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1) The paper Explicit SABR Calibration Through Simple Expansions explains how to calibrate the SABR model in practice. 2) The role of alpha, beta and rho is well explained in the original SABR paper Managing Smile Risk. Beta is most often chosen in advance, to represent a specific dynamic. Although one can find references where people calibrate it to option ...


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Here are a few FX structured product examples: All of these can be notes or swaps, notes will pay back the notional at the end and carry no credit risk (and are normally set so that they are worth 100% at inception - i.e. they'll be worth 99% and the seller will take some profit/hedging costs). Swaps will either be set to be worth 0% (same deal as above, ...


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In short , this claim does not hold under all circumstances. There are a few ways to break down such approximation. The options under consideration have very long expiry, i.e. $T$ is very large As expiration date approaches, the volatility smile becomes more pronounced, i.e. $v$ becomes relatively large. Under extreme market condition, the magnitude of $\...


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This is indeed an important issue if you use SABR in production. If I am correct, you'll need this term $$ \chi(\zeta) = \log \left( \frac{\sqrt{1-2\rho\zeta+\zeta^2}-\rho+\zeta}{1-\rho} \right) $$ and $\zeta=0$ if $F_0=K$. When $\zeta$ is VERY small (e.g., $|\zeta|<10^{-8}$), you can use the Taylor expansion $$\sqrt{1+\varepsilon} \approx 1+\...


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First and foremost it is important to clarify that the underlying is not necessarily normal/lognormal but for the special cases of $\beta$ the underlying is normal/lognormal Conditioned on a realization of the volatility. As mentioned in the answer by @ilovevolatility. Simple stochastic calculus will show the properties you mentioned. For realized ...


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If you want to use the normal SABR ($\beta=0$), my paper, Hyperbolic normal stochastic volatility model (arXiv | SSRN | DOI) might give you a solution. It reports an exact closed-form MC simulation scheme for the normal SABR model. Better than that, it shows that Johnson's $S_U$ distribution ($\sinh$ transformation of normal variate) is a close cousin of the ...


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You don't want to use the SABR (or an extension) to price equity options or FX options. The lag of mean-reversion in the model's volatility dynamics leads to explosive behavior and to a implied distribution that is absolutely not in line with empirics -- especially on longer time horizons. To my knowledge people use it mostly for interest rate derivatives. ...


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I can confirm there is no error in @Sanjay graph. I obtain the same plot with Obloj correction for the SABR formula. In fact, the popular SABR approximation formulas (Hagan or the further corrections) use as hypothesis a small vol of vol. In your case, the vol of vol $\nu$ is very large ($\nu=7$) and it is not too surprising that the approximations break ...


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The SABR model itself is arbitrage-free even for high vol of vol. The question is whether the Hagan et al formula for implied volatility under the SABR model is arbitrage free - it isn't actually. For very low strikes arbitrage can occur using the Hagan et al formula for implied volatility, and perhaps also for very high vol of vol. Question: how do you ...


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Your volatility also depends on your forward level, so does the value of your derivatives so a more accurate definition of your delta under a variable volatility is: $$ \dfrac{\partial V(F,\sigma(F))}{\partial F } = \dfrac{\partial V(F,\sigma)}{\partial F } + \dfrac{\partial V}{\partial \sigma}\dfrac{\partial \sigma(F)}{\partial F} $$ This is because, ...


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No, the simulation is not exact in general, precisely for the reason you mentioned. By "exact", it is meant that there is no discretization error in time. Of course, there will always be a Monte-Carlo sampling error. For the Black-Scholes model, the simulation is exact if you simulate the log asset, as it is a standard arithmetic Brownian motion, and then ...


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performed an experiment myself, using Bachelier's Black Model and coded shifted-SABR normal model. I observe the following We perform a series of experiments, that tests for  Given different z-shifts, what are the SABR parameters to calibrate to the target-set of implied vols  What parameters are necessary and how do they change. With that, we ...


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my two cent's worth. It depends ultimately what you are trying to achieve, and calibrate to. The key tests for implied vols on option prices, relate to their pdf (probability density functions). The Hagan expansion allows us to have a analytical form, by which one can compute the implied vols, and subsequently feed that into a Black76 equation. If the ...


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The answer given so far, by Mats Lind, to the first question is not in the spirit of the paper. I am referring to Question: In the paper they say one should really integrate from $y$ to $\infty$... What they mean is that you should not use any information on what happens from $-\infty$ to $y$, which would correspond to the standard cumulative distribution ...


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