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Stochastic-Local Vol (SLV) is an attempt to mix the strengths and weaknesses of both Stochastic Vol and Local Vol models. Below, I'll quickly summarise each model and their strengths and weaknesses, and then discuss how SLV tries to improve things. Although there are many stochastic vol models, I limit the discussion here to the Heston model to keep things ...


7

Further notes: One shouldn't build an implied volatility surface just from call prices or just from put prices. One should build it from liquid instrument quotes and, if necessary, some less liquid ones. Some markets, like FX option one, quote package prices (butterfly, risk reversal, ATM straddles). Deciding how to parameterize the implied volatility ...


7

In practice, things are actually quite different and a bit more subtle. You really need to differentiate between the underlying being an index or e.g. a single stock. I will try to provide some insight: Index options are, in general, of European type. The market quotes prices for calls and puts and you can back out the implied vols via the usual BS formula. ...


6

A model that reflects the volatility smile is one with dynamics that approximate pricing that yields an implied volatility smile. However, your question makes me suspect you are fuzzy on some of these pieces, so let's go through this in more detail. Implied Volatilities $\implies$ Correct Price? You mention that implied volatility in the Black-Scholes model ...


6

It's probably important that we're talking about IV of an index. From "Volatility Trading" by Euan Sinclair: In equity indexes the skew will be more pronounced than in the individual stocks that make up the index. The volatility of an index, $σ$, is related to the volatility of the components, $σ_i$, by: $$σ^2=\sum_{i=1}^N w_i^2 σ_i^2+2\sum_{i=1}^{...


5

Call and a put of the same strike have the same I.V, in theory. The ONLY reason for this to differ is the limits to arbitrage on call put parity. Now this is a static strategy that has no rebalancing - so the only problem here is transaction costs in buying/shorting the stock. So if you have reason to believe that this strategy is difficult to implement, ...


4

I’m guessing you are finding that your model overvalues Bermudan receiver options and probably undervalues Bermudan payer options. The rationale for this has more to do with supply and demand than theory. That’s because every time a callable bond is issued and swapped, dealers buy Bermudan receiver options, so there’s a huge supply. For Bermudan payers ...


3

Yes it should preserve positivity. However due to numerical noise you may observe very small negative values on the edges of the lattice, that you can truncate to zero. If you solve using Fokker-Planck you may want to start from $t=\delta t$ using a gaussian approximation for the density on the first step, so as to start from a smooth density. An alternative ...


3

No, they are two different but similar things. For two assets $X$ and $Y$: Implied volatility is the function (and similarly for $Y$) $$\widetilde{\sigma}_X:(X_t,K,T,V_t)\rightarrow f^{-1}(X_t,K,T,V_t),$$ that is the inverse of the Black-Scholes formula $f$ (or Black for normal distributions) given a spot price $X_t$ ($Y_t$ for the other asset), a strike $K$...


3

You can replicate the payout of a binary with a put spread with strike prices which are very close to one another. Higher skew makes the further out of the money put more expensive, which makes the put spread cheaper.


3

There are many forms of vega. For var swaps, you can directly differentiate its strike to get a vega that is the sensitivity against the var strike. In LV, you can get a vega by a parallel bump in the entire volatility surface (beware of arbitrage though). People seldom calculate vega as a partial derivative against the implied vol of a certain strike and ...


3

Here is a snip that will create and plot a Heston vol surface import numpy as np import QuantLib as ql from matplotlib import pyplot as plt from mpl_toolkits.mplot3d import Axes3D # Utility function to plot vol surfaces (can pass in ql.BlackVarianceSurface objects too) def plot_vol_surface(vol_surface, plot_years=np.arange(0.1, 2, 0.1), plot_strikes=np....


3

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 explain this. It is merely a nice way to make the SVI parameterization have some stochastic root. If you read Timothy Klassen papers, they suggest that the SVI ...


3

Nice question. My interpretation is via the concept of a risk premium (i.e. risk adversity of market participants). Let me introduce the concept of a risk premium first via US corporate bonds: one can observe that the credit spread of these bonds increases as the credit quality decreases. However when looking at actual historical realized defaults of ...


2

It sounds like you haven't filtered away static arbitrage strategies (such as butterfly spreads, call spreads and calendar spreads) from your data. To keep my answer short and concise, there's a paper by Carr & Madan (2005) that establish the structure of a finite set of tests (a filtering procedure) on your option quotes. When you are left with options ...


2

The relationship between theta and gamma is the Black-Scholes PDE. Let's take normal B-S dynamics with $r=0$: $dS_t = \sigma S_t dW_t$ The pricing PDE for a derivative $g(S_T)$ is (with terminal condition $g$): $\frac{dp}{dt} = \frac{1}{2}\sigma^2S^2 \frac{d^2p}{dS^2}$ Or $\Theta = \frac{1}{2}\sigma^2S^2 \Gamma$ This PDE has a solution (Feynman-Kac Theorem): ...


2

Forward-Start Options are very interesting securities, you can find a lot about them on the internet. It turns out that there is an explicit pricing formula for them in Black-Scholes, the nicest derivation I can find is given in this paper - the pricing formula is given by: As for the forward implied volatility, it turns out there are a few ways to define ...


2

I agree that the above mentioned eSSVI extension is a very efficient and elegant method for calibration purposes. Arbitrage free slices and interpolations can easily be created by making use of the criteria in the papers. It is also described in great detail in the thesis "Extending the SSVI model with arbitrage-free conditions" (google it). It ...


2

For the first question, there have been a number of improvements on SVI/SSVI that are much more flexible than SSVI and also come with easy-to-impose no-arb conditions. See below: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2971502 https://arxiv.org/pdf/1804.04924.pdf Paper 2 builds on Paper 1 and comes with a robust fitting procedure. One thing to ...


1

Yes, it is very very common that implied vol, and particularty that of OTM puts, increases after crashes, even after limited losses. A market maker will have a tendency to increase its volatility on the left side of a smile if a stock price drops, because the risk inherent to that stock is considered higher. Hope it helps.


1

You are competing against thousands of firms, many of them doing this professionally and employing people like the ones you see in the Vola Dynamics link I provided in the comments. So my answer is, no you will not find trading opportunities. I go even further and claim you probably never will (on your own). If you use vendors liker Bloomberg, SuperD, and ...


1

Other than forward start options/straddles, this might help too: there is an interesting analogy between $[S,T]$-forward interest rate exposure related to short $S$-maturity zero coupon bond/long $T$-maturity zero coupon bond position and local volatility related to long calendar spread/short butterfly spread position in this Derman, Kani, and Kamal paper ('...


1

Smile arbitrage is the presence of a butterfly spread arbitrage in a given maturity of your surface, i.e. if your call prices are non-convex leading to an arbitrage. An easy way to spot the arbitrage is to build the call prices and check for strictly convex prices in strike. If you have a parametrisation of the implied volatility $\sigma(K)$ then you can ...


1

The standard way is to fit to a parametric curve and then sample the curve at the strike of interest. In order for call and put vols to match you need to have the correct forward. Finding the appropriate forward presents several challenges. For example, in the case of equity options market a) the underlier can be not in sync with the options' snapshot, b) ...


1

You will want to buy the 10 delta calls and puts vs selling the at-the-money calls and puts when implied volatility is high vs your view on realized. You are selling vol which means that your view is that volatility will be lower than that implied by the markets (your maximum profit will be if the underlying doesn't move (no volatility) and you end up ...


1

just to add to the other answers, the smile is essentially theoretical, in practice since the 87 crash, investors value more downside protection and the demand is higher for out of the money puts, leading to a volatility skew/smirk. also bear in mind that these effects are more pronounced closer to the maturity, at inception the IV curve is essentially flat.


1

The VIX, has a concave shape for its option's Implied volatility.


1

Do you mean actual margins for listed products or clearing on say CME? I am not familiar enough with this but a quick research shows that the CME seems to simply use 3 calls with a +1:-2:+1 configuration. Strikes need to be equidistant in this definition; strike2 – strike1 = strike3 – strike2 Maybe I am wrong but I am assuming you simply mean smile ...


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