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11

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. ...


10

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} = ...


10

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 ...


9

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} ...


8

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 ...


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.


7

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 ...


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 ...


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

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

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 ...


6

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 ...


6

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, ...


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

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

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 ...


4

VIX is a measure of implied volatility, specifically, model-free implied volatility, a concept originally developed by Demeterfi et. al. at Goldman Sachs in the 1990s. One of my recent questions, How to extrapolate implied volatility for out of the money options?, addressed some aspects of MFIV, and the papers mentioned in the question and answers will give ...


4

Generally, you should ignore deep in-the-money option prices because they are far less liquid (due to their low leverage). That lets you use just the calls on the high strikes and just the puts on the low strikes. For your purposes, you can cap the volatility at, say, the highest mid-market volatility of all puts having both a bid and an offer. This ...


4

You have to ask yourself what the ultimate purpose of this parameterization is. In their case, they imply the "end-goal is martingale pricing or maximum-likelihood estimation", both of which are ultimately about capturing long-period dynamics rather than intraday or interday behavior. For this reason, the fact that intraday variance may, ahem, vary around ...


4

First we must define what we mean by implied volatility. Let $c_{BS}(t,S(t),K,T;\sigma)$ denote the price of the call option with strike price $K$ and maturity $T$ in the Black-Scholes model with the volatility $\sigma$ (emphasized in the argument). Furthermore, let $c_{MA}(t,S(t),K,T;\sigma)$ denote the corresponding price on the market. The volatility ...


4

Almost. If you compare the price of a \$38 put option on a security worth \$40 and the price of a \$28.50 put option on a security worth \$30, then the price of the second option is indeed 3/4 of the price of the first option (assuming the other parameters are all the same). The reason is that the unit of money does not matter. If the price of a put option ...


4

I'm going to go ahead an assume the spread you were looking at involved exchange traded options. As you presumably know, the actual implied volatility on your screen is a number derived from option prices by running the Black-Scholes model "backwards" from quoted option price to volatility. Higher prices imply higher volatility. That last statement is the ...


4

Actually, closing options prices can be downloaded from the exchange, so the data necessary to get the skew is available. If for some reason you don't want to use those closing prices, it is possible to obtain a vol skew from VIX and SKEW. You would need to fit the parameters of a stochastic volatility model (such as Heston's) to the VOL and SKEW data. ...


4

There is a known expansion of implied volatility in moments (I'll find the reference) \begin{equation} \textrm{IV} = \textrm{vol} * (1 + \frac{\textrm{skew}}{6} * \textrm{LMM} + \frac{\textrm{kurt}}{24}*(\textrm{LMM}^2-1)) \end{equation} where log-moneyness is \begin{equation} \textrm{LMM} = ...


4

I suggest you avoid using the VIX for implied vols. Why? One has to consider that the VIX is not simply solely dependant on the dynamics on the S&P 500 anymore because the VIX can be traded via options, etc. Thus many more parameters affect the trajectory of the VIX. The VIX has to equal the ATM option vol because this is where arbitrage assumption ...


4

Options on almost all Korean equities today present flat implied volatility, as well as options on some Japanese equities, especially in 60-90 days maturity. Here how the smile looks for T&D Holdings (ISIN:JP3539220008):


4

You can see concavity in mean-reverting underlying assets where the option tenor is comparable to the characteristic reversion time of the asset. For a geometric brownian motion, all underlying prices are possible, so any mean reversion or other limitation on large changes that might occur in reality would ultimately appear as a skinny tail and negative ...



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