The Exact Implied Volatility Smile for Exponential Lévy Models

2012 ◽  
Author(s):  
Matthew Lorig
Keyword(s):  
Implied Volatility ◽  
Volatility Smile ◽  
Exponential Lévy Models

scholarly journals Smile Asymptotics II: Models with Known Moment Generating Functions

2008 ◽  
Vol 45 (1) ◽  
pp. 16-32 ◽  
Author(s):  
Shalom Benaim ◽  
Peter Friz
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Implied Volatility ◽  
Moment Generating Function ◽  
Tail Asymptotics ◽  
Volatility Smile ◽  
Tauberian Conditions ◽  
Black Scholes ◽  
Moment Generating Functions ◽  
Exponential Lévy Models

The tail of risk neutral returns can be related explicitly with the wing behaviour of the Black-Scholes implied volatility smile. In situations where precise tail asymptotics are unknown but a moment generating function is available we establish, under easy-to-check Tauberian conditions, tail asymptotics on logarithmic scales. Such asymptotics are enough to make the tail-wing formula (see Benaim and Friz (2008)) work and so we obtain, under generic conditions, a limiting slope when plotting the square of the implied volatility against the log strike, improving a lim sup statement obtained earlier by Lee (2004). We apply these results to time-changed exponential Lévy models and examine several popular models in more detail, both analytically and numerically.

scholarly journals Smile Asymptotics II: Models with Known Moment Generating Functions

2008 ◽  
Vol 45 (01) ◽  
pp. 16-32 ◽  
Author(s):  
Shalom Benaim ◽  
Peter Friz
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Implied Volatility ◽  
Moment Generating Function ◽  
Tail Asymptotics ◽  
Volatility Smile ◽  
Tauberian Conditions ◽  
Black Scholes ◽  
Moment Generating Functions ◽  
Exponential Lévy Models

The tail of risk neutral returns can be related explicitly with the wing behaviour of the Black-Scholes implied volatility smile. In situations where precise tail asymptotics are unknown but a moment generating function is available we establish, under easy-to-check Tauberian conditions, tail asymptotics on logarithmic scales. Such asymptotics are enough to make the tail-wing formula (see Benaim and Friz (2008)) work and so we obtain, under generic conditions, a limiting slope when plotting the square of the implied volatility against the log strike, improving a lim sup statement obtained earlier by Lee (2004). We apply these results to time-changed exponential Lévy models and examine several popular models in more detail, both analytically and numerically.

scholarly journals Pricing and hedging Asian options under Levy processes and robust long-term investing with learning about stock returns

2021 ◽  
Author(s):  
Andrew Na
Keyword(s):  
Stock Returns ◽  
Implied Volatility ◽  
Calibration Method ◽  
Recursive Utility ◽  
Asian Options ◽  
Volatility Smile ◽  
Bivariate Model ◽  
Risky Assets ◽  
Fourier Cosine

In this work we propose a parametric model using the techniques of time-changed subordination that captures the implied volatility smile. We demonstrate that the Fourier-Cosine method can be used in a semi-static way to hedge for quadratic, VaR and AVaR risk. We also observe that investors looking to hedge VaR can simply hold the amount in a portfolio of mostly cash, whereas an investor hedging AVaR will need to hold more risky assets. We also extend ES risk to a robust framework. A conditional calibration method to calibrate the bivariate model is proposed. For a robust long-term investor who maximizes their recursive utility and learns about the stock returns, as the willingness to substitute over time increases, the equity demand decreases and consumption-wealth ratio increases. As the preference for robustness increases the demand for risk decreases. For a positive correlation, we observe that learning about returns encourages the investor to short the bond at all levels of u and vice versa

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