scholarly journals Asymptotic Implied Volatility at the Second Order with Application to the SABR Model

2015 ◽  
pp. 37-69 ◽  
Author(s):  
Louis Paulot
Keyword(s):  
Implied Volatility ◽  
Second Order ◽  
Sabr Model

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.

PRICING TIMER OPTIONS: SECOND-ORDER MULTISCALE STOCHASTIC VOLATILITY ASYMPTOTICS

2021 ◽  
pp. 1-19
Author(s):  
XUHUI WANG ◽  
SHENG-JHIH WU ◽  
XINGYE YUE
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Option Price ◽  
Second Order ◽  
Approximation Formula ◽  
Stochastic Volatility Models ◽  
Regular Perturbation ◽  
Volatility Models ◽  
Order Expansion ◽  
High Level

Abstract We study the pricing of timer options in a class of stochastic volatility models, where the volatility is driven by two diffusions—one fast mean-reverting and the other slowly varying. Employing singular and regular perturbation techniques, full second-order asymptotics of the option price are established. In addition, we investigate an implied volatility in terms of effective maturity for the timer options, and derive its second-order expansion based on our pricing asymptotics. A numerical experiment shows that the price approximation formula has a high level of accuracy, and the implied volatility in terms of its effective maturity is illustrated.

Mass at Zero and Small-Strike Implied Volatility Expansion in the SABR Model

2015 ◽  
Author(s):  
Archil Gulisashvili ◽  
Blanka Horvath ◽  
Antoine Jacquier
Keyword(s):  
Implied Volatility ◽  
Sabr Model

FORWARD AND FUTURE IMPLIED VOLATILITY

2011 ◽  
Vol 14 (03) ◽  
pp. 407-432 ◽  
Author(s):  
PAUL GLASSERMAN ◽  
QI WU
Keyword(s):  
Implied Volatility ◽  
Transition Density ◽  
Option Prices ◽  
Piecewise Constant ◽  
Model Based ◽  
Model Free ◽  
Out Of Sample ◽  
Sabr Model ◽  
Using Data ◽  
Better Than

We address the problem of defining and calculating forward volatility implied by option prices when the underlying asset is driven by a stochastic volatility process. We examine alternative notions of forward implied volatility and the information required to extract these measures from the prices of European options at fixed maturities. We then specialize to the SABR model and show how the asymptotic expansion of the bivariate transition density in Wu (forthcoming) allows calibration of the SABR model with piecewise constant parameters and calculation of forward volatility. We then investigate empirically whether current option prices at multiple maturities contain useful information in predicting future option prices and future implied volatility. We undertake this investigation using data on options on the euro-dollar, sterling-dollar, and dollar-yen exchange rates. We find that prices across maturities do indeed have predictive value. Moreover, we find that model-based forward volatility extracts this predicative information better than a standard "model-free" measure of forward volatility and better than spot implied volatility. The enhancement to out-of-sample forecasting accuracy gained from model-based forward volatility is greatest at longer forecasting horizons.

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).

Pricing timer options: second-order multiscale stochastic volatility asymptotics

2021 ◽  
Vol 63 ◽  
pp. 249-267
Author(s):  
Xuhui Wang ◽  
Sheng-Jhih Wu ◽  
Xingye Yue
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Option Price ◽  
Second Order ◽  
Approximation Formula ◽  
Stochastic Volatility Models ◽  
Regular Perturbation ◽  
Volatility Models ◽  
Order Expansion ◽  
High Level

We study the pricing of timer options in a class of stochastic volatility models, where the volatility is driven by two diffusions—one fast mean-reverting and the other slowly varying. Employing singular and regular perturbation techniques, full second-order asymptotics of the option price are established. In addition, we investigate an implied volatility in terms of effective maturity for the timer options, and derive its second-order expansion based on our pricing asymptotics. A numerical experiment shows that the price approximation formula has a high level of accuracy, and the implied volatility in terms of its effective maturity is illustrated. doi:10.1017/S1446181121000249

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