learning options from scratch

Moontower Munchies #84

Friends,

A shortcoming I feel in my writing is none of my posts make for a good calling card. There’s no obvious “banger”. A friend described moontower as a “slow burn”. If you read it consistently you feel like you know me (and you do…I don’t have the energy or Huberman-esque levels of productivity to get me in trouble so yea WYSIWYG).

To me, it feels, in a good way, like one long conversation. (I hope the more you read it the more you’re attached to it. A literary stuffie.)

But because I often feel like I’m picking up where we left off, I realize I don’t address some basic info. For example, we cover a lot about options and trading. But unless you are part of the 40 people who first started reading moontower 5+ years ago you’d never know that I haven’t covered the absolute basics of options.

The closest thing to it is a non-exhaustive list of Arbitrage Identities. Even this is not hockey-stick diagram basic.

My recommendation for basics is to check out all the free educational stuff on the Option Industry Council website:

www.optionseducation.org

Within, you shall find the OIC Academy

You’ll need to create a login and password to access the educational materials, but it's free and excellent.

The OIC education initiative is led by Mat Cashman. We’ve had several conversations recently because I’ve been looking for a place to point others for a good foundational education.

Mat has 20 years experience as options as a trader and market-maker. (We started the same year).

He started his career on the trading floor of the Chicago Board of Options Exchange in 2000 and has since traded multiple asset classes across a wide array of exchanges including the CME, CBOT, and the Eurex Exchange. In 2005, Mat helped launch the London trading desk of DRW, a Chicago-based options trading firm, and was instrumental in building the DRW presence in the London trading community. After his time in London, Mat returned to Chicago to join Toro Trading, and was quickly named a partner, overseeing all aspects of their growing U.S. Index Options trading business. During Mat’s time in the options industry, he has always been heavily involved in training and overseeing new traders and has created multiple options education programs along the way to share his knowledge of trading and optionality with people new to the industry. At OCC, Mat is responsible for providing support to a comprehensive options resource center that provides information and education about options. In addition to his responsibilities with OCC, Mat also serves as an instructor of The Options Industry Council (OIC), conducting option seminars and presenting online webinars to all segments of the investing community, including registered representatives and advisors as well as individual investors.

He knows options inside-out. And his mandate is to be a resource for the option industry. You can hit him up on LinkedIn. You can tell him I told you to bug him.

If you’re interested, today at 3:30 pm CT Mat’s hosting a webinar to compare index to etf options. Free signup

The rest of today’s post is not totally basic but it’s very approachable. The most complicated bit of math is an exponent.

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Options Riddle

I saw a familiar type of riddle on Twitter that was directed at fundamental PMs. I gave a lazy answer and later improved it with a better answer after my half-assed-ness gnawed enough at me.

I’ll reprint the riddle and the better answer here but spelling out the steps in greater detail than I did on twitter.

Question:

Estimate the price of a $180 call (20% OTM) on a $150 stock with 50% volatility, 3 months to expiry

150 Call Calculation (The ATM option)

We start by estimating the at-the-money (ATM) call value using:

ATM straddle = .8 * stock price * implied vol * √(Time to expiry in years)

ATM Call = .4 * stock price * implied vol * √(Time to expiry in years)

ATM Call = 0.4× $150 × 50% ×√1/4= $15

180 Call Calculation (The OTM option)

The 150/180 call spread links the 150 call to the 180 call.

Call Spread Value Breakdown

The call spread’s total probability of expiring ITM is around 45%. This is another estimate off the top of my head.

Although you'd expect 50%, option models assume lognormal stock distributions because returns are compounded. Compounded or geometric returns are subject to "volatility drain"— pulling median price expectations lower than the forward price.

You can think of the expected value of the $150/$180 call spread in two parts:

1. The probability that it expires worth its maximum value of $30. This is P(S>$180)

2. The value on average when the stock expires between $150 and $180.
   
   This is 45% - P(S>180)

Computing P(S>180)

Note that the straddle is simply 80% of a standard deviation.

The $180 call is conveniently $30 OTM or .80 standard deviations OTM

We know that 1 standard dev encompasses 68% of a distribution, so at a z-score of +1.0 the one-tailed CDF must be 16%

Spelling that out: 100% - 68% = 32% but we only care about the “up” case when the call is ITM, so we cut that in half to 16%.

Since this exercise is supposed to be all mental math, I’ll guess that a Z-score of 0.80 gives a one-tail CDF of ~ 20%, meaning there's a 20% chance this call will expire in the money (ITM).

We will assume the 180 strike has P(ITM) = 20%

Expected Value Calculation for the 150/180 call spread

1. The case where stock > 180
   
   E(call spread | S>180) = Max value x P(S>180) = $30 x 20% = $6
   

2. Case where S is between $150 and $180
   
   E(call spread | 150<S<180) = Average value of the call spread when s is between the strikes x P(stock between 150 and 180) =

   
   $15 x 25% = $3.75
   
   💡Why $15?
   
   The average roll of a die is 3.5
   
   The average roll of a die given that the roll is greater than ‘3’ is 5. This assumes a uniform distribution over that range.
   
   This same style of approximation works well enough for the call spread. Assuming the stock expires between 150 and 180, the call spread is worth $15 on average. The probability it expires between those strikes is the total probability of the stock expiring higher than $150 which I estimated earlier as 45% minus the probability of it roofing above $180 which we estimate at 20%. So the probability of the stock being between 150 and 180 is about 25%
   
   Hence, $15 x 25%

We sum all scenarios where the call spread expires ITM (ie when the stock is above $150):

Call spread estimate: $6 + $3.75 = $9.75

If the 150 call is worth $15 and the 150/180 call spread is worth $9.75, then the 180 call is worth $5.25

Recapping key bits:

1. Knowing the ATM straddle approximation .8SV√T

2. Guessing that the probability of a >.8 standard deviations ~ 20%

3. Estimating that the probability of the stock going up is less than 50% in a Black Scholes price process (and that at 50% vol that probability is lower than say at 16% vol — in fact the drag is proportional to vol squared)

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In the twitter discussion, a great link from 2012 emerged:

Calculating option prices in your head (7 min read)

The Hardy Decomposition offers a handy way to estimate OTM option prices in your head. By breaking down an option’s price into intrinsic value and a HardyFactor (which depends on how far you are from the strike, measured in standard deviations), you can quickly approximate the time value of the option.

The following comes from the post:

Option Price = Intrinsic + ATMPrice*HardyFactor

The HardyFactor is:

\(\exp\left(-0.5 \cdot d_1^2\right) + \frac{d_1}{0.4} \cdot N(d_1) - \max\left(\frac{d_1}{0.4}, 0\right) \)

Which is approximated by:

\(\text{HardyFactor} \approx e^{-1.4|d_1|} \)

The post recommends remembering just a few values.

d1 is just how many standard deviations you are from the strike.

⚠️Looking at a quant forum it looks like the HardyFactor approximation is for options being priced with the ‘normal’ distribution version of the B-S model as opposed to the more commonly used lognormal version

Revisiting the riddle

If we revisit the riddle, we know the 180-strike has a d1 = .8 standard devs

If we linear interpolate between .5 and 1 we get a HardyFactor = 40%

Option Price = Intrinsic + ATMPrice*HardyFactor

180 call = 0 + $15 * 40% = $6

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My call spread method yielded $5.25

The HardyFactor method (quickly) got us to $6.00

Sound like we have a decent market!

I put into an option calculator:

https://www.cboe.com/education/tools/options-calculator/

Pretty fun stuff. If the OTM call IV is discounted by 1 vol point (so -2% skew vs the 50% ATM IV @ the .27 delta option) then the theoretical call value would be $5.616 - .2575 (ie the vega) ~ $5.36

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If you want more reinforcement on this I wrote a thorough twitter thread explaining vertical spread comprehension in detail.

Finally, Josh’s post from a few months ago was a real-time demo of thinking about bets based on what the put spreads were implying:

Interestingly the options market has priced the odds of the market closing between the June low and the April-May lows in December at approximately 6-to-1. These odds are not much changed from prior to the recent downturn and increase in VIX. This may seem strange but it is due to the increase in put skew that we have seen over the past few weeks. When the VIX was around 12 the out-of-the-money (OTM) puts were not trading much higher in implied volatility than the at-the-money (ATM) puts. This meant that put skew was flat but it also meant that the put spreads were relatively expensive because a put spread sells the OTM puts and buys the ATM puts. When put skew increases the OTM puts increase in price more than the ATM puts lowering the price of the put spreads, all else being equal. If we combine the ~4% drawdown from the high with the increase in put skew we find that the put spreads from the June low to the April-May low have only increased slightly in price despite the notable short-term change in the market.

If the options market is offering roughly the same odds as it was prior to the excess high then that implies that options traders are discounting the short-term change we have seen in the market or put skew has increased enough to offset the price appreciation you would expect in the put spreads from the directional move lower in the market. The 10 delta puts in December are trading approximately 150% of ATM puts as of Friday's close and we have seen real money buying of these puts since the market turned lower. Those puts are used by real money investors as either crash protection since the 10 delta puts in December represent a strike roughly 15% lower from current levels or they are used as a bet on higher implied volatility since those puts respond more to changes in implied volatility than they do changes in the index price. What this means is that the real money flows buying OTM puts has kept the line for the market taking out the June low stable despite the receipt of new information from the short term timeframe and real money investors.

If you can think of market risk as the distribution of potential outcomes across time you can see how the odds must sum to 100% for any given timeframe and as different flows move different parts of the distribution it can result in certain outcomes being repriced by default. What we are pointing out is the difference between market generated information revealed in our cursory overview of the daily and monthly bar charts and the pricing of risk in the options market. We’ve seen flows that have repriced volatility and skew and we have seen a change in the short term market dynamics. The market has repriced volatility and skew but it has not materially changed the line on certain parts of the distribution whose probabilities have increased slightly based on market generated information.

Up to this point all we have done is identify short-term change in the market based on market-generated information, laid out the framework for understanding how to measure the durability of that change from this point forward and observed that the options market has not moved the line on the first milestone for downside continuation. This may mean that the recent sell off is a good buying opportunity or it may mean that the OTM put buying flows are providing a good opportunity to bet on a break of the June lows before the end of the year. This is where the rubber meets the road in terms of investment decision making. Your timeframe, risk tolerance, and goals are all unique to yourself personally or professionally and they determine how you make a decision based on the information we have laid out here.

Let’s leave it there for today.

Stay Groovy

☮️

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