There are many stochastic volatility modelling approaches that fit each use cases for instance, DD (displacement diffusion) is used modelling inflation, CEV (constant elasticity of variance) used modelling FX, interest rates, LMM, SABR etc., There isn’t one model that fits all. Therefore trader needs to make judgement based on the market forecast and implied prices how he/she perceives the future and uses the appropriate model to quote the price and hedge the risk. Note that pricing is easy, as one can always take the market quoted prices and calibrate given model to replicate the plain vanilla prices. But most important aspect of any good modelling theory is how well it implies the future hedging cost, as therefore predicts if not accurately but approximately the future slope of the underlying asset movement.
We will discuss one such model, proposed in early 2002 by Hagan et all, SABR, called Stochastic Alpha Beta and Rho model. This model is proposed to alleviate the dynamics of local volatility model which was proposed by Dupire, Derman and Kani which while calibrates today’s plain vanilla option prices perfectly but wrongly predicts the dynamics of future smile.
The original SABR model shifts the underlying, hence the delta risk changes which is improved by Bernlett’s new improved SABR model. We will see below how to include the new modified delta and vega risk.
SABR ATM vol, 
![\sigma_{ATM} = \frac{\alpha_0}{f^{1-\beta}}\{ 1 + [\frac{(1-\beta)^2}{24}\frac{\alpha_0^2}{f^{2-2\beta}} + \frac{\rho \beta \nu \alpha_0}{4 f^{1-\beta}} + \frac {2 -3\rho^2}{24} \nu^2] T \}](https://s0.wp.com/latex.php?latex=%5Csigma_%7BATM%7D+%3D+%5Cfrac%7B%5Calpha_0%7D%7Bf%5E%7B1-%5Cbeta%7D%7D%5C%7B+1+%2B+%5B%5Cfrac%7B%281-%5Cbeta%29%5E2%7D%7B24%7D%5Cfrac%7B%5Calpha_0%5E2%7D%7Bf%5E%7B2-2%5Cbeta%7D%7D++%2B+%5Cfrac%7B%5Crho+%5Cbeta+%5Cnu+%5Calpha_0%7D%7B4+f%5E%7B1-%5Cbeta%7D%7D++%2B+%5Cfrac+%7B2+-3%5Crho%5E2%7D%7B24%7D+%5Cnu%5E2%5D+T+%5C%7D&bg=ffffff&fg=000000&s=0&c=20201002)
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