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 ...

The answer given is mostly wrong: @msitt uses a convoluted way without explicitly mentioning it (put-call symmetry) to actually give the price of a USD Put, not of a USD Call as requested. Here is a ...

Ikonen and Toivanen don't say that the LCP is solved exactly, they simply say that the modified back-substitution is a valid algorithm to solve the LCP. A numerical error may arise around the ...

No, and this is wrong. The implied vols (from market prices) are actually not necessarily convex but yet may be still arbitrage-free, there are many examples of this for various equities. Furthermore, ...

It depends what you exactly call Dupire's formula. If you take the original formula, valid under zero interest rates and dividends (or equivalently, considering undiscounted option prices on the ...

You might want to differentiate between the growth rate $\mu$ and the discount rate $r$. @Gordon's solution is the most logical thing to do, given the question. However, in practice, it is not ...

The main issue with the answer from @quantuple is that the price does not converge to the Black-Scholes price when rho=0 or when the quanto adjustment is negligible. The question is answered in ...

Zhu makes sense to me. The vega cash in Black-Scholes corresponds to a shift of the vol surface by 1%. If you bump only $v_0$ in Heston, you bump only the short maturities, and if your structure is ...

The issue has much more to do with the SVI parameterization per se, and not with any arbitrage constraint. The fact that Heston as $T \to \infty$ becomes close to SVI is not very useful either to ...

Pat Hagan describes this well in the famous SABR paper Managing smile risk. An approximate relation given in equation (B.64) reads $$\sigma_N \approx \sigma_B \frac{f-K}{\ln f/K}\left(1-\frac{\sigma_B^... View answer 3 votes The formula for pricing a swaption under normal volatility is simply the Bachelier formula. It may be found in many papers (for example, Le Floc'h Fast and accurate basis point volatility), and is ... View answer Accepted answer 3 votes There is no real "risk-free" rate. Now to answer your question, r is time-dependent and should correspond to the repo rate corresponding to the maturity of your forward. In I, dividends should ... View answer 3 votes I think this is the old accrual methodology, historically used for the banking book. I believe it is not market standard anymore and regulators require an MTM (mark-to-market) valuation. Here is an ... View answer 3 votes This is a well known issue. There are three possible tricks: I am surprised that none of the answers so far mention the work of Lord and Kahl Optimal Fourier Inversion in Semi-Analytical Option ... View answer 3 votes They are not traded, even Over-The-Counter (OTC). Asian options with arithmetic averaging are traded. The geometric Asian may be used to derive a closed-form approximation for the arithmetic variety,... View answer 2 votes The market will quote Call and Put options prices within a bid-ask spread. In order to imply the volatility, one may choose to use the bid, the ask, or the mid. Although the mid is a better idea in ... View answer 2 votes 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 ... View answer 2 votes 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)... View answer Accepted answer 2 votes 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 ... View answer 2 votes Such a question really invites me to recommend my own book Applied Quantitative Finance for Equity Derivatives, which you can buy on Amazon. The book devotes 200 pages to the subject of volatility. ... View answer 2 votes Brian B gives the overall idea. But the use of a simple polynomial will not be appropriate in general. The paper Model-free stochastic collocation for an arbitrage-free implied volatility: Part I ... View answer 2 votes 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... ... View answer Accepted answer 1 votes The flexible forward contract is very much like an American option: at each exercise date, you have the choice to receive the payoff (S-K) or not. The difference with a regular option is that you ... View answer 1 votes Another sketch of proof: If you move to the equivalent PDE (using Feynman-Kac), you can assume that S is positive, find the solution by log-transfomation. Then as the solution is unique given initial ... View answer Accepted answer 1 votes Some of the assumptions here are wrong. The issue here is that$$S_0 \neq e^{-rT} E[S],$$but$$F = E[S]. And thus Z should be Z=V-theta*(VC-exp(-rT)*F). If you output mean(VC) it's very clear. It ...

The model is interesting, but rarely used in practice. One main reason is the choice of the range $\sigma_{min}, \sigma_{max}$. As Martini and Jacquier explain in their article The uncertain ...

In general, you don't want to build one arbitrage-free bid surface and one arbitrage-free ask surface, since such surfaces would have no practical use. As you mention, any hedging will involve both ...

This is not the typical Heston stochastic differential equation (SDE). In the original Heston paper, the SDE is defined without $\lambda$, that is $\lambda=1$ and $v(0)=v_0$ not necessarily 1. In ...