maturity. A novel representation of the classical CEV price is used to investigate the asymptotics of the average implied volatility. It is shown that the average implied volatility of the at-the-money call option is lower and of deep out-of-the-money call options, under appropriate sufficient conditions, greater than the implied CEV volatilities. Unlike in the classical CEV model, the shape of the out-of-the-money tail can be both downward and upward sloping depending on the tails of random time. The model is implemented in limit lognormal time. Its multiscaling law is shown to imply a term structure of implied volatility that is qualitatively more sensitive to changes in the time to maturity than is the classical CEV model.

BLACK–SCHOLES–MERTON IN RANDOM TIME: A NEW STOCHASTIC VOLATILITY MODEL WITH PATH DEPENDENCE

2007 ◽  
Vol 10 (05) ◽  
pp. 847-872 ◽  
Author(s):  
DMITRY OSTROVSKY
Keyword(s):  
Brownian Motion ◽  
Implied Volatility ◽  
Time Derivative ◽  
Sufficient Conditions ◽  
Random Time ◽  
Average Price ◽  
European Options ◽  
No Arbitrage ◽  
Volatility Model ◽  
Black Scholes

A generalized Black–Scholes–Merton economy is introduced. The economy is driven by Brownian motion in random time that is taken to be continuous and independent of Brownian motion. European options are priced by the no-arbitrage principle as conditional averages of their classical values over the random time to maturity. The prices are path dependent in general unless the time derivative of the random time is Markovian. An explicit self-financing hedging strategy is shown to replicate all European options by dynamically trading in stock, money market, and digital calls on realized variance. The notion of the average price is introduced, and the average price of the call option is shown to be greater than the corresponding Black–Scholes price for all deep in- and out-of-the-money options under appropriate sufficient conditions. The model is implemented in limit lognormal random time. The significance of its multiscaling law is explained theoretically and verified numerically to be a determining factor of the term structure of implied volatility.

Martingales versus PDEs in finance: an equivalence result with examples

2000 ◽  
Vol 37 (04) ◽  
pp. 947-957 ◽  
Author(s):  
David Heath ◽  
Martin Schweizer
Keyword(s):  
Stochastic Volatility ◽  
Term Structure ◽  
Sufficient Conditions ◽  
Stochastic Volatility Model ◽  
Structure Model ◽  
Term Structure Model ◽  
Volatility Model ◽  
Equivalence Result

We provide a set of verifiable sufficient conditions for proving in a number of practical examples the equivalence of the martingale and the PDE approaches to the valuation of derivatives. The key idea is to use a combination of analytic and probabilistic assumptions that covers typical models in finance falling outside the range of standard results from the literature. Applications include Heston's stochastic volatility model and the Black-Karasinski term structure model.

On the Performance of the Comonotonicity Approach for Pricing Asian Options in Some Benchmark Models from Equities and Commodities

2017 ◽  
Vol 20 (01) ◽  
pp. 1750005
Author(s):  
Jilong Chen ◽  
Christian Ewald
Keyword(s):  
Upper Bounds ◽  
Stochastic Volatility Model ◽  
Asian Options ◽  
Convenience Yield ◽  
Yield Model ◽  
Cev Model ◽  
Constant Elasticity ◽  
Constant Elasticity Of Variance ◽  
Hedging Strategies ◽  
Volatility Model

In this paper, we investigate the applicability of the comonotonicity approach in the context of various benchmark models for equities and commodities. Instead of classical Lévy models as in Albrecher et al. we focus on the Heston stochastic volatility model, the constant elasticity of variance (CEV) model and Schwartz’ 1997 stochastic convenience yield model. We show how the technical difficulties of inverting the distribution function of the sum of the comonotonic random vector can be overcome and that the method delivers rather tight upper bounds for the prices of Asian Options in these models, at least for strikes which are not too large. As a by-product the method delivers super-hedging strategies which can be easily implemented.

scholarly journals Option Pricing under the Jump Diffusion and Multifactor Stochastic Processes

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Shican Liu ◽  
Yanli Zhou ◽  
Yonghong Wu ◽  
Xiangyu Ge
Keyword(s):  
Diffusion Process ◽  
Option Pricing ◽  
Stochastic Volatility ◽  
Term Structure ◽  
Implied Volatility ◽  
Jump Diffusion ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Volatility Model ◽  
Jump Diffusion Process

In financial markets, there exists long-observed feature of the implied volatility surface such as volatility smile and skew. Stochastic volatility models are commonly used to model this financial phenomenon more accurately compared with the conventional Black-Scholes pricing models. However, one factor stochastic volatility model is not good enough to capture the term structure phenomenon of volatility smirk. In our paper, we extend the Heston model to be a hybrid option pricing model driven by multiscale stochastic volatility and jump diffusion process. In our model the correlation effects have been taken into consideration. For the reason that the combination of multiscale volatility processes and jump diffusion process results in a high dimensional differential equation (PIDE), an efficient finite element method is proposed and the integral term arising from the jump term is absorbed to simplify the problem. The numerical results show an efficient explanation for volatility smirks when we incorporate jumps into both the stock process and the volatility process.

Empirical Performance of the Constant Elasticity Variance Option Pricing Model

2009 ◽  
Vol 12 (02) ◽  
pp. 177-217 ◽  
Author(s):  
Ren-Raw Chen ◽  
Cheng-Few Lee ◽  
Han-Hsing Lee
Keyword(s):  
Option Pricing ◽  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Pricing Model ◽  
Implementation Cost ◽  
Pricing Models ◽  
Cev Model ◽  
Constant Elasticity ◽  
Volatility Model ◽  
Option Pricing Model

In this essay, we empirically test the Constant–Elasticity-of-Variance (CEV) option pricing model by Cox (1975, 1996 ) and Cox and Ross (1976), and compare the performances of the CEV and alternative option pricing models, mainly the stochastic volatility model, in terms of European option pricing and cost-accuracy based analysis of their numerical procedures. In European-style option pricing, we have tested the empirical pricing performance of the CEV model and compared the results with those by Bakshi et al. (1997). The CEV model, introducing only one more parameter compared with Black-Scholes formula, improves the performance notably in all of the tests of in-sample, out-of-sample and the stability of implied volatility. Furthermore, with a much simpler model, the CEV model can still perform better than the stochastic volatility model in short term and out-of-the-money categories. When applied to American option pricing, high-dimensional lattice models are prohibitively expensive. Our numerical experiments clearly show that the CEV model performs much better in terms of the speed of convergence to its closed form solution, while the implementation cost of the stochastic volatility model is too high and practically infeasible for empirical work. In summary, with a much less implementation cost and faster computational speed, the CEV option pricing model could be a better candidate than more complex option pricing models, especially when one wants to apply the CEV process for pricing more complicated path-dependent options or credit risk models.

A Risk-Neutral Stochastic Volatility Model

1998 ◽  
Vol 01 (02) ◽  
pp. 289-310 ◽  
Author(s):  
Yingzi Zhu ◽  
Marco Avellaneda
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Monte Carlo Study ◽  
Stochastic Volatility Model ◽  
No Arbitrage ◽  
Volatility Model ◽  
Black Scholes ◽  
Risk Neutral ◽  
The Difference ◽  
Log Normal

We construct a risk-neutral stochastic volatility model using no-arbitrage pricing principles. We then study the behavior of the implied volatility of options that are deep in and out of the money according to this model. The motivation of this study is to show the difference in the asymptotic behavior of the distribution tails between the usual Black–Scholes log-normal distribution and the risk-neutral stochastic volatility distribution. In the second part of the paper, we further explore this risk-neutral stochastic volatility model by a Monte-Carlo study on the implied volatility curve (implied volatility as a function of the option strikes) for near-the-money options. We study the behavior of this "smile" curve under different choices of parameter for the model, and determine how the shape and skewness of the "smile" curve is affected by the volatility of volatility ("V-vol") and the correlation between the underlying asset and its volatility.

Martingales versus PDEs in finance: an equivalence result with examples

2000 ◽  
Vol 37 (4) ◽  
pp. 947-957 ◽  
Author(s):  
David Heath ◽  
Martin Schweizer
Keyword(s):  
Stochastic Volatility ◽  
Term Structure ◽  
Sufficient Conditions ◽  
Stochastic Volatility Model ◽  
Structure Model ◽  
Term Structure Model ◽  
Volatility Model ◽  
Equivalence Result

We provide a set of verifiable sufficient conditions for proving in a number of practical examples the equivalence of the martingale and the PDE approaches to the valuation of derivatives. The key idea is to use a combination of analytic and probabilistic assumptions that covers typical models in finance falling outside the range of standard results from the literature. Applications include Heston's stochastic volatility model and the Black-Karasinski term structure model.

GENERALIZED BN–S STOCHASTIC VOLATILITY MODEL FOR OPTION PRICING

2016 ◽  
Vol 19 (02) ◽  
pp. 1650014 ◽  
Author(s):  
INDRANIL SENGUPTA
Keyword(s):  
Option Pricing ◽  
Stochastic Volatility ◽  
Gaussian Distribution ◽  
Implied Volatility ◽  
Stochastic Volatility Model ◽  
Inverse Gaussian ◽  
Martingale Measures ◽  
Structure Preserving ◽  
Volatility Model ◽  
Equivalent Martingale

In this paper, a class of generalized Barndorff-Nielsen and Shephard (BN–S) models is investigated from the viewpoint of derivative asset analysis. Incompleteness of this type of markets is studied in terms of equivalent martingale measures (EMM). Variance process is studied in details for the case of Inverse-Gaussian distribution. Various structure preserving subclasses of EMMs are derived. The model is then effectively used for pricing European style options and fitting implied volatility smiles.

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