This post starts with a response to a reader question but leads to a deeper question — do arbitrage-free prices present opportunities to fundamental or directional investors?

One thing I’ll be doing more of is sharing my answers to reader questions. Here’s one that channels a topic that eternally confuses option investors when they learn about skew (emphasis mine):

In your excellent article Lessons from the .50 Delta option, you wrote that by bidding up the put skew, the market makes call spreads more expensive. Combining this with your insights on the deeper understanding of vertical spreads, implied distributions, and thinking of spreads as odds, I was left wondering how the steepness of the skew relates to outright buying or selling options.

Here’s my question: Let’s say the SPX Dec24 call skew (100-105 moneyness) is descending. If I were to buy a call spread with 100/105 moneyness, I’d probably measure it as slightly more expensive than a 100/95 put spread. This aligns with what you’ve been writing. However, what I don’t understand is: if call spreads are more expensive (because the 105 vol is lower than ATMF), wouldn’t it be better to buy 105 calls outright due to the low vol? In my mind, this contradicts the perception we get from the spreads. The market seems to favor upside scenarios and bid up call spreads, yet at the same time, outright calls get cheaper?

Could this have something to do with IVs, deltas, or VRP?

It’s such a great and common question.

My response:

I know it feels like a contradiction. I’ve even framed it that way because it grabs your attention…it really feels like one!

But ultimately it’s not.

Think of option skew creating vertical spread prices that imply a distribution which “corrects” the Brownian motion assumption — namely, that the market is positively skewed but more likely to go down than up.

This is exactly the opposite of empirical results…the market is negatively skewed and a favorite to go up!

The upside call is cheaper because there’s less positive long-tail skew than Black Scholes assumes but the call spread is more expensive because the stock is more likely to go up.

We shift probability to the right but truncate the magnitude of the upside. It’s like betting on a favorite in sports…you’ll probably be right but the upside isn’t as large.

The main point is that the distribution is like a sculpture — the volatility skew moves some clay from the right tail to the left tail, and shifts the whole sculpture rightwards a smidge.

The paywalled Thursday Moontower posts generally dig deeper into investing concepts practically. They’ve been very option-centric lately. However, the most popular one I’ve written recently is not directly about options. It drove 20 paid subs which is an unusually high amount.

[Incidentally, the topic of paywalls, monetization, and so forth is something I’m happy to discuss. In fact, I have a backlog item to come up with a pricing model for “Should I paywall content or not” mostly to itemize most of the major inputs into such a model and their sensitivities or greeks. In any case, the topic of monetization feels under-discussed at least from my aperture. I wonder if that’s because people are hesitant to share. In any case, I see no advantage in being opaque on how to think about monetization. At the end of the day, some amount of what I write is paywalled. A lot of effort goes into but I don’t subscribe to Marx’s Labor Theory of Value — if it’s worth paying for, it’ll get paid for, and if it’s not it won’t.]

Here’s the post, with the paywall removed 🙌

🔎Volatility Depends On The Resolution

Despite the word “volatility” in the title, it’s more germane to the topic of risk broadly than options. It’s not a long post, but it’s a lot to ponder for any investor. The heart of it can be summarized in a line:

Realized volatility depends on sampling frequency

This concept has such a broad appeal and because of my bit above in Money Angle about realized distributions vs theoretical lognormal distributions, it seemed like a good one to unlock.

The substack post is a condensed version of the whole post which includes lots of charts.

For example, charts that compare realized daily returns vs what is expected theoretically:

The real-world distribution includes:

1. Fatter tails (note that greater than 3 st devs is one giant bucket so that’s why even the Gaussian expectation is slopes back up a touch)
2. A higher peak (more small moves than theory predicts)
3. Positive drift (in theory the drift is approximately [RFR-div yield]/365…a much smaller number than the actual drift has been. Incidentally, people really struggle with the Black-Scholes assumption that a stock has no expected return premium — but that’s actually not the assumption UNLESS you are talking about a specific context — pricing options as a replicating portfolio. Financial Hacking has a great clarification on this.)

The options market adjusts to this empirical reality by:

1. Increasing downside put IVs or put skew. This extends the distribution of the left tail but also increases the implied probability of a positive stock return by pushing up the value of ITM calls and therefore call spreads. This shifts the center of the distribution to the right.
2. The wing options on both the calls and puts can smirk up increasing the mass in the tails. In price space, this pushes the value of the OTM options relative to the ATM straddle which increases the value of the butterfly (the wings up, relative to the “meat” of the fly). If you think a stock is going to have more small moves than the Gaussian distribution of logreturns predicts then you want to buy a butterfly that’s short the ATM straddle — that is exactly the structure whose value increases. 

The volatility skew creates option spread prices that conform to empirical actuarial odds. They “correct” the assumptions baked into the underlying Black-Scholes distribution.

One of the most provocative bunny trails from this conversation is how a long-term call option price that is priced using the RFR for drift will look wildly cheap (and the put expensive) to an investor who assumes higher rates of return because the stock is risky. But if they bid the calls and offered the puts with their return assumptions in the model they would offer risk-less arbitrage profits to option traders would simply do “conversion” strategies.*

If this intrigues you, check out the Warren Buffet section in Real World vs Risk-Neutral Worlds. An option that looks cheap to a fundamental investor focused on the near-term can simultaneously look expensive to an arbitrageur who has no opinion about the underlying but is in the business of squeezing profits from replicating that option for less. Both the directional investor and arbitrage trader can win.

(The loser in a zero-sum paradigm is all the noise traders flipping the stock back and forth in the interim to the delta-rebalancing option trader).

*conversion example

Stock = $100

Interest rate =3%

Theo value of synthetic future (ie long 100 strike call/short 100 strike put) = $103

The call will be worth $3 more than the put. C – P = $3.

Buying the call and selling the put is the same as “buying the 100-strike combo for $3 which is equivalent to buying the synthetic future for $103 — why? Whether you exercise the call in a year or get assigned on the put, you will have paid $100 for the stock. Plus you paid $3 today for that structure, which is why you can say you paid $103 for the stock to be settled in one year.

Imagine a fundamental investor who thinks the stock will be worth $108 in a year (8% RoR instead of RFR) and is willing to pay $5 for the combo ie $105 for a 1-year future on the stock.

They would be buried by arbitrage traders who would sell the combo at $5 and buy the stock for $100 to hedge. At expiry, the arbs will either be assigned on their short call or exercise their long put, guaranteeing they sell the stock at $100. They already bought the stock today for $100 and will have to pay $3 interest to hold it for a year. So at expiry they will deliver the stock to the fundamental investor (selling the stock they bought for $100 at $100…a wash).

The arbitrageur will have paid $3 in interest…but remember they collected $5 in option premium via the combo. Net result: $2 riskless profit

The market hoovers up riskless profits so the arbs will compete to sell the combo to the fundamental investor all the way down to a price of $3 where the combo is fairly priced compared to the RFR.

It doesn’t matter what the fundamental investor or Warren Buffet thinks. If they pay more for the calls relative to the puts than the cost of carry warrants, they get arbed. This has nothing to do with stock distributions, volatility, or any theoretical nerd stuff. A 10-year-old can draw the cash flows on a “conversion” arbitrage.

A “reversal” arbitrage is the opposite…if the combo is too cheap, say $1, you can buy the synthetic future (buy call/sell put), short the stock and at expiration your short stock will be covered by the stock you buy via the future (ie exercise the long call, or get assigned on the put) and the $3 interest you earn on the short stock proceeds covers the $1 premium you paid for the combo leaving $2 of arbitrage profit.

Practically speaking, reversals and conversions, the first arbitrage trades you learn about in options, still have risk.

• There’s pin risk if the stock expires on the strike and you have to guess if you will be assigned on an option or not.
• Interest rates can change from your initial assumption — the overnight interest rates that you pay for long stock or collect as a short rebate change. They can also change dramatically if a stock becomes hard-to-borrow.
• Dividends can be announced, increased, or cut. Higher divs increase put prices relative to calls so if you have a reversal position and a stock announces a dividend your long combo will drop by the dividend
• In other words, a reversal position is long rho and a conversion is short rho (it’s rooting for either interest rates to drop, the stock to become hard-to-borrow, or the stock to pay a larger div)

Friends,

Today we jump into another large learning post including a chart book and commentary.

In this document, you will:

• learn to compute realized volatility
• appreciate the difference between the sampling window and lookback
• learn to annualize realized volatility
• examine market data to see how realized volatility varies with the sampling period

[bong rip]

Let’s cook…

I was able to fit the whole post except the chartbooks in the body of this email but here’s the link to the full post. It’s better formatted and colored there. Again, the chartbook is too large for substack.

🔎Volatility Depends On The Resolution

Introduction

These short posts explain the relevant computations. If you have ever computed a standard deviation, you are 99% of the way there.

➕Computing Realized Volatility

It’s common to compute realized volatility or the standard deviation of returns.

• The formula commonly uses daily closing prices (ie daily returns).That’s the 1 in the formula. However, we can sample the volatility using longer windows. If you chose 5 instead of 1 you are “sampling the volatility weekly instead of daily”.

• N represents the lookback.The larger the N the larger the sample size.

• Since volatility is only sampled on business days:
  • a lookback that samples the vol weekly will have an N of about 52 in a year while a lookback that samples the vol daily will have 251 samples in a year
  • the longer your sampling period, the smaller the sample size for a given lookback

A word on μ in the realized volatility formula
It is not uncommon for traders to discard μ which is equivalent to setting it to zero. The easiest way to appreciate why is to imagine a stock whose logreturn is exactly 2% per day. The daily deviation from the mean logreturn of +2% would be 0, in turn, rendering the realized volatility measure zero! If a stock went up (or down) 2% per day and we concluded that volatility is zero, then it’s fair to say the measure is broken. By “de-trending” the formula by ignoring μ, you get a less biased measure. In practice, this matters less over shorter lookbacks where the “drift” is a smaller component of the volatility. If your lookback periods are long, the drift becomes significant. If the SP500 has an annual drift of +9% with a standard deviation is 16% the drift is a substantial portion of the volatility. The impact is an order of magnitude smaller for shorter lookbacks. On a daily basis, the drift is 3 bps while the volatility is 100 bps.

🗓️Annualizing Realized Volatility

To be repetitive, take note of the two time periods involved:

1. The sampling windowThis can be daily, weekly, monthly, annual returns. It can even be a return based on tick vols (hourly, minutely…)

2. The lookback periodThis is how far back in time we are sampling returns

For example, we can sample:

• daily returns for the past month or year

• hourly returns for the past day or week

The lookback determined the sample size.

The sampling window determines our annualization factor.

💡Annualization factor examples

• daily = √251

• weekly = √52

• monthly = √12

• hourly = √(251 x 6.5) = √1631.5

*You can go crazy with this if you want. There are days when the markets close early, the number of periods varies with leap years or what day of the week January 1 falls on. But maintain a sense of proportion. Any error do to these differences will be swamped by the fact that all implied vols and realized vols are imperfect measures because time itself is linear while event or voltime is not.

See ⏳Understanding Variance Time

You simply cannot exhaust how deep you can get into this. Where you draw the line depends on your strategy. 99.5% of strategies are robust to the approximations above.

Now we compute realized volatilities on actual historical prices to see what we can learn.

Data Exploration

Setup

Since we want to compare the effect of sampling period on realized volatility we will keep the lookback period constant.

As a reminder, if we computed daily volatility for the past 6 months:

• the sampling period is daily or 1 day

• the lookback is 6 months

We call this “6-month realized sampled daily”

Since volatility tends to “cluster” (low vol periods follow low vol periods and high vol periods tend to follow high vol periods), option markets for short or even medium terms maturities will give more weight to recent realized volatility. If you were pricing a 3-year option then you’d be more inclined to examine longer a longer lookback which smooths out the sharper peaks and valleys.

This idea is echoed in the concept of a “vol cone”

For our data exploration, we will

1. use constant lookback of 40 business days which is close to 2 months worth of trading days. We call this “40-day vol” .

2. sample the vol using daily, 2 day, 5d , 10d, and 20d windows.
   1. This maps to 40-day vol being computed from samples of the following quantities respectively:
      • daily (n = 40)
      • 2 day (n = 20)
      • 5d (n = 8)
      • 10d (n=4)
      • 20d (n =2)

3. compute 40-day vol for each business day from 1/2/2013 until 8/16/2024 (>11.5 years)

Example using AAPL

Let’s step through an example with AAPL.

1. We compute the realized vol for the 40 day lookback at with carious sampling windows.In the snippet below, we also compute the ratio vols from each sampling period to the realized vol computed from daily samples

3. Scatterplot the ratio of 40-day realized vol sampled at different frequencies vs 40-day realized vol sampled daily
   1. A point on the black dotted line means the 40-day vol sampled daily was the same as the 40-day vol at the respective frequency
   2. If a point is above the line, the vol sampled at a slower frequency is higher than the vol sampled daily (you might interpret this as a market that was trending. For example if a market went up 1% per day, then the daily vol would be about 16% but the 5d vol would be approximately 5% x √52 or 36%!
   3. The regressions show that vol sampled at slower frequencies is lower than vol sampled at a higher frequency.

      

      Compute the aggregate stats for over 11.5 years

      There are 2 observations worth noting:

      The longer the sampling period, the lower the vol for the same 40-day lookback

      

      The standard deviation of that ratio grows with the sampling period. If sampling periods for the same lookback differ greatly there will be more variation in the measured realized vols relative to each other!

      The variation in the ratio by sampling period is easily seen with box-and-whisker charts

      

[See the main post for Chartbooks for various tickers]

Concluding thoughts

The most general results applied to every symbol:

• For a given lookback, less frequently sampled vol for a given lookback is biased lower. This echoes an earlier post Risk Depends on the Resolution

• However, in every case, the slower the sampling frequency (ie every 4 weeks instead of every 1 week), the more volatile the realized vol itself is compared to the daily sampled vol.

These results reinforce an intuitive idea: the longer your sampling period, the smaller the sample size for a given lookback, therefore:

The more frequently you sample vol the faster you converge on a better estimate of the volatility

If you only looked at annual returns it would take many years to get a sense of how volatile an asset or strategy is. This is also why evaluating investment managers on monthly returns is dangerous. It hides the risk.

Assorted observations:

• Periods when less frequently sampled vol give us a higher realized vol indicate trending (long option player will have wish they hedged less frequently or on a lower delta and vice versa!)

• Option traders think in straddles for shorter dated options but in vol for longer dated vols. Vol-thinking prompts you to ask “will this thing move X% per day” for the next year vs can this stock move 5% in a month. There’s a puzzle embedded in this revelation — the long gamma trader will conclude they should hedge daily to “capture” the higher vol, but a non-option trader may conclude that this is an argument for mean reversion on shorter time scales.

• An interesting scenario to filter for would a situation where realized volatility sampled slowly (ie weekly) was higher than higher frequency sampled vol for the same look back.
  • Like the earlier example of a stock that goes up 1% a day (16% realized vol sampled daily) vs >30% vol sampled weekly.
  • The reasoning here is: if the vol market is more focused on higher frequency sampled vol to price options it might underprice implied volatility.
  • I used the moontower.ai backend to filter for scenarios in a few names where monthly realized vol sampled every 3 days was higher than monthly realized vol sampled daily. I called the ratio “Trend Premium” so I filtered for situations where it was > 1.
    • I then filtered again where the options were cheap — a VRP < 1 (the RV in VRP coming from daily samples, making this analysis conservative)

This was an example from QQQ over the last 2 years.

The left chart highlights VRP < 1 AND “Trend premium” > 1

The right chart shows how the lagged VRP a month later (so what is the ratio of IV today divided by the realized vol that was eventually realized)

The realized vol underperformed the IV more than it usually does. So much for that. (I actually toggled through a bunch of names in this way but just eyeballing I couldn’t see a bias one way or the other. A sample of 2 years plus using overlapping data as its rolling windows is not exactly a proper study on this idea. The overlapping windows is not an issue for the general study earlier in the post which really just tries to understand the relationship of RV sampled at different frequencies for the same lookback. Overlapping windows does shrink your sample size if you are mining for a signal as I was flailing with this “Trend Premium” filter)

• Finally, here’s a summary table of the 40-day realized vol sampled daily for the past 11.5 years from our studies:

  

Last Sunday, I sent out a paywalled flash post explaining why I thought GLD November volatility looked like a sale based on Friday, 8/16 marks. The trade was still available on Monday morning.

It was a good trade.

[Unfortunately I didn’t do it myself and of course I’m cherry-picking by publishing this thread. As karmic penance, this week, I’ll write up a recent trade I did do that didn’t go so well]

Let’s see what happened.

First, I’ve now unlocked the full post so unpaid subs can also see the thought process Flash Post on GLD Vol but the tl;dr was the 90d or November expiry stood at as being expensive.

Post-mortem after 1 week

The November 232 GLD ATM call price

⇒ On Friday 8/16 (stock reference: $231.89): $9.37

⇒ On Friday 8/23 (stock reference: $232.05): $8.45

The profit is about $.92 or about 10% of the option premium. But this is a naive way to look at the p/l.

Decomposing the p/l

By selling this option you won to 2 different forces. The trade thesis was that the VRP and vol were both high so it’s not shocking to win on both but it’s never a given.

You would have collected:

1) $.58 in vega p/l due to IV falling

The IV on the 232 call fell from 17.2% to 15.9%. The option vega is $.45.

1.3 vol points x $.45 ~ $.58 cents in vega p/l

That’s an 8% change in strike vol over a week on a 90d option. I haven’t crunched the numbers, but based on writing weekly post mortems on my book for about a decade, I’d estimate that’s about a 1 standard deviation change in the IV for gold.

2) Another $.34 due to realized vol underperforming the IV

Theta won this week’s bout with gamma.

What about delta?

I am being a bit loose with the p/l decomp.

a) If you didn’t delta hedge, the stock went down and then rallied again this week. The calls finished the week down $0.92, despite the stock closing basically unchanged (up $0.16 for the week). It’s probably more accurate to say you made a bit more on the theta/gamma battle since you likely lost close to $.08 on delta p/l

b) If you did delta hedge daily, your P&L would have been +$0.84.

This output is from a daily delta hedge simulation assuming you were filled at market mid-prices. Note, it’s from the perspective of a long option holder, so the cumulative P&L is -$0.84 but I just flipped the sign since shorted the option.

It’s pretty random that the hedged and delta-hedged results were similar. The delta-hedged version is in general the best place to locate the P&L drivers because the trade thesis was grounded in vol not direction.

[Note: Selling an ATM call delta neutral is effectively selling a straddle. To be more accurate, if you sell 100 calls and hedge on a .50 delta you must buy 5,000 shares. You can algebraically re-factor you position as “I’m short 50 calls and 50 synthetic puts” which is 50 straddles. This trade is not quite because you hedged on a model delta of .57 instead of a .50 delta but it’s similar enough that the distinction only warrants being in this bracketed text.]

A thought on exit

If you look at the moontower.ai tools that led to the flash post in the first place they haven’t changed much despite the fall in IV. The short still looks like one of the better legs out there but it did make what I’d estimate is a 1 standard dev profit over a week and it’s not quite as obvious as it was a week ago.

As a retail trader, I wouldn’t add. I’d have a bias towards maintaining the short since it’s still looks relatively high although I’d probably care to take a closer look at how much election vol is embedded. (I’m not a registered advisor so I can speak relatively freely but to be clear I’m not suggesting anything other than how I would think about this. It should be very obvious that what is suitable for me or anyone else doesn’t mean it’s suitable for you. I write a newsletter that should be construed as entertainment and that is how you should view all newsletters. That the entertainment overlaps with useful doesn’t change what it is. If I put this under the guise of advice, I’d have to change a lot of stuff but it would also, like, cost money. I personally like that we have different channels for info to exist rather than trying to stuff all expression into regulated grooves. Don’t poison the commons by being a moron.)

If I was in my fund seat, I wouldn’t add to it (this is the market maker inside me who always argues with the position trader Jekyll who lives on the other side of the corpus collosum) but I certainly wouldn’t cover. If I were short 100k vega and my bid was hit for 20k vega and be content with the “screen edge” to mid-market.

[The nature of being in a noisy, unsystematic, repeat business with brokers is messy art….although my thinking risks becoming “quaint” every year that passes en route to 1 giant supercluster replacing the entire trading world which itself will be a pastoral phase to ruminate upon while we await being refined into robot fuel.

Sorry just a joke. Well, unless I’m right. Then I want the credit inscribed on the steel-ribbed drum containing my innards.]

Commentary

If you look at the “funnel” thinking from the flash post showing how I found this expensive option, you’ll see that I thought IV and VRP tailwinds were aligned. That doesn’t happen often. The charts you find in moontower.ai make the “no easy trades principle” really visible. Volatility risk premiums (VRPs) are high when vols are low and vice versa.

Another way of understanding the principle is what I call “distributional edge vs. carry”— aka the problem with buying low and selling high. You can read more about it here: Distributional Edge vs. Carry.

Moontower.ai offers a vol trader’s perspective. If you are professional vol trader running a relatively vol-neutral book there’s always things that look cheap vs expensive. By definition if you define fair cross-sectionally, half the universe is expensive relative to the other half. If this is your mandate you are in a low-margin trading business that has very attractive scaling properties.

But you are not that.

Instead you can just look at look at the tools regularly. You’ll typically conclude: “nothing to see here”. There’s nothing wrong with folding cards especially when there’s no ante. Retail’s advantage is not needing to trade. Buffet’s fat pitch and all that.

Buffet, of course, also keeps studied so he can recognize and act quickly when the opportunity comes. I look at moontower.ai every day. Most days I shrug, meh nothingburger, get back to regularly scheduled life and work. Breaking even on the cost of a sub is making (or saving) a dime on 100 contracts in the course of a year. It’s a low bar. The real cost is in glancing at it at least once a week to develop intuition. I talk about this more in this FAQ question. If that cost is too high, there’s no shame in avoiding options altogether or ‘index and chill’. (This is what most people should do. Our tool is for the self-selected ‘compelled’).

The tools really shine when you blend their insights with your own fundamental and directional biases to fine-tune or discover bets. While it doesn’t have all the features of full-fledged option software yet, it started with “how I want to see things.” It is opinionated. Of course, we care deeply about what users want, but I can’t deny the influence of my own taste. In fact if it wasn’t for me wanting to see things my way we wouldn’t even be this far. I certainly didn’t look at the option software space and conclude “oh this looks like a great way to make money”. It starts with having a differentiated way of seeing options and operationalizing it. Making money on the software will be a byproduct of having an experienced approach and philosophy that offers something different as opposed to making shinier versions of features that can feel disjointed when they aren’t pulled together with a trader’s lens.

Wrapping up

As I mentioned earlier, this trade was cherry-picked for the post-mortem. I wouldn’t even have considered the “victory lap” if I hadn’t built the delta-hedged simulator this week. That’s where I pulled the Python screenshot.

So far the only two flash posts I’ve done are this one and the GME trade—winning two for two is statistical significance, right? Ok, ok. I’ll make an offering to the delta gods by writing up a bad trade I did in IWM later this week using a similar volatility decomposition framework.

The main thing I want you to takeaway is that options are always a vol trade (except for reverse-conversion funding stuff). Put-call parity is a very deep insight.

Your understanding of bets, even those without options, will grow significantly if you use even simple option decomposition to think about path dependence and attribution. I’ll keep hammering this point—I’m pretty sure even a middle-schooler with an attention span (and interest, of course) could learn this stuff if it’s repeated enough.

If the Moontower sphere, the writing and the software, isn’t making it easier, then I’m screwing up.

In Connecting Vol Surfaces To Option P/L, we showed how a position in the 6-month 25d put or 25d call in IWM would have performed if you:

• bought the option and hedged the delta on the close of 7/10/24 with the stock at $202.97 and went on vacation
• returned on 7/16/24 and liquidated both you stock and options position near the close with the stock at $224.32

With the stock up over 10% and IVs higher on the 185-strike put and 223-strike call you made money both on the vol expansion and because you were long gamma — although you started market-neutral, you had a net long delta when you looked at your account.

This week we will examine the same trade but instead of going on vacation we will see what happens if you hedge each day.

The most valuable part of this exercise will be the option p/l decomposition.

The 185 put and the 223 call have a negative and positive delta respectively. Since we are sterilizing the delta or directional p/l of the option with shares we want to ignore that portion of the p/l.

We care about the p/l that we don’t attribute to delta. That’s our vol p/l. We can decompose that p/l into 2 primary buckets:

1. Vega: how much of the p/l is coming from the change in implied vol
2. Realized p/l: what is p/l due to gamma or the change in delta which was not continuously hedged minus the cost of the optionality aka theta.

The p/l decomposition will contain some error. We will give that error a sense of proportion and discuss its source.

In the process, you will learn some trickery around the calculation of realized vol and the concept of “sampling”.

Onwards…

We start with the change in the Jan2025 vol surface from 7/10/24 to 7/16/24

In both examples, we will buy an option on 7/10/24 and hedge its delta once per day until 7/16/24 (5 business days).

As a reminder this was IWM’s price chart for that period in which it rallied >10%:

Buying the 185 put delta-neutral

🕛Stepping through 1 day

7/10/2024

• Near end of day we buy 100 Jan’25 185 puts for $4.75 or 20.7% IV.
• We buy 2,440 shares of IWM for $202.97 to hedge the -.244 delta

7/11/2024

• Stock rallies $8.15 to $211.12
• Our put declines to $3.63, its delta falls to .183, and the strike vol increases 1.5 vol points to 22.2%
• We sell 613 shares to re-establish a delta-neutral position. This is computed by:
  • Change in delta * contract quantity * 100 multiplier
  • -.061 * 100 lots * 100 ~ 613 shares (decimal rounding in the model gives us 613 not 610)

P/L Computation

• Share p/l: You rode 2,440 shares up $8.15 = +$19,889
• Option p/l: The put lost $1.12 in value = 100 lots * 100 multiplier * $1.12 ~ –$11,150

✅Net P/L on the hedged put position = +$8,739

At end-of-day on 7/11/24, you are once again delta-neutral. Long 1,827 shares against your .183 delta puts

📅Stepping through 4 days since initiation

Stepping through the same logic every day yields this table:

Your cumulative p/l is $18,400 with the daily hedged strategy.

⚖️Comparison to only hedging once on 7/16

Had you simply bought the puts on 7/10 and hedged with 2,440 shares, then liquidated the puts on shares on 7/16 your p/l would have been:

Share p/l = 2,440 * $21.35 = $52,102

Option p/l = 100 contracts * 100 multiplier * -$2.24 = -$22,400

✅Net P/L on the hedged put position = +$29,702

It appears that hedging the delta every day incurred a cost. But it also reduced your risk. If the stock had tanked on 7/12 after the big rally you would have been thrilled that you delta-hedged by selling shares at the close of 7/11.

A better way to think about this is in terms of “what realized vol did you sample?”

💡A note on computing realized vol💡

Realized volatility computed from daily returns is the standard deviation of logreturns annualized by √251*

When computing a standard deviation, it’s common to square the distances of each observation from the sample mean. This will understate the volatility in a trending market. If a stock goes up 1% a day, you’ll compute a realized vol of zero.

In this example the logreturn stream is +3.94%, +1.18%, +1.69%, +3.20%. That stream has a standard deviation of 1.29%. Annualized, that’s 20.4% realized vol.

Does that return stream really feel like just 20.4% vol?

Of course not. The issue is the mean return is 2.50% so the deviations are not large.

If we instead skip the step of subtracting from the mean (which is equivalent of saying the mean is 0) then we get a realized vol of 50.1% which feels closer to reality. After all, if we moved 2.5% per day the realized vol would be approximately 2.5% * 16 = 40% vol.

*Don’t forget Juneteenth

In our daily hedging, we sampled a realized vol of 50.1%

(An idea that should be now hitting you over the head is if you hedge deltas at any cadence different than once a day at the close you will sample a different vol than what close-to-close vol readings claim the realized vol is. You created your own history. You’re a god, be careful with all that power!)

In the case, where you didn’t hedge (or close the position) for 4 days what realized vol did you effectively sample?

ln(224.32/202.97) * sqrt (251/4) = 79.2% vol

No wonder you made more money!

Vol P/L Decomposition

The 185 put lost value because of it’s delta. But some of the premium loss was offset by an increase in IV.

Options are about volatility. Write it on the chalkboard like Bart Simpson.

To understand options we need to “watch the film”. We need to map the p/l to the driver of the p/l.

We can strip out the delta. That leaves the option’s return a function of both implied and realized volatility. We will lump both of those under the cleverly named category “Vol P/L”.

The first thing we do is just look at the daily option p/ls in the red box from the condensed view of the table from earlier:

Part of those option losses stem from the mechanical truth that the stock went up and puts have a negative delta. The vol p/l is what’s left over when we remove the losses from the stock rallying.

For example when the stock goes from $202.97, the 185 put falls from $4.75 to $3.63. Since it had a .244 delta and the stock surged $8.15 we expect the option to lose $8.15 *.244 or $1.99 and fall to $2.76 by the end of Day 2 just due to delta.

But it doesn’t.

It only loses $1.12. The option appears to have made $.87 in vol profits! Multiplied by 100 contracts that’s $8,739.

Again the accounting…the actual option p/l was -$11,150. If we break that down, the option lost about $19,900 due to delta but made $8,700 due to vol.

Within this vol p/l there’s an implied vol portion and a realized vol portion.

Vega p/l = implied vol portion

Gamma + Theta p/l = realized vol portion

We estimate the p/ls with the following formulas:

Vega p/l = vega * vol change *contracts * multiplier

Theta p/l = theta * days elapsed * contracts * multiplier

Gamma p/l = 1/2 gamma * (change in stock)² * contracts * multiplier

💡See Moontower On Gamma for the derivation — it’s neat since it’s the same approximation for distance traveled in time t for a given acceleration

Let’s take inventory:

1) We start with the actual real-life option p/l

2) Subtract how much of that p/l comes should come from delta

3) The remainder should equal the sum of our estimated vega, gamma, and theta p/ls

Again:

Now we can zoom in on the decomposition by day for the 185 put:

If we zoom in on the Error section we see that the the approximations for the p/l attributable to the greeks work very well as measured by the error/extrinsic value:

Buying the 223 call delta-neutral

You can test your understanding by following along the table for the 223 call hedged daily. When you initiate the position you buy the call and short the stock to hedge.

Unpacking the error term

The fact that you make more money on the 223 call hedging daily than the 185 put is also a clue as to why there is error at all between our greek-decomposed p/l attribution and the actual vol p/l.

Greeks come from snapshots of time, moneyness, vol, rates, and IV.

The error term is really the sum of minor greeks such as:

• vanna (change in vega as spot changes or change in delta as vol changes)
• volga (change in vega as vol changes)
• charm (change in delta as time elapses)

The 223 call was about 50% more profitable to hedge every day than the 185 put because it “picked up” vega as vol increases and also picked up more gamma as it became closer to at the money.

[But the gamma increase was not as large as it would have been if the vol did not increase. A lot of cross-currents but they mostly fit under a reasonable error term. This is less true as moves and vol changes climb into crazy numbers.]

I’ll close with a final observation…whether you bought the put or call, once the stock started rallying you hedged by selling shares.

That’s long gamma.

Your position, regardless of the contract type, grows in the direction of the move. It’s only about vol because the delta can be whatever you want it to be.

Related reading

• Finding Vol Convexity

Thursday’s paid post was cutting to the heart of vol trading — dynamic hedging to isolate vol mispricing.

There’s a part in the paid section called A word on Option P/Ls that I’ll share here:

On option P/Ls

Once we strip out an option’s delta p/l, we are left with a “volatility p/l”.

Volatility p/l has an implied vol portion and a realized vol portion.

Vega p/l = implied vol portion

Gamma + Theta p/l = realized vol portion

We estimate the p/ls with the following formulas:

• Vega p/l = vega * vol change *contracts * multiplier

• Theta p/l = theta * days elapsed * contracts * multiplier

• Gamma p/l = 1/2 gamma * (change in stock)² * contracts * multiplier
  • 💡See Moontower On Gamma for the derivation — it’s neat since it’s the same approximation for distance traveled in time t for a given acceleration

Let’s take inventory:

1) We start with the actual real-life option p/l

2) Subtract how much of that p/l comes should come from delta

3) The remainder should equal the sum of our estimated vega, gamma, and theta p/ls

Error

We have simplified “vol p/l” to be a function of realized vol (gamma) vs the cost or hurdle embedded in the implied vol (theta). We are ignoring the option price’s sensitivity to changes in interest rates (rho) or implied vol itself (vega).

But there are additional greeks such as volga and vanna that we can attribute some of the change in optin price and therefore p/l. But they are typically much smaller effects.

An easy way to see this is by comparing the hedged option p/l with what we expect from simply estimating the gamma + theta p/l.

That comparison is captured by comparing actual hedged p/l vs the predicted theoretical p/l.

Here’s a chart of the daily prediction errors for the whole year:

Even those spikes remain under 4 tenths of a cent. Overall, the prediction error each day is small percentage of the daily “volatility p/l”

💡There are higher order and “cross” greeks that will “explain” the error between this estimate of the vol p/l and what is actually experienced. But as you see, the error is small. Gamma, theta, and vega do the heavy lifting.

Just based on my local observations, it still feels like the bid-ask on residential real estate is wide.

I wrote Staring Out The Window in October 2022:

Musing #1: Bid-Ask Widening

A year ago the people that paid ridiculous prices for RE were market orders. “Fill me at any price”. Many of them were immediately in the money (ie they probably could have turned around and sold a month later for more. Maybe not net of transaction costs but you get the idea). This isn’t shocking. When optimism turns to euphoria, the rate of change of the returns themselves can explode into a parabolic curve. Of course, such curves are unsustainable. The smug moment of being in the money is short-lived in the same way that a fund that buys a ton of stock going into the close usually gets a favorable mark on their daily p/l. Their sloppy buys drove the price higher in a short period of time. The real sellers didn’t have time to react before the close. But as soon as they check the comps overnight, you can be sure the supply is coming tomorrow morning.

I think of it like water going down a drain…once most of the water is through the drain the remaining liquid swirls quickly around the drain before you hear that sucking sound. Whoosh. The last bid is filled. With maximum punnage — the liquidity is gone.

In the meantime, many other buyers were priced out. You can think of them as limit bids. It’s an imperfect analogy but it will suffice. As things go south now, some of those bidders might be anchored to their original bids which were “cheaper” than where the home traded. However, if they get filled on the way down, they actually have more negative edge even though they got this theoretical house for a cheaper price than the original buyer. You could belabor this with a stylized model but understanding this concept is a big step towards understanding trading.

Anyway, the old limit bids are probably the new ask and the real bid/ask spread is wide. Prospective buyers are adjusting their bids much lower to keep the monthly payment constant or at least manageable, but sellers who likely have cheap financing from the prior low rate regime do not have to cross the spread. If current prices are 5% off their highs but the new mortgage math means homes they should be 20% lower (similar to the stock market) the current listing prices are the “asks” of a wide market.

The current housing market is facing a unique form of illiquidity in the single-family home segment, largely due to the sharp rise in interest rates during 2022 and 2023. The root of this illiquidity lies in the significant bid-ask spread that has emerged as a result of these rate changes.

I’ve been renting the same home since Oct 2020. I have a year remaining on my current lease and have some reason to believe I might not be able to renew again. It’s a bit stressful because renting again without being able to lock is inconvenient. Buying is tough because inventory is lean so it remains a seller’s market even though prices have been flat. Homes locally have been “dead money” for 2 years. (They went down 10% in 2022, recovered in 2023, and flat this year).

Investing instead of owning has been a fairly even proposition on a 4 year lookback but renting for the past 2 years has been a relative windfall. I’m not big on the idea of treating a home as an investment, but the disparity in cost to own vs cost to rent in the past couple years means you’d need opium-level psychic benefits from owning to make it look reasonable.

[My cost to rent is about 1/3 of what it would cost to buy the same house — said otherwise I could rent 3 houses for the cost to own 1. I won’t give exact numbers but it’s equivalent to renting a $1mm house for less than $2,500/m. Crazy when you consider that it’s almost $1k/m for insurance alone out here.

Of course, trying to link home prices to rental cost is only one way of coming up with a valuation. It’s rooted in relative value/opportunity cost type thinking. Another, probably more relevant method, is cost of replacement. Land here is about $2mm acre for average lots and closer to $3mm for premium lots in a flat, desirable location. The cost to build starts at $625/ft. Permitting means you must rent for about 2 years while your home is built. Existing homes trade for about $1k per sq ft.

Throw all of this in a spreadsheet and you’ll find that the IRR to build is terrible…which is another way of saying existing homes are a relative bargain. Especially because building costs only go in one direction which is a safe bet since builder margins suck as it is. Hey, CA is a beautiful place, but it has South America-style disparities, economic dysfunction, and a landed gentry. If I didn’t live here I never would have taken Georgist drugs. ]

Anyway, I’m not asking for sympathy by admitting stress. Part of the ownership premium people is so they don’t have that stress. My inconvenience is baked into whatever I think I’m saving so complaining is just being greedy.

Alas, we have been looking at open houses for the past year as depressing as they are. It got me thinking about this wide bid/ask spread that I believe is present but invisible since there’s no true order book. Listings are up, home sales are down, and prices are up 1% y-o-y according to Redfin.

My sense is houses are worth very different amounts to existing homeowners vs buyers. And since many buyers are homeowners (I’m talking about people changing primary residence not second homes), the split valuation even exists within the same brain. You’re Hyde when you list and Jekyll when you bid.

Let’s walk through it.

Homeowners who secured low-interest-rate mortgages years ago effectively “shorted bonds”. As interest rates rose, the value of these loans plummeted, embedding equity into these “short bond” positions. This means that the mortgage itself has become an asset that is highly valuable to the current owner. It’s like “it’s equity in a mark-to-theo short”. But that equity is trapped. It’s specifically tied to that home. The new buyer doesn’t get it because they must finance at the higher current rates.

This mechanically alters fair value of the asset depending on who owns or doesn’t own it. The homeowner might not be able to articulate it but they have two assets: the physical home and the valuable low-interest-rate mortgage. If they were to sell the home, they would have to buy back the mortgage at its face value, rather than its current impaired value thus losing the accrued profit from the “short bond” position.

Let’s make it concrete with a numerical example.

You bought a $500k house 5 years ago with a $100k down payment. You borrowed $400k at 3%.

What do you owe today?

Step 1: Calculate the Monthly Payment on the Original Mortgage

The formula for the monthly payment of a fixed-rate mortgage:

Where:

• M = monthly payment
• P = principal loan amount = $400,000
• r = monthly interest rate = 3%/12 = 0.0025
• n= number of payments = 360

Monthly payment = $1,686.42

The mortgage still has 25 years (or 300 payments) until maturity.

The remaining balance after 5 years is $355,625 (from this calculator)

In our fake world that’s pretty similar to the real one, a lot has changed in 5 years. Interest rates have doubled to 6%.

What is the value of the outstanding mortgage?

Step 2: Calculate the Present Value of the Remaining Payments

We need to find the present value of these remaining payments, discounted at the current 6% market rate.

The formula for the present value of an annuity is:

• M = $1,686.42 (monthly payment)
• r = 6%/12 = 0.005 (new monthly interest rate)
• n = 300 (remaining payments)

PV = $261,744

The present value of the remaining ~$356,000 mortgage, when discounted at the current 6% market rate, is approximately $261,744.

This significant difference highlights the additional equity embedded in the homeowner’s “short bond” position due to the lower interest rate. The homeowner has an intuitive sense that they are losing when they sell the home because they will have to pay the bank $356k to close the loan when it’s only worth $262k. Eww.

The additional $94k of equity that the homeowner has at prevailing interest rates represents almost 20% of the value of the $500k home!

If mortgage rates fall, the conventional wisdom that marginal demand to buy should increase is a fair assumption. However, rates falling cuts directly into this shadow equity that owners feel compared to a high-rate environment. I suspect this will actually “loosen” a bunch of trapped supply as the bid/ask spread narrows as the homeowners embedded equity in their “bond short” shrinks.

[I’m using the word “shadow” but it’s quite real vs the alternative of buying the same house for $500k at the higher interest rate. It’s “shadow” because the only way to monetize it is to let time elapse until the mortgage eventually goes away. Your lower cost of living relative to someone who doesn’t have a low interest-rate mortgage on the same property is the only way to realize the equity.

Active solutions to this illiquidity trap is allow homeowners to somehow port their mortgage to a new property or allow them to buy back their mortgage at the current value instead of the remaining principal amount.

I already hinted at a passive solution. Let the clock run. As time progresses, homeowners continue to pay down their mortgages. With each payment, the principal balance of the mortgage decreases, and the equity in the home increases. Over time, the impact of the low-interest-rate mortgage diminishes as the remaining balance shrinks. This gradual reduction in the outstanding mortgage balance reduces the value of the “short bond” position, making it less of a factor in the homeowner’s decision to sell. Eventually, as the mortgage balance becomes smaller relative to the home’s value, the embedded equity becomes less significant, narrowing the bid-ask spread.]

If mortgage rates fall in concert with the economy and employment weakening (pretty standard backdrop to falling interest rates), then supply may loosen in combination with general demand shortfall. It feels like a downside risk…but by now I’m also resigned to believing home prices won’t fall. We don’t have enough of them. Lending standards are conservative. There’s nothing frothy about the supply/demand balance. At the same time, it’s illiquid and unaffordable. My selfish position is I’d like to see prices ease but I’d happily settle for a wider selection of homes, even if they are overpriced.

The small cap leg of the rotation in mid-July 2024 can be seen in IWM.

I cherry-picked points before the move and at a peak.

In 4 business days, July 10th to July 16th, IWM rallied >10%

The vol surface for the 6-month option expiry (Jan 2025) had a near parallel shift higher.

That’s a substantial move in a 6-month surface but considering ATM vol was 18.8% before the move, a 10% rally in 4 trading days is 4 standard deviation event. Some breathlessness in the surface seems reasonable.

💡To compute how far that move is plug and chug into this equation:

In this post, we will use these vol curves to compute returns on various options. We’ll use a mix of socratic method and simply making observations to get you in comfortable. This is a set-up for upcoming posts that will go deeper into dynamically hedged option p/ls and option p/l attribution.

We don’t have to go into anything more than arithmetic to cover a lot of ground and understanding.

The pre-work for this post was to use a European-style option calculator with these vol surfaces and assume 0 RFR and divs. These simplifications have no material impact on the intuition we’re building.

Sample of what this looks like on 7/10/24 with the IWM surface snapshotted with the stock at $202.97

We will focus on:

• ~.75d call (.25d put)
• ATM call (~.53d — do you remember why ATM options have deltas > .50?)
• ~.25d call

Outright directional trades

Observations:

• You make the most money per contract on the .76d call because it has the highest delta and the stock rallied >$21
• You make the highest return on the lower delta options. If you were going to invest a fixed number of dollars (say $100k), you would have gotten the most bang for your buck with the 223 calls which had .25d on July 10th.

This is basic, directional option p/l stuff but never hurts to reinforce.

Let’s move on to delta-neutral trades.

Delta-neutral

Suppose we didn’t have a directional bias on IWM but we know that small-caps have been massively underperforming larger stocks. Maybe this feels somewhat unstable and we think there’s a large move coming. You envision 3 scenarios:

1. IWM has a giant catch-up rally
2. The performance spread will widen even further as IWM longs are really just bagholders with a mean-reversion addiction. You think their thesis is grounded in that keep them scaling into value-traps and the realization that they are the fish at the table will lead to abandonment and capitulation — ie IWM is hanging on by a thin string of hope. The 6 month options will give you time to an unexpected destabilizing move.
3. The large caps fall while IWM sputters around not doing much.

You rule out #3, but can’t decide between scenarios #1 and #2. In both of those scenarios, IWM making a large move is the driver of the spread narrowing. You’re going to buy vol delta-neutral (you will sterilize the delta of the option at the initiation of the trade).

Unable to muster a directional bias, you decide to randomly choose a strike to buy. Either the 25d put, the ATM call, or the 25d call. Call vs put doesn’t really matter because you will hedge so that your delta is zero anyway.

It’s late in the day Wednesday, July 10th. The stock is $202.97.

Since OTM options are tighter/more liquid than ITM options you focus on the:

• 185 put
• 203 call
• 223 call

You will buy 100 contracts of one of those options and delta hedge it.

☘️☘️☘️

What luck!

The day after you buy the option delta neutral the stock popped nearly $10 higher to $211.

But since it’s your birthday weekend, you left town and disconnected. You get back just in time for the close on Tuesday, July 16th.

The stock is up >10% since you went on vacation. It is $224.32!

You started delta-neutral, but considering how large the move was compared to the implied move in the options you expect to be up a lot of money on your portfolio of shares plus 100 contracts.

💡Useful rules of thumb

At 16% annualized vol:

• daily implied move = 1% (16%/ √251 OR 16%/~16)
• weekly implied move = 2.2% (16%/ √52 OR 16%/~7)
• monthly implied move = 4.6% (16% / √12 OR 16%/3.5)

We didn’t even need to compute that z-score earlier in the post to know that a 10% move in less than a week was substantial compared to what option premiums implied.

Here’s what happened on a fixed strike basis between the 2 vol surfaces from 7/10 to 7/16:

Things to notice and rotate in your brain:

• The vega, gamma, and theta are highest near the ATM strike
• The strike vols increased in a fairly uniform way
• When we are zoomed out we can easily say “this option trader has a long vol position”. But you’re p/l is a function to what happens to option premium. Talking about vol is useful because it’s how we reason about the relative cheapness or expensiveness of options…but when you get down to the actual inventory you hold in your account you want to be more granular. What happened to the strike vol of the option you actually own? In this example, because of the size of the rally, your long option position would make money even if the strike vol was down. You would have gotten paid on gamma — as the stock climbs, your call delta is getting longer (or your put delta is shrinking) while your physical share position is static. So your portfolio delta is increasing as the stock goes up! You can infer the significance of p/l attribution…your vega p/l could have been negative while your gamma p/l could have more than made up for it. In this case, your vega and gamma p/l’s would be positive. We aren’t going to do attribution today, but we are getting you loose 🙂

A skew detour

In Scatterplot Gallery I talk some about how skew flattens when vols expand to high levels and vice versa.

In this case, the strike vols rose fairly uniformly.

Imagine a vol curve summarized as:

25d put = 20% IV (33% skew relative to ATM)

ATM call or put = 15% IV

25d call = 12% IV (-20% skew relative to ATM)

Ok, let’s say all strike vols uniformly increase by 100 points.

Respectively:

120% IV

115% IV

112% IV

There’s basically no skew in this market.

(Experienced traders will recognize a sleight of hand —> If the vols go up by that much than on a fixed strike basis all of these options will tend towards .50d and we are no longer measuring 25d to 50d skew. We can’t look at the same strikes to measure skew as the vol changes because of the recursion of vol impact on delta…I know the word “vanna” typically is followed by “flows” but this is vanna in its platonic form. Change in delta per change in vol)

Even though I used a sleight of hand in the extreme example, it’s useful to make it memorable that skew flatten as vol ramps.

We can see this even in our IWM example. Above we looked at a fixed strike vol change. This is a floating strike vol change where we compare the vols at fixed deltas instead of strikes.

I hope the detour didn’t add confusion. I just wanted to show the “skew flattening” in the wild and it was a handy device for contrasting fixed strike vs floating strike changes.

[Sometimes these 2 ways of comparing vol changes are referred to as “sticky strike” vs “sticky delta”. Traders will often run models that assume the market locally behaves in one of these ways, but there are hybrid specifications as well. This is definitely in the weeds, so don’t stress out about it unless your income depends on it. If that is the case, stress hard, because this is the type of stuff that you split hairs over in practice.]

Let’s return to delta-neutral land.

What is the p/l for each of the 3 options we could have chosen to buy 100 contracts and then neutralize the delta for on 7/10/24?

I’ll provide summary tables which show the p/l as well as the required rebalance quantities you must trade to get back to:

1. delta neutral
2. your initial vega position (remember, you were definitely long vol, but because the spot price and IVs changed your vega position changed. Time also passed but this has an immaterial influence. We make the assumption that you will trade some amount of ATM options to get back to your initial vol length)

Scenario 1: You bought 100 lots of the 185 puts & hedged on a .24 delta

Notes:

• As your long put is now far OTM (it’s only -.11 delta now), you are not as long vega as you started. You are now only long $2,980 vega so you need to buy about 25 ATM options if you want to maintain the same vol length. Now that the big move occurred you’d need to re-assess your vol axe. Maybe you are happy the vol position is small. Maybe you want to flatten your vega entirely or even go short. The example just shows how even though the vol increased, the option’s moneyness is causing it to have less vega than it did before the move. (There’s a little rabbit hole in there because when vol increases, the vega of an OTM option also increases. In this case, the increase is swamped by the option being low enough delta that the net effect is less vega overall).
• Your long deltas because you owned 2,400 shares but your puts are only spitting off -1,100 deltas, so you must sell 1,300 shares to rebalance to delta-neutral.

Scenario 2: You bought 100 lots of the 203 calls & hedged on a .53 delta

Scenario 3: You bought 100 lots of the 223 calls & hedged on a .25 delta

The main observation for this scenario is your .25 delta call is now a .55 delta call and you are longer vega! You now get to sell options to get back to your original vol length. It’s a high class problem to have a growing supply of ammo as chaos sets in.

The strike vol went from 17.6% to 20.9% while the calls went from .25 delta to ATM and you are longer 25% more vega.

This is worth contrasting with Short Where She Lands, Long Where She Ain’t. In this 223 call example, the stock went to our long strike and we won big. The difference stems from 2 forces:

1. These options are not near expiry. They have plenty of vega in them and vol is roofing. These premiums are just getting bigger.
2. We got to the strike very fast. We almost gapped here. Because you stepped away for 4 days, you didn’t have a chance to sell the stock at $211, $213, and $217 to stay delta-neutral. By being negligent and opting for a tan, you let your long gamma ride a trend.

---

That’s a lot to think about for now but it will keep you limber when we get into option p/l attribution with delta hedging.

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This is part 2 of “building an options chain in your head”

In part 1, we reasoned through the following problem without an option model:

Let’s assume that SPY volatility is 16% and we want to hedge using a 1-year put . That’s in the ballpark of a long-term average.

So the question at hand has 2 parts.

1. What strike corresponds to 1 standard deviation down?
2. What do you think that put costs as a percent of the spot price?

Simplifying assumptions:

• Spot = 100 (round number and it lets us just talk in percents. The 90 strike is the 10% OTM put)
• RFR = 0% (we don’t want to distinguish between spot and forward price. It’s trivial to adjust as needed)
• vol is constant (there’s no volatility skew)
• the stock price is lognormally distributed (this is a basic Black-Scholes assumption. It’s handy because logreturns are normally distributed which allows us to use the bell-curve)

Solution

We stepped through a logical progression that started with a formula that is as well known as the Pythagorean theorem to option traders:

[This handy formula also represents another measure of volatility — the MAD or mean absolute deviation. All of this is covered in the derivation of that approximation.]

Steps to approximating the value of the 1-standard deviation put:

1. Estimate the 1 s.d. OTM strike
2. Estimate the ATM straddle to find the value of the ATM put
3. Estimate the value of the ATM/1 SD put spread
   ATM put – put spread = 1 s.d. OTM put

The post walks through the details.

Solving…

1 s.d. OTM put strike = $85

Estimated value of the $85 put ~ $1.45 or 1.45% of the spot price.

Evaluation

We checked our estimate vs a Black-Scholes model with a constant vol for each strike.

Our estimate vs Black Scholes

100 put: $6.40 vs $6.38

85 put: $1.45 vs $1.19

100/85 put spread: $4.95 vs $5.19

The approximation of the ATM put from the straddle was within 2 cents or 30 bps of the B-S calculation.

However we underestimated the put spread which in turn overestimated the put by $.26 or 21%.

Today, we will cover:

• why we underestimated the put spread
• a way to interpret the price of any vertical spread as a statement that you can bet on. If you want to take a crack at it in the meantime I’ll ask you this: If the 100/85 put spread is trading for $4.95 what specific over/under bet is the market offering you?
• finally, we’ll compare a flat vol put spread estimate with a real-time price to learn how market vols imply a different distribution (the easy part) and why many people interpret it exactly wrong (the counterintuitive part)

This will elevate your interpretation of vertical spreads so you can use them to make risk-budgeted bets on a stock’s destination.

Why did we underestimate the put spread value?

There was a clue in the screenshot of the theoretical option chain.

We estimated that the ATM put had a .50 delta.

The option chain shows that the ATM 100 strike put:

• has only a .468 delta
• has an N(d2) or P(ITM) of 53.2%

The full explanation can be found in Lessons from the .50 Delta Option, but the gist is that lognormal stock distributions are positively skewed. Since the stock is bounded by zero but has infinite upside, then the potential for a large positive return can only balance the spot price (the fulcrum or mean) if the stock is slightly favored to go down. (This is a much less crazy assumption than it sounds btw).

The geometric mean of the distribution is dragged down by 1/2 the variance:

arithmetic expectancy – 1/2 variance = geometric expectancy

Since the RFR = 0, the arithmetic return = 0 (my favorite elucidation of this assumption. If you want my explanation go here.)

In continuous terms, we compute the expected stock price net of the volatility drag:

$100e^(.5 * .16²) = $98.73

That’s the 50/50 point of the distribution.

Zoom in on the chain to see the magic:

Recall, back-of-the-envelope reasoning gave us:

• 100 put value of $6.40
• 100/85 put spread of $4.95

But with the probability of the stock closing below $100 being higher than 50% the put spread is worth more. By assuming it was only 50% we underestimated the put spread.

The theoretical chain says it’s worth $5.19 which pushes the value of the 85 put down (ie the spread between the 100 put and the 85 put is wider than our original estimate).

A deeper understanding of the put spread

The next part of the exploration is to elevate our understanding of the put spread (or any vertical spread).

A vertical spread that is dynamically hedged is a vanna trade. It’s a bet on how vol changes with the stock price. Sometimes called stock-vol correlation. There are people who do this. They don’t love themselves.

I’m going to address those of you who affirm life. Those of you who treat a vertical spread as a distributional bet on the stock price. They are friendly structures. You can risk-budget them which means:

“I am going to risk X to make [distance between the strikes] minus X”

It’s important that you stare at that statement a bit to understand it.

You are making a simple bet with a defined risk.

• You buy a $5 wide call spread for $1, you are getting 4-1 odds.
• You sell a $10 wide put spread at $4 you are laying 3-2 odds.

Our Black-Scholes chain said the put spread is worth $5.19.

I computed the put spread in another way that I will show below. Because of rounding and some discrete/continuous bucketing I get a value of $5.17.

We will use the $5.17 value to really dive into the meaning of the put spread price.

If the 100/85 put spread is $5.17 and its maximum value is $15, then if you buy it, you are getting 9.83 to 5.17 odds or 1.9 to 1.

But what are you getting 1.9 to 1 odds on?

When we use odds there’s typically a proposition at hand.

“I’m getting 3 to 1 on the Warriors winning.”

“I’m getting even money that the total score of the game will be greater than 202 points”

“I’m getting [X to Y odds] on [event occurring]”

If you buy the put spread for $5.17 you are getting 1.9 to 1 odds on the stock doing what?

I’ve seen people answer “I’m getting 1.9 to 1 on the stock expiring below $85”.

That’s not quite right.

In training, we learned the shorthand:

The [value of the spread] / [distance between the strikes] implies the probability of the stock expiring below the midpoint of the spread.

In this case, the put spread for $5.17 implies we are getting 1.9 to 1 odds on the stock expiring below $92.50.

I don’t remember the derivation of that interpretation.

(I do remember sitting thru 4 hours of deriving the assumptions of B-S and the next day another 4 hours of the formula’s derivation. But I only remember the sitting. 20 minutes into the first lecture I gave up on taking notes. Lost. MIT kids were just yawning with ennui so I was feeling pretty f’d in this crowd.)

But fear not…I was able to derive a visual interpretation this past week by pricing the put spread discretely.

The expected value of the put spread is the probability of the stock at expiring at each of the 1,500 prices between $100 and $85 times the probability of that price. Because we know the N(d2) or probability of the stock expiring below any particular strike, we can compute the probability density between any 2 strikes by taking the difference. [That was the last column in the chain screenshot from earlier].

The following chart is worth internalizing.

Take special note of the following :

1. The value of the put spread = weighted payoff (ie the sumproduct of density * expiry value)
2. The [put spread value] / [distance between the strikes] ~ the cumulative probability of the stock expiring below the midpoint!

The big takeaway is your ability to define explicit bets that sound like:

“I’m getting 3-1 odds of this stock expiring above X”

Interpreting real-life option skews

A Black-Scholes chain with a flat volatility across the strikes is a great tool for understanding how a lognormal distribution would relate option prices.

In reality, we go in reverse — we use the implied smile to show us the implied distribution. A stock index is not lognormally distributed. It’s not “more likely to go down but counterbalanced by a chance of mooning”. They are more likely to actually go up (“risk premium” is the common explanation), but if they fall suddenly, the distance can be quite far. This is the negative skew we witnessed dramatically during the dot-com bust, the GFC, and most recently the Covid shock of 2020.

The belief that market indices have negative skew coupled with the net desire for investors to hedge large downside exposure animates the bid for OTM put options. These downside strikes typically trade at a premium implied volatility to ATM or upside.

If we use higher implied volatilities for downside puts, in our example above the 100/85 put spreads would be cheaper. Said otherwise…the OTM puts would be closer in value to the ATM puts with the higher implied vol partially offsetting the discount you get because the option is OTM.

Our understanding of put spreads as being a bet on the probability of the stock being below the midpoint might create confusion.

“So you’re telling me there is negative skew but the put spread is cheaper and the probability of the stock falling is implied to be…lower?”

Exactly. It sounds unintuitive until you remember that the positive skew that characterizes a pure lognormal distribution is the same idea in the opposite direction — “the stock has volatility drag and is a favorite to fall, but upside is greater.”

Put skew says: “This index is a favorite to rise, but if it falls, the distance is further than a lognormal distribution would suggest”.

Lots of put skew makes put spreads cheaper which implies the probability of the stock going up, as represented by the odds embedded in spread price/distance between strikes, is higher!

In other words, put skew pushes the left probabilities higher making the distribution look more normal than lognormal. It undoes or counterbalances the lognormal distribution.

Let’s use a real-life 1-year vol skew to see how our put spread chart changes.

On June 20, 2024 I looked at the closing surface for the SPY June 20, 2025 (1-year) expiry.

The spot price at the snapshot was $547.43 but the forward price (where the call and put are equal) was approximately $565. To compare this skew with our toy example we will re-center the SPY strikes by dividing them all by the $565 forward price. Mapping to our toy example, the $565 strike becomes the 100 strike and the $480 strike becomes the 85 strike.

Instead of using the at-the-forward IV of 14.5% for every strike, we use the real-life IV surface to compute the values of the options at the re-centered strikes.

The lower ATM vol (14.5% instead of 16%) pushing down the value of the 100 strike (ie ATM) put to $5.78 from $6.38 in our toy model. But the 100/85 put spread falls far more than $.60. It’s now worth only $3.82 vs $5.19. The difference is the 85 put is worth far more than our toy model with the 19.2% vol at the 85 strike reflecting a hefty skew premium.

This crushes the value of the put spread, lowers the implied probability of the index falling compared to what a flat vol lognormal skew would imply.

Remember:

Put spread / distance between strikes ~ p(stock<midpoint of strikes)

We can use that same identity to compute the density between strikes. Then we compare the implied skewed distribution to the flat lognormal distribution.

We will divide every put spread by the distance between adjacent strikes. We then use the butterflies (the spread of adjacent probabilities) to estimate the density at the middle strike.

[Aside: With the flat distribution we could simply use the put’s N(d2) but because the market effectively kluges a distribution by setting a different vol for each strike, it’s not consistent to consider a singular distribution when our N(d2)’s are being generated by different strike vols.

A pitfall of having different strike vols is the possibility of an arbitrageable vol surface…see this made-up example:

In this demonstration, a “large” jump in strike vol over nearby strikes has made the lower strike MORE likely to go ITM and worth more than the put of the strike above it. In other words, the 390/385 put spread has a negative price. I’m pretty sure we could muster a zero bid for it though.]

After computing implied densities from the put spreads, now compare the lognormal vs skewed downside distributions.

The negatively-skewed distribution (blue) has less probability mass near-the-money and a longer left tail than the lognormal (grey) distribution.

Flow creates opportunity

Holding a stock price constant, the market uses the options market to bid for downside in 2 ways.

1. It can bid for put skew. This will increase the implied left tail in exchange for saying “a small sell-off” is less likely. The shape is more negatively skewed.
2. It can bid for vol but not necessarily the put skew. This will increase the range of probabilities but stuff most of that probability into the meat of the distribution. This will increase the ATM put faster than the OTM put thus increasing the put spreads and increasing the implied probability of the stock falling a medium amount at the expense of the right tail. This makes the shape look more lognormal.

Bearish options order flow is nuanced. It alters the probability mass between the intermediate and far downside. If you are bearish but the market bids too much for the left tails, you can sell the way OTM puts and buy nearer puts legging put spreads for a cheap price (ie an attractive binary bet).

If vols explode in a selloff but the skew flattens you can own the tail and be short the high premium options. As the vol comes in, the skew can reflate, putting a tailwind at both your option leg p/l’s.

There’s no programmatic rule for what works but by using vertical spreads to see what odds the market is giving you for various scenarios you can find ways to bet with the market at attractive prices even if you share the same directional view so long as you have a differentiated view on how that directional view is distributed across strikes.