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

2013 ◽  
Vol 16 (08) ◽  
pp. 1350047 ◽  
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).

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

2015 ◽  
Vol 18 (02) ◽  
pp. 1550013 ◽  
Author(s):  
ARCHIL GULISASHVILI
Keyword(s):  
Asymptotic Formula ◽  
Implied Volatility ◽  
Asset Price ◽  
First Hitting Time ◽  
Left Wing ◽  
Cev Model ◽  
Constant Elasticity ◽  
Constant Elasticity Of Variance ◽  
Price Distribution ◽  
New Formula

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.

scholarly journals EXACT PRICING AND LARGE-TIME ASYMPTOTICS FOR THE MODIFIED SABR MODEL AND THE BROWNIAN EXPONENTIAL FUNCTIONAL

2011 ◽  
Vol 14 (04) ◽  
pp. 559-578 ◽  
Author(s):  
MARTIN FORDE
Keyword(s):  
Stock Price ◽  
Implied Volatility ◽  
Large Time ◽  
Transition Density ◽  
Closed Form Expression ◽  
Modified Model ◽  
Exponential Functional ◽  
Price Distribution ◽  
Zero Correlation ◽  
Sabr Model

We derive a closed-form expression for the stock price density under the modified SABR model [see section 2.4 in Islah (2009)] with zero correlation, for β = 1 and β < 1, using the known density for the Brownian exponential functional for μ = 0 given in Matsumoto and Yor (2005), and then reversing the order of integration using Fubini's theorem. We then derive a large-time asymptotic expansion for the Brownian exponential functional for μ = 0, and we use this to characterize the large-time behaviour of the stock price distribution for the modified SABR model; the asymptotic stock price "density" is just the transition density p(t, S0, S) for the CEV process, integrated over the large-time asymptotic "density" [Formula: see text] associated with the Brownian exponential functional (re-scaled), as we might expect. We also compute the large-time asymptotic behaviour for the price of a call option, and we show precisely how the implied volatility tends to zero as the maturity tends to infinity, for β = 1 and β < 1. These results are shown to be consistent with the general large-time asymptotic estimate for implied variance given in Tehranchi (2009). The modified SABR model is significantly more tractable than the standard SABR model. Moreover, the integrated variance for the modified model is infinite a.s. as t → ∞, in contrast to the standard SABR model, so in this sense the modified model is also more realistic.

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

2000 ◽  
Vol 03 (04) ◽  
pp. 661-674 ◽  
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.

scholarly journals SHORT MATURITY ASIAN OPTIONS FOR THE CEV MODEL

2018 ◽  
Vol 33 (2) ◽  
pp. 258-290 ◽  
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.

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

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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