options

Constant Maturity Data

I’ve been asked multiple times why/when I use constantly maturity data for research and modelling. I thought I’d cover it here on my blog since its been a while. I hope to post more in the coming months/future as it has been a good way for me to organize my thoughts and share what I’ve been working on.

Constant maturity (CM) data is a way of stitching together non-continuous time series just like the back adjusted method. It is used heavily in derivative modelling due to the short-term time span a derivative (options, futures, etc) is listed/traded.

What is it and how is it used?

The CM methodology is essentially holding time constant. Various derivative contracts behave differently as time approach expiration so researchers developed this method to account for that and study the statistical properties through time.

I’ll provide a couple of usages.

In options trading, we know that time is one of the major factors that affect the price of an option as it approaches expiry. Options that expire further out in time are more expensive than options that expire closer to today. The reason for this is due to the implied volatility (IV). Researchers who want to study IV across time but not take the expiration affect in to account needs to hold time constant. For example, the study of how IV changes as a stock option approach earning announcements.

In futures, the CM methodology can be used to model the covariance matrices for risk analysis. For example, if you are trading futures under the same root (Crude) across various expirations, this method has shown to be rather useful is managing portfolio level risk.

For cash, the standout examples are the recent proliferation of the volatility ETPs. Most of these products are structured in a way to maintain a constant exposure to a given DTE. They will buy/sell calendar spread futures daily to rebalance their existing position.

How do you calculate it?

I’ve come across multiple ways of doing this. I will show you the most basic way and readers can test out which suit them best. The method I’ve used in the past is a simple linear interpolation given points. So assuming you are calculating IV for 30 days but you only have IV for a 20 and 40 DTE ATM option the equation is:

cm.pt = ( (target.dte – dte.front) * price_1 + (dte.back – target.dte) * price_2 ) / (dte.back – dte.front)

Here target DTE is the expiration you want to calculate. DTE.front should be < DTE.back as the front signifies it expires before the back. This is not the only way; there are other ways just like non-linear interpolation, etc. Carol Alexanders books provide more examples and much better explanations than I ever can!

Hope this helps!

Mike

Advertisements

Vertical Skew IV

Vertical Skew is the shape of the implied volatility (IV) term structure for a single options chain maturity. There is also something called a horizontal skew which is the IV across maturities. The movement of the vertical skew structure has been of interest to me recently when analyzing some of my option positions.

I had a long put butterfly position on with center short strikes 20 points below market. At trade initiation my greeks were as follows: Delta: -56, Gamma: -1.04, Theta: 117.9, Vega -343.8. This is generally what you’d expect for a short vol position. Delta is slightly negative due to the fly being bearishly positioned. My expectation was that for any decline in the markets you’d expect reduced pnl decline due to the fact that the delta will partially offset vega. As it turns out this is not the case, my position actually gained money on a price decline which was opposite to what my greeks are telling me. To understand we must look at IV Skew.

Below are the IVs for RUT September 2015 expiration put options. I specifically picked this period to illustrate the transition from high vol to low vol environment. If you look closely, you will notice that the green line is steeper than the red line. The second graph is the difference between the two line – its increasing. As we go from OTM options to ITM options, the rate of change of IV increases. Another words in our graph, ITM options (right side) will decline more in value (steepens) then OTM and ATM option when IV drops (vice versa for increases in IV – flatten). This phenomenon is not captured by the Black Scholes model as it assumes fixed volatility during the lifetime of the option.

IV

Diff

 

Now how does this help me understand what happened to my position? Well, my right wing (highest strike) put option within my fly is an ITM option. When I was analyzing my position, I assumed that the long put wings of my fly had equal pnl contribution but that’s not the case. My right wing benefited the most from a given IV increase which means the overall negative effect of vega was over-estimated. In fact it may (and I am not 100% sure) be that both delta and vega was working in my favor assuming that the pnl contribution of the long wings was greater than the losses incurred from the short center strikes of my fly. Armed with this information, I think people can incorporate this in to their adjustments and maybe create some ways to exploit this. Open to any ideas!

Pretty cool eh?