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1

As @Jesper Tidblom already stated in his comment, the quant finance problem is not in inverting observed prices to estimate the implied volatility; this is a well understood and, admittedly, simple problem these days. Finding (model) prices for very complex derivatives products is a potential field for applied ML. Especially in counterparty and market risk ...


1

I could also add that options at more extreme strikes can be very insensitive to the volatility. Unless you use a ridiculous number of decimals for the option prices in that situation, those prices would then look the same on the market screen, while the volatilities for the options might differ more substantially. So for this numerical reason it is also ...


4

To add to @Jan Stuller answer , ATM options are pretty close to linear in volatility in the BS model (and exactly linear in the normalized Bachelier model). Options away from the strike are positively convex in volatility (note that OTM vs ITM makes no difference , just distance from strike). The exception is that in BS at very high lognormal vols, there ...


9

I think it's interesting to look at this problem graphically also. I get a different answer, depending on whether the option is ITM, ATM, or OTM. In the plot below, all options have 1-year expiry, rates are 0.01 and spot is 100. The ITM call has strike 80, the ATM call has strike 100 and the OTM call has strike 150. I added a linear function (y = 40* vol) ...


3

The reason for people quoting in IV is because spot is moving! If someone asks for a quote in 50 delta SPX, that will move by the millisecond. But if you just quote it in vol terms then that is pretty static. It just makes getting a trade done simpler.


4

IV quotes let you compare prices of options on the same underlying with different strikes, expirations and types. It is hard to say if 2.50 for 200@45dte is more or less than 3.70 for 150@90dte. Their implied volatility is directly comparable. Some claim that you can also compare IVs for options with different underlyings but I’m less sure about that.


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 ...


0

Interactive brokers have it. https://interactivebrokers.github.io/tws-api/tick_types.html You need data subscription.


8

For a vanilla call option, the price of the option increases monotonically with implied volatility. For functions like this, newton's method works really nicely, and it's not very sensitive to the choice of starting parameter I've borrowed an image from this webpage, detailing the technique: If you think of the red line as being the price of your option ...


1

In short, because 1) the assumption of lognormal returns does not hold in real life--the markets have more skewness and kurtosis and 2) writers of protection want to be compensated more for writing insurance on low probability but high cost events.


1

Bear in mind that the IV you see quoted is Black Scholes IV. The only takeaway can be that the BS model is not the correct model to ACCURATELY price options. Differing IVs are the "fudge" to get better pricing and that option quoting (at the market maker level) really occurs through IV and is just expressed as price. When you look at the ...


3

Pat Hagan describes this well in the famous SABR paper Managing smile risk. An approximate relation given in equation (B.64) reads $$\sigma_N \approx \sigma_B \frac{f-K}{\ln f/K}\left(1-\frac{\sigma_B^2 T}{24}\right),$$ where $\sigma_N$ is the normal (or Bachelier) vol, $\sigma_B$ is the Black-Scholes volatility, $f$ is the forward price, $T$ the option time ...


0

This classical article by Patrick Hagan might help you: http://janroman.dhis.org/finance/Norm%20-%20LogNorm/Hagan%20Normvol.pdf


1

Martin, I believe this is your research paper that you mentioned? https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3644295 For my time-series class at DePaul, I chose for my group project to study the relationships between the SPX, VIX, VIX9D, VIX3M, etc. One of the things we're wrestling with is trying to match the realized volatility of the SPX with the ...


0

"I believe 1st shows k days volatility, since it will be equal to 0 if spot came back to the same value . However, what does 2nd (total variance) actually represent in this case ?" Your first formula is simply absolute % change over k-days. This is sometimes used to compare against the breakeven on an options position (e.g. a straddle) if you aren'...


3

For option pricing in the classical Black-Scholes model, you assume the underlying stock follows Geometric Brownian Motion: $$S_t = S_0 + \int_{h=0}^{h=t} S_h \mu dh + \int_{h=0}^{h=t} S_h \sigma dW_h = S_0 \exp \left( \mu t + 0.5 \sigma^2 t + \sigma W(t) \right)$$ Take the log of the solution above and you get: $$ \ln\left( \frac{S_t}{S_0} \right) = \mu t + ...


3

As stated on the VIX9D page (see the link from noob2): The CBOE S&P 500 9-Day Volatility Index SM (VIX9D) estimates the expected 9-day volatility of S&P 500® stock returns. Similar to VIX®, VIX9D is derived by applying the VIX algorithm to options on the Standard &Poor's 500 Index (SPX options), but it uses SPX options with expiration dates that ...


1

I also got a bit confused about understanding intuitively what happens in the limit with the Black-Scholes Call and Put values as the volatility goes to infinity, but after analyzing the derivation of the formula it becomes clear. My confusion was about why the contribution of the $-K$ term disappear in the limit. My faulty reasoning was a bit like this : If ...


0

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

Premium decay is non linear and speeds up as time passes. Therefore, for other than high delta options, the cost per day will be less for longer dated options. Simply determine the time premium and divide by the number of days until expiration. If you're buying very high delta calls, the time premium won't matter much. Where you'll probably get ripped is ...


1

Deep ITM options will not have much time value left and will experience very little time decay. They will trade like the stock as the delta will be very close to 1. All else being equal (same strikes) one week options will have faster time decay than one month options. However, one month options will have rapid time decay as well. By using one week ...


1

For what date is the chart derived? One definition for implied volatility skew is: (25 delta put implied volatility - 25 delta call implied volatility) / 50 delta. Can you test to see if this calculation for the options maturities is consistent with the values on your graph for the date in question? With respect to the vol beta, this appears to be a ...


2

$(p_t,c_t)$ are respectively related to the put/call slopes of the total implied variance, not variance $$ w(k,t)=\sigma^2(k,t) t $$ Under SVI $$ w(k) = a + b \left(\rho(k-m) + \sqrt{(k-m)^2 + \sigma^2} \right) $$ such that $$ \frac{\partial w}{\partial k}(k) = b \left( \rho + \frac{k-m}{\sqrt{(k-m)^2+\sigma^2}} \right) $$ and $$ \lim_{k \to \pm \infty} \...


1

Echoing @noob2 's comments. Additionally, one of the things you might want to be aware of is there is a time to maturity difference between VIX and your calculation of historical volatility. While you are using a constant time frame (30 day) for your volatility calculation, VIX utilizes the near term options contracts for its calculation. As options have ...


1

The VIX methodology is rather involved but in essence it uses all of traded near the money (as it turns out they need not be near the money as long as they are within a continuum of traded options strikes), near dated S&P 500 options to arrive at the VIX (A detailed explanation of the calculation can be found on the CBOE website, http://www.cboe.com/...


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