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Let's relabel this as What (TF) is SABR? Alpha, Beta and Rho are the point of the model. So explaining them is explaining the model. A model of two processes Unlike earlier models in which the volatility was modelled as a constant (Vasicek, Hull-White, etc), SABR assumes that as well as the price of the thing being stochastic, so is its volatility. That ...

6

I think you did something wrong in translating the input to numerics. As pointed out by dm63 normal vols are quoted in basis points. Using equation A.67a) from the Hagan paper you linked we see (setting $\beta = 0$) $$\sigma_N(K) = \alpha\frac{\xi}{x(\xi)}\left[1+\frac{2-3\rho^2}{24}\nu^2\tau_{exp}\right]$$ where $\tau_{exp} = 0.25$ in your example and ...

5

If I got this right, you get a SABR model from fitting market implied volatilites (from market price via Black) to SABR volatilites (from SABR parameters via formula above). Then you step back and think the SABR distribution needs improvement because it is not arbitrage free. Instead you use the collocation method to replace it with its projection onto a ...

4

The SABR process is a strict martingale for all values of beta < 1 (in particular, negative betas are fine). If beta = 1, the process is a strict martingale if and only if rho < 0. Under all other circumstances, i.e. beta > 1, or beta = 1 and rho >= 0, the SABR process is a local martingale but not a martingale (it may explode in finite time).

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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 ...

3

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, ...

3

In fact, this is a confusion caused by a sloppy notation. The rigorous version of the setup should be $$A(K)\rightarrow \epsilon A(K).$$ Then we let $x:=\frac{f-K}\epsilon$. The rest is the usual singular perturbation operation.

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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 $\... 3 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+\... 3 The simplest answer is that prices are not derived from the models. Prices are a result of trading in the market and express the sum of the information and opinions of all market participants at the time. Despite what many academics would like you to believe, markets participants are not always perfectly rational and not always 100% motivated by the maximum ... 3 Here you can refer to the previously asked question regarding the advantages and disadvatanges of the volatility surface:- What are the advantages/disadvantages of these approaches to deal with volatility surface? To my knowledge, the main advantage of using SABR in comparison to other models like Heston is that, SABR assumes lognormal distribution whereas ... 2 You would simply calculate the prices of various strike options using your parameters, then calculate the black scholes implied vol of each option. Did I miss the point of your question ? 2 I would say that SABR is overkill for inflation options, because due to the scarcity of prices for inflation options, there isn't enough information to fit all the SABR parameters. It is probably better to adopt a simple Normalized model of inflation rates, which you can calibrate by looking at historical normalized volatility. This will also take care of ... 2 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. 2 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 ... 2 Try to redesign your object function, your difvol, so it's a function of a two dimensional vector dilvol<-function(X){ rho<-X[1] nu<-X[2] ##type in your function here and use rho and nu normally.. } nlm(difvol,c(0.1,0.1) This might work 2 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 ... 2 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. ... 2 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 ... 2 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 ... 2 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 ... 1 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 ... 1 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 ... 1 Given (conditional on) a realisation of the volatility, it is normal for$\beta = 0$and lognormal for$\beta = 1\$

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The SABR parameters are typically calibrated daily. Intraday recalibrations may be required on particularly volatile days. The choice beta = 1 is a popular choice when SABR is used in FX or equity markets, because of the distribution characteristics of the respective asset returns (matched best by beta = 1). In the rates markets, for which SABR was ...

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If you have a few options quotes, preferably a few around ATM, and a few in the call and put wings, then you might want to try using SVI for the smoothing of the IV smile (and hence get a value for the ATM IV). SVI calibration is also relatively quick, so you don't need to use stale parameters, you could update the parameters intra-day to the available ...

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The cross derivative FD formula is wrong to start with. If you take out the factor 4 from the denominator, it would become a valid (1st order) formula. As is, it's plain wrong and may well be the reason for your problems (though of course there may be more reasons). By the way, your other spatial FD approximations are second order, so not sure you intended ...

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The prices come from the market. One common source is Reuters. For items not traded in open markets you might assemble a set of bids from various counter parties. Volatilities are derived from the market prices. Many places use the market prices as inputs to calibrate shocks to their models. Traders might get "correct" prices from their models which ...

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