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13

I have worked on this topic extensively (pricing and calculating IV in production) and believe can offer an informed opinion. First of all Mathworks - the company that creates Matlab is not a trading firm so you should probably not rely on their advice so much. There are few closed form options pricing models, and all have practical shortcomings. ...


13

Consider a more financially plausible model than Black-Scholes: one where the stock can suddenly go bankrupt due to fraud, and the volatility varies over time. Neither model is perfect, but the new one (call it SVJ) will be "less wrong". Mathematically, we no longer have the Black-Scholes SDE based on a single stochastic generator $W$ $$ \frac{dS}{S} = ...


12

You can directly imply a probability distribution from a volatility skew. Note that, for any terminal probability distribution $p(S)$ at tenor $T$, we have the model-free formula for the call price $C(K)$ as a function of strike $K$ \begin{equation} C=e^{-rT} \int_0^\infty (S-K)^+ p(S) dS \end{equation} Therefore we can write \begin{equation} e^{rT} ...


12

You may want to first broadly categorize volatility models before comparing between them within each class, it does not make sense to compare standard deviation models with an implied vol model. I would broadly classify as follows: Historical realized volatility: Those include standard deviation (sum of squared deviations), realized range volatility ...


11

The Black-Scholes option pricing model provides a closed-form pricing formula $BS(\sigma)$ for a European-exercise option with price $P$. There is no closed-form inverse for it, but because it has a closed-form vega (volatility derivative) $\nu(\sigma)$, and the derivative is nonnegative, we can use the Newton-Raphson formula with confidence. Essentially, ...


11

Setting aside, that it's not pure riskless arbitrage, but rather statistical arbitrage: You can extract the profit by performing continuous delta hedging. If you constantly adjust your hedge position you gain/lose money by delta hedging. Being long option (gamma long), you sell at higher prices and buy at lower ones. Over the course of time you realize ...


11

It seems that you are thinking of the volatility as some sort of standard deviation of your stock price. It is not. In the BS model, $\sigma\sqrt{T}$ is the standard deviation of the log-return $\log(\frac{S_T}{S_0})$. There is no mathematical upper bound to its standard deviation. There is also no mathematical problem with returns being negative either. ...


11

Let $t_0, t_1, \ldots, t_n$ be observation dates, where $0=t_0 < \cdots < t_n = T$, and $\{S_t \mid t \geq 0\}$ be the equity price process without dividend payments. Then the realized variance is defined by \begin{align*} \frac{252}{n}\sum_{i=1}^n \ln^2 \frac{S_{t_i}}{S_{t_{i-1}}}. \end{align*} Note that, for sufficiently small $x$, \begin{align*} ...


9

Implied volatility is the volatility implied by some model. You will have a skew if your model is implying different volatilities for different strikes. However, the realized volatility of the underlying will be the same for all strikes. So, when you are dealing with realized vol, you can drop the "moneyness" axis. Volatility cones can help you compare ...


9

There is no "plain Black Scholes implied surface" because implied volatilities come from options market prices (calls and put). If you had a whole continuum of call prices $C : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$, $(T,K) \mapsto C(T,K)$ you would get a implied volatility function $\sigma_I : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ ...


8

OptionMetrics uses a kernel smoothing algorithm to interpolate the volatility surface. Their assumptions tend to be based on the academic consensus and have become somewhat industry standard, so the real answer to your question may be that there really is no good functional form.


8

The way market makers mark their volatility curves is by using models which 'fill in the gaps', i.e. they will make a price for a given option even if they do not believe this option is going to get a lot of volume. They are still willing to go long/short because they have a strategy to hedge their overall position (i.e. by managing their greeks and ...


8

It is a very simple procedure and yes, Newton-Raphson is used because it converges sufficiently quickly: You need to obviously supply an option pricing model such as BS. Plug in an initial guess for implied volatility -> calculate the the option price as a function of your initial iVol guess -> apply NR -> minimize the error term until it is sufficiently ...


8

Brenner and Subrahmanyam (1988) provided a closed form estimate of IV, you can use it as the initial estimate: $$ \sigma \approx \sqrt{\cfrac{2\pi}{T}} . \cfrac{C}{S} $$


7

Skew is indeed a widely used word and can represent one of the following: Skew(ness) - 3rd standardized moment that represents assymetry of the distribution (olaker metioned it his answer). (Volatility) skew - is observable property of implied volatility surface that can be seen on the market after the 1987 crash. It shows that OTM puts (high demand) are ...


7

The skew is almost always bid for puts on the stock market. When stocks go down, people tend to panic and volatility goes up as a result. Since the puts get more vega when the market goes down, they trade at higher vols. Read up on stochastic volatility for a more in-depth explanation.


6

That implied volatility you are observing was calculated using the standard Black-Scholes model (BSM). As we all know, no model is a perfect representation of reality. The variation (or skew) you observe is a consequence of the model being wrong. Let's think about the implications of the BSM not being exactly correct and everybody knowing that fact. ...


6

It can be shown using a combination of calendar and butterfly that one can lock now the future variance conditionally to the spot being around some specific level (local vol). So if you bought it and it gets realized higher and the spot is there, you get money. if the spot is not there, you are neutral. Another way to look at the dependency of spot level and ...


6

Well as far as I know it is a really hard but interesting question. Asymptotics of smile in the strike direction is not known in a model free way as far as I know. I think I can remember that nevertheless you have upper and lower bounds if you know something about the underlying dynamics and especially the first moment of explosion. I can't remember the ...


5

The "industry standard" for calculating implied volatility is OptionMetrics. Chapter 3 of their reference document contains details of how they calculate all the inputs to the standard Black-Scholes model. They also have a white paper just on dividend yield forecasting, which can potentially be a major issue. However, much of the data they use is far from ...


5

First, note that there are actually quite a few implied volatility curves...I am afraid there is no "the" volatility curve. Right off the bat I can think of The put and call bid and offer curves The put and call midmarket price curves The put and call midmarket vol curves The out-of-the-money bid, offer, midmarket price and midmarket vol curves so that ...


5

At strikes distant from the forward value, pretending that options have some meaningful implied volatility gets kind of silly. Options really have prices (both bids and offers), and we all just translate that to volatility because doing so provides a convenient normalization. Just to take one example, discrete price quoting completely obfuscates the ...


5

N(d2) is near to the probability the option will expire in the money; I have a video showing how d2 is similar to distance to default in the Merton here on youtube. N(d1) is the delta. The technical issue is that N(d2) is a risk-neutral probability; the input in d2 is the riskfree rate, although the theory is more involved. But, if you replace the ...


5

I guess if your American-style option is in no-exercise region, you can use exactly the same bisection method as for European option.The implied volatility will be different, but the method is still the same. See for example, here, chapter 9.3.3. The applicability of bisection method for American-style options is discussed in the book "Binomial Models in ...


5

Implied volatility has very little to do with any particular pricing model, especially not much with BS. BS is a translation tool between prices and volatility, with its own many model deficiencies. I won't get into such model assumptions because my point is an entirely different one. Even the smile/smirk is entirely unrelated to the Black-Scholes model and ...


5

In that white paper itself they quote where it came from: “More than you ever wanted to know about volatility swaps” by Kresimir Demeterfi, Emanuel Derman, Michael Kamal and Joseph Zou, Goldman Sachs Quantitative Strategies Research Notes, March 1999. This is a classic article which you should definitely read if you are trading volatility. While there might ...


5

First of all, may I point out two big misperceptions that you may have: Implied Volatility (IV) is the input to any vanilla option pricing model (not just Black Scholes (BS) that impacts the pricing the most. You can verify this by flipping through the different risk exposures (greeks and higher order sensitivities) and study mean volatilities in such risk ...


5

My try to answer this question with some other questions: Is the BS model right? No. Is it useful: yes. Taking a traded price and the BS Model there is only one input factor that is not given by the market: the implied volatility. It is a measure to compare options across time and strike. Are there better models? yes. Those that you mention: The local vol ...


5

No. Implied volatility isn't a historical measure of standard deviation. Implied volatility is used to relate a market price to some model, be that Black-Scholes or something more sophisticated. Another way to phrase it, implied vol is that single vol input into a model, such that the model reproduces the market prices. Different models will have ...


4

Scott Mixon argues in What Does Implied Volatility Skew Measure that among all measures of implied volatility skew, the (25 delta put volatility - 25 delta call volatility)/50 delta volatility is the most descriptive and least redundant (volatility is Black-Scholes implied volatility). His paper, recently published in the Journal of Derivatives, gives a ...



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