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.

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.

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.

scholarly journals A Stochastic Volatility Alternative to SABR

2008 ◽  
Vol 45 (4) ◽  
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.

scholarly journals Capturing the volatility smile: parametric volatility models versus stochastic volatility models

2016 ◽  
Vol 5 (4) ◽  
pp. 15-22 ◽  
Author(s):  
Belen Blanco
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Jel Classification ◽  
Option Prices ◽  
Expiration Date ◽  
Volatility Smile ◽  
Stochastic Volatility Models ◽  
Volatility Models ◽  
Black Scholes ◽  
Option Pricing Model

Black-Scholes option pricing model (1973) assumes that all option prices on the same underlying asset with the same expiration date, but different exercise prices should have the same implied volatility. However, instead of a flat implied volatility structure, implied volatility (inverting the Black-Scholes formula) shows a smile shape across strikes and time to maturity. This paper compares parametric volatility models with stochastic volatility models in capturing this volatility smile. Results show empirical evidence in favor of parametric volatility models. Keywords: smile volatility, parametric, stochastic, Black-Scholes. JEL Classification: C14 C68 G12 G13

LOCAL STOCHASTIC VOLATILITY WITH JUMPS: ANALYTICAL APPROXIMATIONS

2013 ◽  
Vol 16 (08) ◽  
pp. 1350050 ◽  
Author(s):  
STEFANO PAGLIARANI ◽  
ANDREA PASCUCCI
Keyword(s):  
Characteristic Function ◽  
Stochastic Volatility ◽  
Fourier Space ◽  
Stochastic Volatility Models ◽  
Fourier Methods ◽  
Analytical Approximations ◽  
Volatility Models ◽  
Approximation Formulas ◽  
Real Market ◽  
Lévy Jumps

We present new approximation formulas for local stochastic volatility models, possibly including Lévy jumps. Our main result is an expansion of the characteristic function, which is worked out in the Fourier space. Combined with standard Fourier methods, our result provides efficient and accurate formulas for the prices and the Greeks of plain vanilla options. We finally provide numerical results to illustrate the accuracy with real market data.

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