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Option Pricing

Discussion in 'Options' started by Average Joe, Feb 19, 2016.

  1. Average Joe

    Average Joe Guest

    Hello,

    I am relatively new to options trading. Things are relatively hazy right now with various terms thrown around when I read stuff.

    For example, CEV (Constant Elasticity of Volatility). Are there any laymans explanation in this and how it is calculated and utilized in option pricing inside OptionVue? I'd love to implement it in a programming language to get a feel for how it is used.

    Additionally, in OV it says it uses a Variable Volatility Model which is based on Horizontal Skew. How is this implemented?

    Thanks,
     
  2. Average Joe

    Average Joe Guest

    Additionally, how are True Delta and True Gamma calculated? Is there an equation?
     
  3. Kevin Lee

    Kevin Lee Well-Known Member

    Hi Average Joe,
    Let's see if I can help with your questions :

    1. CEV - when the underlying moves up or down, the IVs tend to go up or down as well. For example - for equities, the usual case is underlying and IV goes in opposite direction, ie underlying down, IV up and underlying up, IV down. Not all underlying behaves that way though. Some has direct relationship between underlying and IV. CEV tries to capture this behavior into the pricing and greek modeling. OV will calculate the CEV number based on historical data and update the settings automatically. Different underlying will have different CEV numbers.

    So in layman terms, CEV is the factor that estimates how much the IV curve will move vertically up or down as the underlying moves.

    2. Horizontal skew is a bit more complicated. The basic concept is that the sensitivity of IV depends on how far away is the expiration. IVs of shorter dated options are usually more volatile than longer dated ones. Therefore, you don't see a single CEV number in OV but rather 15 in total. If you look at the right side of the table, you see buttons "1", "5", "10". Representing 1, 5 and 10 days. That's trying to model the IV sensitivity when market makes a fast move in 1 day or moderate move in 5 days or slow move in 10 days. Example - if market moves X pts in 1 day, IV will jump a lot more than if the market moves the same X pts in 10 days. That's what these 1, 5, and 10 attempt to model. The 30d, 60d etc... are DTEs. Therefore, in total, there 3 sets of 5, or 15 CEV values.

    upload_2016-2-20_2-46-32.png


    The following videos go into a lot more depth on this topic :





    Go to 49 mins and 1 hr mark for a good explanation of how IV "climbs up and down" the VI smile as market moves up or down. This is a very critical concept to grasp in order to fully understand the IV modeling.


    3. Lastly - True Delta. First of all, let's understand the process of calculating option greeks. OV starts with option price. That's given - it's a traded value obtained from market. Given Bid and Ask, they can calculate mid price. From mid price, they derive MIV. The MIVs of all the strikes will form a curve, and usually that's a jagged curve. To smoothen it, OV uses a "Best-Fit" algorithm to create a Projected IV curve, which you can see from "Vertical Skew" tab - "Skew Graph". Then apply the CEV factors to the Projected IVs, you can calculate the "True IV" for each strikes. Depending on how far out in time you project, the "True IV" for any individual strike will turn out to be different (again based on the 15 CEV values). Lastly, using the True IVs as input, you can now calculate the future dated option prices and the option greeks. From there you can then draw the T+n line.

    Delta that is calculated this way is called True Delta. What's not "True Delta" then? That's if you assume static MIVs and project forward without any curve fitting nor CEV factoring.

    I hope my understanding is correct. If I mis-interpreted anything, please correct me.

    4. Unfortunately, OV does not take into consideration the change in the shape and the slope of IV curve. They assume those are static, but we know they are not in the real world. That's the primary reason why the T+0 line can sometimes slope down or change its shape all out of a sudden.

    I think OV is working on improving this, but my view is that's very difficult to model. There are some general patterns of how the shape and slope change, but sometimes they change unpredictably. An error in modeling will cause significant issue as we experienced couple months ago. Ironically, having a better "best-fit" curve, which was basically what OV did in the problematic versions, actually resulted in more inaccurate greeks and projected prices. That has to do with the fact that OV assumes an IV curve with static shape and slope.

    This is an area of interest for me to research as I believe a better grasp of IV skew change can help in our trade. Anyone who has similar interest, let me know. Perhaps we can jointly figure this one out.
     
    Last edited: Feb 20, 2016
    Bruno, RayM, GreenZone and 5 others like this.
  4. Balazs

    Balazs Guest

    Hi Kevin.

    Great response, as usual ;-) Just to comment on your last lines. I am also very interested in IV skew. I have started (only a week ago), to start making a database of IV skews for the first three monthly esxperation cycles (I don't keep track of weeklies). Basically, Looking at my end of they ATM, 10, 20 and 30 points below the market 50 points wide butterflies and look at how the skew relate to the short strikes. I express this the way that Uwe built his tool for IB costumors. The skew of the long strikes will be expressed in a percentage of the Short strike. So if the IV of the lower long is 27,5% and the short strike has an IV of 24,3 then I express the skew as 113,1%.
    I also record the VIX of that moment and the butterfly price. In this way, I hope to get a grasp of, how prices relate to a certain VIX and with what skew. Also in what environments what skews are typical and how they affect the butterfly prices. Is there anything else that would be interesting to record?

    Perhaps we can exchange information.

    Balazs
     
    uwe likes this.
  5. Bryan Doyle

    Bryan Doyle Well-Known Member

    Kevin, great explanation and attachments! Thanks.
     
  6. Kevin Lee

    Kevin Lee Well-Known Member

    Hi Balazs

    I have been monitoring the IV curve for a couple years now and have developed some feel of how the skew changes. But my experience is limited to RUT and my data collection is not consistent nor systematic. I used Excel DDE in the past. But Excel is kind of limited. After a while, there'll be too many rows and excel vb is not very stable. Some times it just crashes and data collection stops. Now I'm using the OV export function to collect data instead. Again this is good for day to day snapshot analysis to help my trading but not good to analyze long term trend.

    So, i think the first step is really to build a consistent data base of option IV and pricing data. Perhaps for multiple indexes, equities and futures options so that we have flexibility to expand our analysis in the future. I look forward to partnering with someone with the technical skill to be able to do that. I did consider buying historical options data, but what I saw were all too costly.

    I'm not at the stage to design better trades out of this knowledge yet. That'll be the eventual goal, but I think l'm probably several steps away.
     
  7. Average Joe

    Average Joe Guest

    Hello Kevin,
    Thanks for the detail reply. It really helped with my understanding of how option models gets put together.

    In your above response, I follow everything up until you mentioned applying CEV factors. Is there an equation for this? (like do you multiply the derived numerical factors to the projected IVs)? I am unsure if I am understanding correctly but does CEV moderate the rate of change of IV for a given change in the underlying? I think I get the basic idea, just unsure how it actually gets done. I found a matlab function attached that seems to calculate something called Constant Elasticity of Variance (not volatility). Are they the same?

    With regards to modelling the IV curve I am not directly aware of any particular models used in options (I am a noob :() but from futures, we use to model the forward curve via PCA. The way we used to do it is taking constant maturity futures contracts and estimate the cov matrix. From there we do an eigen decomposition to derive the eigenvectors and eigenvalues. Each eigenvector can be thought of as beta exposure which we tend to attribute risk to. We tend to focus on the first 3-4 vectors as they explain most of the underlying processes variance (cumulative eigenvalues). The first vector/factor explains parallel shifts (up or down) of the forward curve since. The second factor explains the rate of change (slope) of the forward curve and the third explains the curvature. I'm thinking there may be something useful here, but not sure how it can help with what you are interested in.

    Anyhow I am totally new to options. M3 has been something I am poking around with but am itching to understand how stuff gets put together in option pricing/modelling. I don't like black boxes.
     

    Attached Files:

  8. vega4mike

    vega4mike Well-Known Member

    I am no expert in this field but isn't the CEV just another implied volatilty model of which there are many. It just so happens to be the one OV use. I'm not sure which one TOS use & I'm sure the institutions use various other ones, a quick google search will produce quite a view. Which one is the best, well, even the academic literature on the topic does not have a consensus view.
    Here's another one which is popular amongst practioners, ---the Volga Vanna model. I've attached a paper & a spreadsheet, which I found during my casual search.
    I am not an expert on this & to be honest, I did stuff like this back in my academic days, my brain now just goes for the simple stuff.
    My point is I wouldn't get hung up on the CEV model, its just another implied vol model.
     

    Attached Files:

    Last edited: Feb 20, 2016
    Bruno, uwe and Kevin Lee like this.
  9. Kevin Lee

    Kevin Lee Well-Known Member

    I'm not exactly sure about the formula to translate the CEV numbers to the actual change in IV. But I know that CEV < 1 means IV and underlying price have negative relationship. CEV > 1 means positive relationship. Perhaps someone from OV can answer that question.

    On Constant Elasticity of Variance - I don't think that's the CEV in OV, despite the name. Constant Elasticity of Variance is a generic mathematical concept that describes the volatility of the stochastic component of the Brownian Motion model. OV's CEV as I understand is empirically derived. Again... I'm not 100% sure but this is my understanding.
     
    Last edited: Feb 21, 2016
  10. Capt Hobbes

    Capt Hobbes Well-Known Member

    A bit more precisely, CEV is an extension of the original Black-Scholes model in which the variance of the stochastic term changes with the price of the underlying, instead of being constant as in the original BS. The SDE of the model is shown here:

    https://en.wikipedia.org/wiki/Constant_elasticity_of_variance_model

    Defining it that way with gamma in the exponent makes the variance a function of S, with constant elasticity equal to gamma. Hence the name. My guess is that the CEV parameter in OV is the gamma from the formula, with values selected to fit historical data.
     
  11. Average Joe

    Average Joe Guest

    ahh I think I get it. so essentially, OV estimates the gamma factor (wonder how they do it with historical data...linear regression??).

    So lets me try and summarize the procedure. Derive MIV from bid and ask for select options. Options must meet some criteria (listed in the help file page on OV) before it is selected. With MIV for selected options, we smooth it. With the newly smoothed IVs we can derive the greeks with the CEV black scholes model. Hope this sounds right.

    To take in account horizontal skew, we calculate different CEV numbers for different expirations. So when we calculate the greeks above, we choose the cev gamma number that is closes to the options expiration.

    This is pretty cool. I am in the process of replicating this in the R language. Will report back when I have something solid.:)
     
    N N likes this.
  12. N N

    N N Well-Known Member

    Check out www.quantgo.com - they might have more affordable historical equity options pricing.
     
  13. Average Joe

    Average Joe Guest

    Yea I was considering them once before but then the problem is you don't get to download the data. You work out of their amazon EC2 environment (which you gotta pay yourself) and then you upload your program on to that environment and query their data and pipe it in to your program for analysis. They have R and python pre-installed I remember.

    I mean you can download option data from OV. It works and I've been using it for my analysis with decent results.
     

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