8

One possible reason could be jumps. Over the longer maturity, there could be more jumps so the jumps average out in a way; whereas over the short term, a jump can make a bigger difference and hence the risk of jump increases demand. This reasoning is used to justify Stochastic volatility with jumps models in some books.


7

The reason is that, as shown in Proposition 2.1 of that paper, in order to exclude static calendar arbitrage, the total variance has to be strictly increasing in forward moneyness. See also the below to links for details on this result. The intuition is that for European options, only the distribution of the terminal spot price is relevant. Furthermore, $F_t^...


7

I have also currently started to learn about the subject. This is some of the material I have encountered: Many people recommend the book "The Volatility Surface: A Practitioner's Guide" by Jim Gatheral. It is a standard reference in the area (even though I personally found it a bit confusing and a bit unclear at some parts). The author also have ...


6

You are an investment bank. You trade a multitude of vanilla and exotic options. You want to make sure the option prices you quote as a client are arbitrage-free with respect to liquid option prices quoted in the market $-$ and also consistent between the different trading desks within your bank. Basically you want to avoid other market participants taking ...


6

Maybe it would help you to think of it the following way. The strike $\sigma^2(T)$ of a fresh-start variance swap of maturity $T$ in the Heston model only depends on parameters $(v_0,\theta,\kappa)$, see related question here. More specifically \begin{align} \sigma^2(T) &= \Bbb{E}_0^\Bbb{Q}\left[ \frac{1}{T} \langle \ln S\rangle_T \right] \\ &= \...


6

Long story short: yes both might introduce static arbitrage opportunities if performed blindly. There are 3 types of static arbitrage to consider: Calendar arbitrage: total (implied) variance should be an increasing function of time for fixed (forward) moneyness. Vertical arbitrage (or call spread arbitrage): call spreads should have a positive price ...


5

Let's take a step back to look at what implied volatility (IV) really is. If we know the price of a call option, the interest rate (we can use the spot rate corresponding the option maturity) then Implied volatility is that level of volatility that will result in the option price when putting into the Black-Scholes formula for a call option value. If we ...


5

This is not quite true, in either direction. If you have an arbitrage free implied vol surface, you might not have a well-defined local vol surface. An example comes from a discrete model. Consider a spot dynamics where the spot is a martingale that jumps up or down by integer amounts. The spot distribution is discrete, with zero density in between ...


5

I'll answer both of your questions in one go: Your ideas are correct. If the Black-Scholes model was true, the implied volatility surface would be flat but it is not in real life. Thus, the geometric Brownian motion as stock price model is misspecified and we need more sophisticated models (sto vol, jumps etc), in particular if we want to price more ...


4

This is a big industry, but here are some alternatives(as usual, the best choice depends on purpose and desired accuracy): Fit a quadratic in delta space: $\sigma_{\Delta}=a + b \left( \Delta - \Delta_{ATM} \right) + c \left( \Delta - \Delta_{ATM} \right)^2$. When you have fitted this equation, you can input delta, and the function will return ...


4

You can convert the implied volatility to local volatility using this formula: $\sigma^2 \left(T,y\right)=\frac{\frac{\partial w}{\partial T}}{1 -\frac{ y}{w} \frac{\partial w}{\partial y}+\frac{1}{2}\frac{\partial^2 w}{\partial y^2}+\frac{1}{4}\left(\frac{ y^2}{w^2}-\frac{1}{w}-\frac{1}{4}\right)\left( \frac{\partial w}{\partial y}\right)^2}$ Where y is ...


4

In the end I found that fitting a SABR smile to each tenor (borrowing a result from this answer) was sufficient to build a local vol surface that was smooth and well-behaved enough to build a variance surface worked nicely. I also fitted a Heston model to it, and the two surfaces do look fairly similar. Here is the final code and the fits generated (the long ...


4

If you use constant strike, the moneyness changes as the underlying changes. Out of the money equity options tend to trade at a premium to at the money options (smiles/skew). Therefore, the moneyness is used to take into account the movement of the underlying. Yes, if you are trying to price an option with a strike whose moneyness is in between the ...


4

First off, there are different types of moneyness one can use when constructing a volatility surface. Each have their own advantages. Absolute-moneyness: using absolute spot-strike comparison as a measure of moneyness. ATM would correspond with S=K. This has a simplistic interpretation when looking at option payoff diagrams at maturity. Simple-moneyness: ...


3

Here are my thoughts. Let's take for example the pair EURUSD and USDEUR. The fx rate for EURUSD will be $X_t$ and USDEUR $1/X_t$. Now assume that $d{X_t} = \mu{X_t} dt + \sigma{X_t} dW_t $ then thanks to Ito's Lemma you have $d\bigl(\frac{1}{X_t}\bigr) = 0dt -\frac{1}{X_t^2}dX_t +\frac{1}{2}\frac{-2}{X_t^3}(\sigma X_t)^2 dt = -\frac{1}{X_t^2}dX_t -{X_t}\...


3

The answer in Implied Vol vs. Calibrated Vol as suggested by noob2 is more complete. But it may be slightly misleading in your last example. I've been a vanilla option market maker for ten years, so I'll chime in on what I would mean by that. If a market maker says he's calibrating his vols to the market it means exactly what you're saying: getting prices ...


3

Yes, that's what we wish to see from the correctly-specified model. Now, let me try to answer your 2nd and 3rd questions together as they are based on the same confusion. There are two different concepts: model-implied volatility and model-implied BSIV (Black-Scholes Implied Volatility). I think you are confused because of mixing them up. So yes, people ...


3

I tried something along these lines in Quantlib python a few weeks ago. Slightly more simple compared to your approach I think: start with a standard delta quote convention for FX vols (10D puts, 25D puts,ATM,25D call, 10D call) calculate the moneyness of the options to obtain the strike set (this will be a large strike set since each option maturity will ...


3

This is a really good observation. Warrants issuers systematically overprice their products compared to the listed options market. Different issuers will show different degrees of overpricing in similar products. This depends on a few parameters but the main one is the respective issuer's outstanding position in that or similar products. Contrary to listed ...


2

I just checked this one. I saw the picture you are referring to and I think the frown is not real. Much of it looks like noise created by wide bid ask spreads and extremely high implied vols. The data here is very poor and the options are spread very far apart. The underlying is $4.60 and the options are struck a dollar apart. That would be like SPX ...


2

In black-scholes world, correlation between volatility and spot is zero. From the above details you can estimate how the implied volatility for a given option (note options have FIXED strikes) might change for a given move in spot. If when stock goes up, the option's implied vol goes down, this would be a violation of the black-scholes model (which scenario ...


2

The skew/smile of long term options is flatter than short term options, the reason for this can be explained in several ways. The Vega of a shorter-dated option is smaller than a longer-dated option. Vega is the dollar value of a 1% change in implied volatility. i.e., 30d ATM option, $65 strike, .31 ivol = VEGA .07 30d 25 delta option,31% ivol = ...


2

You should use the out of the money options on either side of the curve as they carry the most information about the optionality part of the price.


2

A good reference book for FX conventions can be found from the book Foreign Exchange Option Pricing by Iain Clark. The 25% delta risk-reversal quote $\sigma_{25-RR}$ satisfies the system of equations \begin{align*} \begin{cases} \Delta_{call}(k_{25-call}, \sigma_{25-call}) &\!\!\!= 0.25\\ \Delta_{put}(k_{25-put}, \sigma_{25-put}) &\!\!\!= -0.25\\ \...


2

If your barrier is american and your market has any sort of volatility skew then trying to map some sort of moneyness measure to the vol surface will almost certainly fail. That is due to the fact that barriers are sensitive to forward skew, and you need a model to capture that as a continuum of vanilla option prices sadly tells you nothing about the forward ...


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 ...


1

I think your list covers the approach quite well. What I would add to point 3(i) is that there is a (generally positive) spread of implied vols to realized vols. In this case what might be useful is to combine point 2 with point 3(i) i.e. ascertain the implied/realized spread from the proxy market and apply that to the realized vol obtained from your ...


1

The values of implied volatility are deduced from option prices on the market. It is a market view of how the Black-Scholes prices should be modified due to various factors like fat tails in the probability distribution, supply and demand etc. If the prices are higher than those from BS, then sure the prices include a premium. The implied volatility ...


1

The concept behind showing volatility vs expected return is that a risk averse investor will wish to minimise risk, and maximise return. However, how good a proxy is volatility for risk? Given a normal distribution, risk and return (sigma and mu) alone will suffice. But especially for non-normally distributed returns (as alluded to by the mention of higher ...


1

Swaption vol can have 3 dimensions: option expiry, underlying tenor and strike. In your example, if nothing is said, then it's probably ATM (at the money) volatility which means it's the vol for a Swaption with a strike equal to the forward of the underlying. So if you only have a surface, and not a cube, you probably don't have exactly the information ...


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