scholarly journals Spectral Decomposition of Option Prices in Fast Mean-Reverting Stochastic Volatility Models

2011 ◽  
Vol 2 (1) ◽  
pp. 665-691 ◽  
Author(s):  
Jean-Pierre Fouque ◽  
Sebastian Jaimungal ◽  
Matthew J. Lorig
Keyword(s):  
Stochastic Volatility ◽  
Spectral Decomposition ◽  
Option Prices ◽  
Stochastic Volatility Models ◽  
Volatility Models

scholarly journals A Stochastic Volatility Alternative to SABR

2008 ◽  
Vol 45 (04) ◽  
pp. 1071-1085
Author(s):  
L. C. G. Rogers ◽  
L. A. M. Veraart
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Closed Form ◽  
Stochastic Volatility ◽  
Option Prices ◽  
Stochastic Volatility Models ◽  
Underlying Asset ◽  
Volatility Models ◽  
Sabr Model ◽  
Modified Dynamics

We present two new stochastic volatility models in which option prices for European plain-vanilla options have closed-form expressions. The models are motivated by the well-known SABR model, but use modified dynamics of the underlying asset. The asset process is modelled as a product of functions of two independent stochastic processes: a Cox-Ingersoll-Ross process and a geometric Brownian motion. An application of the models to options written on foreign currencies is studied.

Capturing implied correlation skew from options prices via multiscale stochastic volatility models

2020 ◽  
Vol 07 (04) ◽  
pp. 2050042
Author(s):  
T. Pellegrino
Keyword(s):  
Stochastic Volatility ◽  
Calibration Procedure ◽  
Second Order ◽  
Model Parameters ◽  
Option Prices ◽  
European Options ◽  
Stochastic Volatility Models ◽  
Volatility Models ◽  
Order Expansion ◽  
Implied Correlation

The aim of this paper is to derive a second-order asymptotic expansion for the price of European options written on two underlying assets, whose dynamics are described by multiscale stochastic volatility models. In particular, the second-order expansion of option prices can be translated into a corresponding expansion in implied correlation units. The resulting approximation for the implied correlation curve turns out to be quadratic in the log-moneyness, capturing the convexity of the implied correlation skew. Finally, we describe a calibration procedure where the model parameters can be estimated using option prices on individual underlying assets.

A Tractable Framework for Option Pricing with Dynamic Market Maker Inventory and Wealth

2019 ◽  
Vol 55 (4) ◽  
pp. 1117-1162
Author(s):  
Mathieu Fournier ◽  
Kris Jacobs
Keyword(s):  
Stochastic Volatility ◽  
Risk Premium ◽  
Reduced Form ◽  
Market Maker ◽  
Option Prices ◽  
Stochastic Volatility Models ◽  
Variance Risk Premium ◽  
Volatility Models ◽  
Variance Risk ◽  
Index Option

We develop a tractable dynamic model of an index option market maker with limited capital. We solve for the variance risk premium and option prices as a function of the asset dynamics and market maker option holdings and wealth. The market maker absorbs end users’ positive demand and requires a more negative variance risk premium when she incurs losses. We estimate the model using returns, options, and inventory and find that it performs well, especially during the financial crisis. The restrictions imposed by nested existing reduced-form stochastic-volatility models are strongly rejected in favor of the model with a market maker.

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.

CLOSED-FORM APPROXIMATION OF PERPETUAL TIMER OPTION PRICES

2014 ◽  
Vol 17 (04) ◽  
pp. 1450026 ◽  
Author(s):  
MINQIANG LI ◽  
FABIO MERCURIO
Keyword(s):  
Asymptotic Expansion ◽  
Closed Form ◽  
Stochastic Volatility ◽  
Option Prices ◽  
Stochastic Volatility Models ◽  
Volatility Models ◽  
Expansion Technique ◽  
Black Scholes ◽  
Approximation Formulas ◽  
Closed Form Approximation

We develop an asymptotic expansion technique for pricing timer options in stochastic volatility models when the effect of volatility of variance is small. Based on the pricing PDE, closed-form approximation formulas have been obtained. The approximation has an easy-to-understand Black–Scholes-like form and many other attractive properties. Numerical analysis shows that the approximation formulas are very fast and accurate, especially when the volatility of variance is not large.

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