scholarly journals Forecasting the Variability of Stock Index Returns with Stochastic Volatility Models and Implied Volatility

2003 ◽  
pp. 71-97 ◽  
Author(s):  
Eugenie M. J. H. Hol
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Stock Index ◽  
Stochastic Volatility Models ◽  
Volatility Models ◽  
Stock Index Returns

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.

scholarly journals TIME-CHANGED FAST MEAN-REVERTING STOCHASTIC VOLATILITY MODELS

2011 ◽  
Vol 14 (08) ◽  
pp. 1355-1383 ◽  
Author(s):  
MATTHEW LORIG
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Perturbation Techniques ◽  
European Option ◽  
Stochastic Volatility Models ◽  
Multiple Factors ◽  
Price Process ◽  
Volatility Models ◽  
Time Changes ◽  
Implied Volatility Surfaces

We introduce a class of randomly time-changed fast mean-reverting stochastic volatility (TC-FMR-SV) models. Using spectral theory and singular perturbation techniques, we derive an approximation for the price of any European option in the TC-FMR-SV setting. Three examples of random time-changes are provided and are shown to induce distinct implied volatility surfaces. The key features of the TC-FMR-SV framework are that (i) it is able to incorporate jumps into the price process of the underlying asset (ii) it allows for the leverage effect and (iii) it can accommodate multiple factors of volatility, which operate on different time-scales.

ON THE ASYMPTOTICS OF FAST MEAN-REVERSION STOCHASTIC VOLATILITY MODELS

2007 ◽  
Vol 10 (05) ◽  
pp. 817-835 ◽  
Author(s):  
MAX O. SOUZA ◽  
JORGE P. ZUBELLI
Keyword(s):  
Asymptotic Expansion ◽  
Stochastic Volatility ◽  
Implied Volatility ◽  
Correction Term ◽  
Option Prices ◽  
Stochastic Volatility Models ◽  
Black Scholes Model ◽  
Volatility Models ◽  
Asymptotic Formulae ◽  
Black Scholes

We consider the asymptotic behavior of options under stochastic volatility models for which the volatility process fluctuates on a much faster time scale than that defined by the riskless interest rate. We identify the distinguished asymptotic limits and, in contrast with previous studies, we deal with small volatility-variance (vol-vol) regimes. We derive the corresponding asymptotic formulae for option prices, and find that the first order correction displays a dependence on the hidden state and a non-diffusive terminal layer. Furthermore, this correction cannot be obtained as the small variance limit of the previous calculations. Our analysis also includes the behavior of the asymptotic expansion, when the hidden state is far from the mean. In this case, under suitable hypothesis, we show that the solution behaves as a constant volatility Black–Scholes model to all orders. In addition, we derive an asymptotic expansion for the implied volatility that is uniform in time. It turns out that the fast scale plays an important role in such uniformity. The theory thus obtained yields a more complete picture of the different asymptotics involved under stochastic volatility. It also clarifies the remarkable independence on the state of the volatility in the correction term obtained by previous authors.

Computing the implied volatility in stochastic volatility models

2004 ◽  
Vol 57 (10) ◽  
pp. 1352-1373 ◽  
Author(s):  
Henri Berestycki ◽  
J�r�me Busca ◽  
Igor Florent
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Stochastic Volatility Models ◽  
Volatility Models
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