# Tag Info

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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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I have found the answer to my own question during the last month where the question have been unanswered The main question: Look at page 9 from "the ODE can be rearranged to $f'(y)=....=F(y,f)$". For given $\alpha, \beta, rho, \epsilon$ and $\gamma$ we have all the relevant information to solve this ODE, hence finding $f(y(t))$ which is denoted as $f(y)$ ...

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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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The relationship between the two models is described in details in Implied Volatility Formulas for Heston Models by Hagan et al. In particular an expansion of the implied volatility under the Heston model that matches the one of a SABR model is described. It gives an explicit correspondence between the parameters of each model.

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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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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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You are observing the same underlying $S_t$, therefore it has to be one set of parameters for all maturities. You could add a term structure to the parameters , however , since you are using SABR, I assume you use Hagan expansion to generate the implied vols, and for this approximation, the parameters must be constant.

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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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I am not going to answer all of your questions, but let me give it a go. I don't have a qualified answer to this one. In practice however $\beta$ is always bounded such that $\beta \in [0,1]$ (0 and 1 both included). see SABR chapter in Derman & Miller (2006). But as far I know, $\beta$ is not bounded in the original paper so in theory it can take any ...

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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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Given your regression relationship between atm IV and forward price, as long as beta <1, atm IV and forward price are negatively correlated which is usually consistent with the market observations - the higher the forward price (longer maturity), the lower the atm IV. If beta is greater than 1, rather, ATM IV and forward price are positive correlated, ...

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