LEFT-WING ASYMPTOTICS OF THE IMPLIED VOLATILITY IN THE PRESENCE OF ATOMS

The paper considers the asymptotic behavior of the implied volatility in stochastic asset price models with atoms. In such models, the asset price distribution has a singular component at zero. Examples of models with atoms include the constant elasticity of variance (CEV) model, jump-to-default models, and stochastic models described by processes stopped at the first hitting time of zero. For models with atoms, the behavior of the implied volatility at large strikes is similar to that in models without atoms. On the other hand, the behavior of the implied volatility at small strikes is influenced significantly by the atom at zero. De Marco, Hillairet, and Jacquier found an asymptotic formula for the implied volatility at small strikes with two terms and also provided an incomplete description of the third term. In the present paper, we obtain a new asymptotic formula for the left wing of the implied volatility, which is qualitatively different from the De Marco–Hillairet–Jacquier formula. The new formula contains three explicit terms and an error estimate. In the paper, we show how to derive the De Marco–Hillairet–Jacquier formula from the new formula, and compare the performance of the two formulas in the case of the CEV model. The graphs included in the paper show that the new asymptotic formula provides a notably better approximation to the implied volatility at small strikes in the CEV model than the De Marco–Hillairet–Jacquier formula.

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THE LARGE-MATURITY SMILE FOR THE SABR AND CEV-HESTON MODELS

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024913500477 ◽

2013 ◽

Vol 16 (08) ◽

pp. 1350047 ◽

Cited By ~ 11

Author(s):

MARTIN FORDE ◽

ANDREY POGUDIN

Keyword(s):

Implied Volatility ◽

Large Time ◽

Heston Model ◽

Large Time Asymptotics ◽

Cev Model ◽

Constant Elasticity ◽

Constant Elasticity Of Variance ◽

Time Estimates ◽

Time Asymptotics ◽

Sabr Model

Large-time asymptotics are established for the SABR model with β = 1, ρ ≤ 0 and β < 1, ρ = 0. We also compute large-time asymptotics for the constant elasticity of variance (CEV) model in the large-time, fixed-strike regime and a new large-time, large-strike regime, and for the uncorrelated CEV-Heston model. Finally, we translate these results into a large-time estimates for implied volatility using the recent work of Gao and Lee (2011) and Tehranchi (2009).

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Modelling and Forecasting LKR/USD Monthly Exchange Rates using Constant Elasticity of Variance (CEV) Model

International Journal of Scientific and Research Publications (IJSRP) ◽

10.29322/ijsrp.8.4.2018.p7620 ◽

2018 ◽

Vol 8 (4) ◽

Author(s):

W. M. H. N. Weerasinghe

Keyword(s):

Exchange Rates ◽

Cev Model ◽

Constant Elasticity ◽

Constant Elasticity Of Variance ◽

Modelling And Forecasting

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THE PRICING OF EUROPEAN OPTIONS UNDER THE CONSTANT ELASTICITY OF VARIANCE WITH STOCHASTIC VOLATILITY

Fluctuation and Noise Letters ◽

10.1142/s0219477513500041 ◽

2013 ◽

Vol 12 (01) ◽

pp. 1350004 ◽

Cited By ~ 1

Author(s):

BOUNGHUN BOCK ◽

SUN-YONG CHOI ◽

JEONG-HOON KIM

Keyword(s):

Numerical Experiment ◽

Stochastic Volatility ◽

Hybrid Model ◽

Multiscale Analysis ◽

Asset Price ◽

European Options ◽

Barrier Option ◽

Price Model ◽

Constant Elasticity ◽

Constant Elasticity Of Variance

This paper considers a hybrid risky asset price model given by a constant elasticity of variance multiplied by a stochastic volatility factor. A multiscale analysis leads to an asymptotic pricing formula for both European vanilla option and a Barrier option near the zero elasticity of variance. The accuracy of the approximation is provided in a rigorous manner. A numerical experiment for implied volatilities shows that the hybrid model improves some of the well-known models in view of fitting the data for different maturities.

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CONSTANT ELASTICITY OF VARIANCE OPTION PRICING MODEL WITH TIME-DEPENDENT PARAMETERS

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024900000814 ◽

2000 ◽

Vol 03 (04) ◽

pp. 661-674 ◽

Cited By ~ 41

Author(s):

C. F. LO ◽

P. H. YUEN ◽

C. H. HUI

Keyword(s):

Option Pricing ◽

Algebraic Approach ◽

Time Dependent ◽

Model Parameters ◽

Cev Model ◽

Constant Elasticity ◽

Constant Elasticity Of Variance ◽

Pricing Options ◽

Term Structures ◽

Option Values

This paper provides a method for pricing options in the constant elasticity of variance (CEV) model environment using the Lie-algebraic technique when the model parameters are time-dependent. Analytical solutions for the option values incorporating time-dependent model parameters are obtained in various CEV processes with different elasticity factors. The numerical results indicate that option values are sensitive to volatility term structures. It is also possible to generate further results using various functional forms for interest rate and dividend term structures. Furthermore, the Lie-algebraic approach is very simple and can be easily extended to other option pricing models with well-defined algebraic structures.

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SHORT MATURITY ASIAN OPTIONS FOR THE CEV MODEL

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964818000165 ◽

2018 ◽

Vol 33 (2) ◽

pp. 258-290 ◽

Cited By ~ 2

Author(s):

Dan Pirjol ◽

Lingjiong Zhu

Keyword(s):

Analytical Approximation ◽

Time Averaging ◽

Asian Options ◽

Option Prices ◽

Asymptotic Results ◽

Cev Model ◽

Constant Elasticity ◽

Constant Elasticity Of Variance ◽

Practical Applications ◽

Rigorous Study

We present a rigorous study of the short maturity asymptotics for Asian options with continuous-time averaging, under the assumption that the underlying asset follows the constant elasticity of variance (CEV) model. The leading order short maturity limit of the Asian option prices under the CEV model is obtained in closed form. We propose an analytical approximation for the Asian options prices which reproduces the exact short maturity asymptotics, and demonstrate good numerical agreement of the asymptotic results with Monte Carlo simulations and benchmark test cases for option parameters relevant for practical applications.

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Option pricing in a subdiffusive constant elasticity of variance (CEV) model

International Journal of Financial Engineering ◽

10.1142/s242478631950018x ◽

2019 ◽

Vol 06 (02) ◽

pp. 1950018

Author(s):

Kevin Z. Tong ◽

Allen Liu

Keyword(s):

Financial Markets ◽

Analytical Formula ◽

Planck Equation ◽

Option Prices ◽

Cev Model ◽

Constant Elasticity ◽

Constant Elasticity Of Variance ◽

New Process ◽

Stable Variable ◽

Classical Constant

In this paper, we extend the classical constant elasticity of variance (CEV) model to a subdiffusive CEV model, where the underlying CEV process is time changed by an inverse [Formula: see text]-stable subordinator. The new model can capture the subdiffusive characteristics of financial markets. We find the corresponding fractional Fokker–Planck equation governing the PDF of the new process. We also derive the analytical formula for option prices in terms of eigenfunction expansion. This method avoids the evaluation of PDF of an inverse [Formula: see text]-stable variable and also eliminates the need for numerical integration to calculate the option prices. We numerically investigate the sensitivities of the option prices to the key parameters of the newly developed model.

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Optimal Reinsurance-Investment Problem under a CEV Model: Stochastic Differential Game Formulation

Mathematical Problems in Engineering ◽

10.1155/2020/7265121 ◽

2020 ◽

Vol 2020 ◽

pp. 1-19 ◽

Cited By ~ 1

Author(s):

Danping Li ◽

Ruiqing Chen ◽

Cunfang Li

Keyword(s):

Differential Game ◽

Exponential Utility ◽

Insurance Companies ◽

Stochastic Differential Game ◽

Investment Strategies ◽

Small Company ◽

Cev Model ◽

Constant Elasticity ◽

Optimal Reinsurance ◽

Constant Elasticity Of Variance

This paper focuses on a stochastic differential game played between two insurance companies, a big one and a small one. In our model, the basic claim process is assumed to follow a Brownian motion with drift. Both of two insurance companies purchase the reinsurance, respectively. The big company has sufficient asset to invest in the risky asset which is described by the constant elasticity of variance (CEV) model and acquire new business like acting as a reinsurance company of other insurance companies, while the small company can invest in the risk-free asset and purchase reinsurance. The game studied here is zero-sum where there is a single exponential utility. The big company is trying to maximize the expected exponential utility of the terminal wealth to keep its advantage on surplus while simultaneously the small company is trying to minimize the same quantity to reduce its disadvantage. In this paper, we describe the Nash equilibrium of the game and prove a verification theorem for the exponential utility. By solving the corresponding Fleming-Bellman-Isaacs equations, we derive the optimal reinsurance and investment strategies. Furthermore, numerical examples are presented to show our results.

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Hedging with Liquidity Risk under CEV Diffusion

Risks ◽

10.3390/risks8020062 ◽

2020 ◽

Vol 8 (2) ◽

pp. 62

Author(s):

Sang-Hyeon Park ◽

Kiseop Lee

Keyword(s):

Discrete Time ◽

Nonlinear Partial Differential Equation ◽

Liquidity Risk ◽

Discrete Version ◽

Cev Model ◽

Constant Elasticity ◽

Constant Elasticity Of Variance ◽

Multi Scale ◽

Replication Scheme ◽

Scale Perturbation

We study a discrete time hedging and pricing problem in a market with the liquidity risk. We consider a discrete version of the constant elasticity of variance (CEV) model by applying Leland’s discrete time replication scheme. The pricing equation becomes a nonlinear partial differential equation, and we solve it by a multi scale perturbation method. A numerical example is provided.

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