Path: How Compounding Alters Return Distributions

Compounded returns experience “variance drain”. This idea captures the fact that typical result of compounded returns is lower than if you compute arithmetic returns even though the expected value is the same. We mostly care about compounded returns. This describes the situation in which your bet size or allocation is a fixed percent of your wealth, savings, or bankroll.

This is in contrast to keeping your bet size fixed (ie if you invested $10,000 in the stock market every year regardless of your wealth).

The distinction is critical because as humans we experience the path of our investments so we care about the distribution of returns in addition to the expected value.

Let’s back up for moment.

Recapping Intuition

  • What land are we in?
    • Compounding Land

      If you bet 1% of your wealth on a coin flip and win then lose, you are net down money. This is symmetrical. If you lose, then win, still down money.

      1.01 * .99 = .99 * 1.01

    • Additive Land

      In additive or non-compounding land we bet a fixed dollar amount regardless of wealth.

      So if I start with $100 and win a flip, then bet $1 again and lose the flip I’m back to $100. The obvious reason is the $1 I bet when my bankroll increased to $101 is less than 1% of my bankroll.

  • The order of win then lose, or lose then win leaves you in the same place in both worlds.

    The order does not matter if we are consistent about how we size the bet (so long as we are consistent to the style whether it’s fixed dollar or fixed percentage).

So is fixed percentage somehow “bad” in that it opens you up to volatility or variance “drag”? 

Well in the last example we used an alternating paths. Win then lose or vice versa. Let’s look at the case where instead of alternating wins and losses, we trend. Win-win or lose-lose.

  • In the additive case, we are either up 2% or down 2%
  • In the compounded case we are up 2.01% or down 1.99%

Wait a minute. In the compounded case, we are better off both ways! So the compounded case is not always worse.

The compounded case is better when we trend and worse when we “chop”.

If bet a fixed percent of our bankroll fair coin toss game we are in compound return land.

Compounding is not “bad”, it just alters the distribution of our terminal wealth

Your net compounded return in the coin-flipping game is negative more often than it’s positive, even though the game has zero expectancy.

So why is the median outcome negative?

It goes back to the trend vs the chop. Compounding likes trending and hates chopping as we saw earlier.

  •  Chopping happens more 𝐨𝐟𝐭𝐞𝐧 so you get a negative median
  • …but this is balanced by a larger trending bonus due to compounding.

Let’s illustrate.

2 Coin Flips

There’s 4 actual scenarios:

2u (trend)
1u, 1d (chop)
1d, 1u (chop)
2d (trend)

Zoom in on “compounding bonus/drag”:

Observations:

  • Chop and trend happen equally.
  • The magnitude of the boost/drag is also equal.

3 Coin Flips

There’s 8 total outcomes, but again order doesn’t matter. So there’s really just 4 outcomes.


The “chops” are bolded. They represent compounding “drag”

Note:

  • You drag 75% of the time!
  • The larger positive boost magnitudes make up for the frequency.

Now that you have the gist, let’s do 10 flips.

10 Coin Flips

  • 65% of the results are chop giving you compounding drag.
  • The times you trend though crush your performance if you only bet fixed dollar!

Visualizing “The Chop”

Let’s take a look visually at paths where N=10  to see the “chop”.

Pascal’s Triangle is a quick way to to get the coefficients of a binomial tree. The coefficients represent combinations which are weighted by the probabilities in the binomial expansion.

I enclosed the “chop” or drag paths

100 Coin Flips

  • The negative median now becomes very apparent in the “cumulative probability” column.
  • The chop occurs in 68% of paths. The median return is -.50% after 100 flips though the expectancy is still zero.
  • In additive world if you win 50 $1 bets and lose 50 $1 bets your p/l is zero.
  • In compounding world, where you bet 1% each time you are down 50 bps in that scenario.
  • The negative median associated with compounding is balanced by better outcomes in the extremes.

Both the maximum and minimum returns in simulations are better than the fixed bet case. This simulation by Justin Czyszczewski (thread) shows just how substantial the improvement is in those less probably trending cases:

Lessons From Compounding Coin Flips

  • Your overall expectancy is zero because the common chop balances the rare but heavily compounding trends.
  • Paths affect distribution of p/l even if they don’t affect expectancy.

Since we actually experience “path” and all its attendant emotions, it pays to think about the composition of expectancy and returns.

Making Property Taxes Apples to Apples

You will be working from home more often. Not all of you but many of you. That means browser tabs devoted to Zillow searches in Austin, Nashville, Vegas, Denver, and Miami. Geo-arbitrage won’t be as dramatic as software devs had hoped since the big companies will cut your pay when you leave, but in some of these places you could sustain a 20% pay cut and still be better off (at least if you’re leaving SF).

One of the biggest inputs into cost-of-living comparisons are so-called SALT (state and local) taxes. Since 2018, SALT deductions are limited to $10,000. They were previously uncapped. This has created even larger disparities in cost-of-living between states. CA, IL, NJ, and NY have income taxes that get a bit handsy with their residents.

Beyond state income taxes, one needs to consider property taxes for a more complete picture. Texans enjoy zero state income tax but hefty property taxes. NJ residents are assaulted from both ends — above average state income taxes and punitive property taxes. How about CA? The state income tax, gas tax and the cost of renewing a vehicle registration are nothing short of sunny weather ransoms.

But what about CA property taxes? The answer to this is sneaky and can be used to understand the impact of property taxes in general. But I’d go further and say that if you have not walked through the math the way we are about to, then you may be walking around with some very mistaken impressions about the cost of housing.

Property Taxes: Apples to Apples

The effect of property taxes depends on 2 core variables. The property tax rate and the assessed value. If you are weighing a house in CA to a house in NJ you want to make an apples-to-apples comparison. How do you do that when the rates are different and the methods of assessing value are different?

Let’s isolate each effect.

[Obviously the cost to buy a home has many factors that can mostly be tucked under the headings of supply and demand. Yet the effect of property taxes is significant so it’s worth isolating. It’s also worth noting that since a primary residence is most people’s largest asset, a property tax is a defacto, albeit incomplete, wealth tax. Economically it’s passed-thru to renters so it hits everyone]

Assessed Value Effect

Property taxes are waged on assessed value. In NJ, assessed value resets whenever a home trades. So if you buy a $1,000,000 home and the property tax rate is 1% you owe $10,000 per year in property tax. As the estimated market value of your home changes, your assessed value changes. So if your market value jumps 15% in one year you can expect a big increase in your tax bill. It may lag the full market return but the idea is the assessed value tracks the value of the home. Downturns in prices require homeowners to plead their case that the home’s value has declined if they want relief on their taxes.

Like NJ, CA assessed value resets to the purchase price after a transaction. But then CA diverges from other states. A month before I was born, in June 1978, CA passed Prop 13, a ballot proposition that has created distortions in wealth that few could have foreseen. Prop 13 froze assessed values at 1976 levels for homes which have not since traded. It also limits increases in assessed value to a cap of 2% per year.

Combined with a NIMBY attitude to permitting new construction, CA features a lopsided sight to behold — multi-million dollar homes with single-digit thousand tax bills. Nice for those owners but not socially desirable.

Consider:

  • The flipside of having seniors be able to stay in their homes is that it limits worker mobility by poorly allocating big homes to people who don’t need them. It basically keeps rooms off the market. If you are a senior citizen on a fixed income you are not going to sell the home you’ve outgrown to buy a condo with much higher property tax than the big house you leave behind. And that’s after you pay a huge cap gains bill.
  • Prop 13 starves the state of tax revenue that needs to come from somewhere. So the state income tax can be seen as a wealth transfer from young, working Californians to older, entrenched Californians.

In a state that has seen generational wealth built on a loop of buying real estate, and cash-out refis it’s easy to see how Prop 13 has contributed to the party. Let’s pretend you buy a home in CA and NJ.

Assume:

  • Each home costs $1,000,000
  • Each has a property tax of 2.5%. We are isolating the assessed value effect so need to hold the tax rate constant.
  • Each home has a real (inflation-adjusted) return of 2% per year.
  • The only difference is the CA home is assessed only when you buy it, but the NJ home is assessed each year.

The CA home’s IRR will be .14% after-tax while the NJ home’s IRR is -.52%. The CA home outperformed the NJ home by .66% per year over 30 years. On a $1mm home that’s over $275,000 simply because the NJ home is re-assessed every year.

It gets crazier. The effect actually explodes with higher appreciation rates. If we double the appreciation rate to 4% per year, the CA homes nets you $700,000 more than the NJ home. Remember that the tax rates are the same! We are just isolating the impact of fixing the assessed value at the purchase price.

The main takeaway is Prop 13 is a call option on inflation. Your home is much less of an inflation hedge than you think if its assessed value increases in-step with the market value.

[This year Prop 15 is on the CA ballot. Prop 15 would repeal Prop 13 for commercial properties only. Based on the examples above, it’s obviously something RE investors are highly concerned about.]

Rate Effect

What if you wanted to compare the price of homes in 2 places with different property tax rates? Let’s pretend CA no longer had Prop 13. Like NJ, it’s property taxes were re-assessed annually. This allows us to simply isolate the impact of differing tax rates.

Let’s assume:

  • Each home costs $1,000,000
  • CA tax rate is 1%
  • NJ tax rate is 2.5%.
  • The homes do not appreciate over 30 years (just to keep it simple)

Let’s explore 2 methods of comparison:

The Mortgage Method

If the homes do not appreciate then their assessed value remains fixed at $1mm. This makes it easy — the CA home owes $10,000/yr in taxes and the NJ home owes $25,000. On a monthly basis, the NJ home costs an extra $1,250. If mortgage rates are 3% we can find that a $300,000 30-year mortgage corresponds to a $1,250 monthly payment. So we can say that a $1mm house in CA costs the same as a $700,000 house in NJ since the $700,000 plus an additional $300,000 mortgage would equate to the cost of the CA home.

The IRR Method

The IRR on your home’s value will approximately differ by the spread in the tax rates. In the table below, we see that the CA home returns 1.44% more (close to 1.50%) over 30 years. If we use an inflation rate of 3% to keep consistent with what I chose as a mortgage rate, we find that the NJ home costs you $300,000 more over the 30 year holding period than the CA home, matching the result from the mortgage method.

Combining Effects

To compare the price of a home in CA to a home in NJ you need to account for both the difference in property taxes and how assessed values are treated. Let’s combine the results in one model with more realistic numbers:

  • A 4% annual home appreciation in both markets
  • A 2% inflation rate
  • CA tax rate is 1%
  • NJ tax rate is 2.5%
  • CA assessed values do not increase, NJ is re-assessed annually

CA, due to Prop 13 and a lower property tax rate, has an almost 2% edge in annual return (3.29% vs 1.34%). Since these are nominal returns and inflation is 2% per year we see that the NJ home end up actually losing value in real terms. The fact that the home is re-assessed every year means that even though the home’s value is growing faster than inflation the taxes are also growing very quickly.

I don’t want to have you miss the point — these CA and NJ homes were assumed to grow at the same rate of 4% per year and yet the CA home earned you an extra $900k in present value vs the NJ home. This is strictly due to lower property taxes and Prop 13.

We know that home appreciation in CA has been faster than NJ (my family considered moving to CA in late 70s, early 80s so we are very sensitive to the comparison). The difference in property tax policies has a staggering delta in terminal wealth when applied to CA real estate boom over the past 50 years.

Wrapping Up

Having grown up in NJ and now lived in CA, I have noticed a massive divide in how people have earned their money and wealth. You cannot live here and not notice the wealth built in real estate and not think about how policy has enabled it. When you start comparing apples-to-apples, the headline prices of CA homes are not as relatively expensive as they appear. Don’t hate on Californians though. Those SALT taxes are still burying all of us who still work for a living.

In sum:

  • Prop 13 allows homes to be a call option on home appreciation/inflation
  • High property taxes on homes that are re-assessed require rapid appreciation to not render the home ‘dead money’
  • Compare homes with different property taxes by amortizing the difference in monthly payments into a mortgage

Sending a thanks to @econompic who I discussed these topics with. As another NJ to East Bay transplant he has given these ideas plenty of thought as well. And on the math side, he gave me the idea to use IRRs instead of CAGRs. CAGRs are simpler because they are compounded returns which require no more than a start value, ending value, and time period. They are commonly used when calculating a return for a stock or fund that you buy and hold.

In this case, IRRs or NPVs are preferable since there are many cashflows.

Straddles, Volatility, and Win Rates

One of my favorite follows on #voltwit is @SqueezeMetrics. The account more colloquially known as “the Lemon” has a personal crusade against using implied vol to refer to option prices. Recall, volatility is just the asset’s standard deviation of returns. It’s usually an annualized number. So if the SPX has a 15% volatility that just means you expect the SPX to return +/- 15% about 68% of the time1

“Lemon” prefers using the average expected move, more commonly known as the straddle.

Thus tweeted the Lemon:

I think the convention of turning the straddle price into an annualized standard deviation is obfuscatory. Straddle gives you the average move that’s priced in. Why complicate that?

I can see how the distinction between average move (aka the “straddle”) and standard deviation (aka the “vol”) is “obfuscatory”.

So let’s clear it up.

Expect to learn:

  • The math relationship between the straddle and the volatility
  • How the distinction relates to win rates and expectancy
  • Why the spread between the straddle and volatility can vary in turn altering win rates
  • My own humble opinion on the matter

Turning Volatility Into A Straddle and Vice Versa

A handy formula every novice trader learns is the at-the-money straddle approximation2:

Straddle = .8Sσ√T

where S = stock price
σ = implied volatility
T = time to expiry (in years)

Ok, let’s pretend the SPX is $100, there’s 1 year to expiry, and implied volatility is 15%. Plug and chug and we get a straddle value of $12 or 12%. Pretty straightforward.

Straddle/S = .8σ√T

If we want to simply speak in annualized terms then we can assume T = 1 and can simplify:

Straddle as % of Spot = .8 x σ

Which of course means if you know the annualized straddle price as a percent of spot you can go in reverse to get the volatility:

σ = Straddle as % of Spot x 1.25

When is this useful?

Let’s say based on a stock’s past earnings move you see that it usually moves 5% per day. In other words, the earnings day straddle should be 5%. Then, you can find the standard deviation:

5% x 1.25 or 6.25%

The standard deviation is a volatility which you can annualize to plug into an options model which will spit out a 5% straddle price.

6.25% x 252 = 99.2% vol

Knowing the 1-day implied volatility is useful when you are trying to estimate a term volatility for a longer period that includes the earnings day (topic for another time).

What’s the practical difference between straddles and volatility?

Volatility is a number you stick into a model to generate a price for an instrument you actually trade. In this case, a straddle. If you input 15% vol into our above example, you will find that a 1-year straddle will cost you 12% of spot.

If you buy this straddle your return is equal to:

Absolute value of SPX return – 12%

Your worst case scenario is the SPX is unchanged and you lose your entire 12% premium. You are “long volatility” in that you want the SPX to move big one way or another.

So let’s talk about what we really care about — expectancy and win rates.

Expectancy

The point of the model is to generate a price that is fair for a given volatility. 12% was the fair theoretical value for a 15% vol asset.

If you pay 12% for the straddle on a 15% vol asset you have zero expectancy.

But that’s not the whole story.

Win Rates

Expectancy and win rate are not the same. Remember that the most you can lose is 12% but since there is no upper bound on the stock, your win is theoretically infinite. So the expectancy of the straddle is balanced by the odds of it paying off. You should expect to lose more often than you win for your expectancy to be zero since your wins are larger than your losses.

So how often do you theoretically win?

A fairly priced straddle quoted as percent of spot costs 80% of the volatility. We know that a 1- standard deviation range encompasses about 68% of a distribution. How about a .8 standard deviation range?

Fire up excel. NORMDIST(.8,0,1,True) for a cumulative distribution function. You get 78.8% which means 21.2% of the time the SPX goes up more than .8 standard deviations. Double that because there are 2 tails and voila…you win about 42% of the time.

So in Black-Scholes world, if you buy a straddle for correctly priced vol your expectancy is zero, but you expect to lose 58% of the time!

Outside Of Black-Scholes World

The Black Scholes model assumes asset prices follow a lognormal distribution. This leads to compounded or logreturns that are normally distributed. This is the world in which the straddle as percentage of spot is 80% of the annualized volatility.

In that world, you lose when you buy a fairly priced straddle 58% of the time. Of course fairly priced means your expectancy is zero. What happens if we change the distribution?

I’m going to borrow an example of a binary distribution from my election straddle post:

  • 90% of the time the SPX goes up 5.55%
  • 10% of the time the SPX goes down 50%

    Expected move size = 90% x 5.55% + 10% x 50% = 10%

Expected move is the same as a straddle. The straddle is worth 10% of spot. Your expectancy from owning it is 0.

If this was Black-Scholes world, we would say the volatility is 1.25 x 10% = 12.5% (not annualized). But this is not Black Scholes world. This is a binary distribution not a lognormal one. What is the standard deviation of this binary asset?

We can compute the standard deviation just as we do it for coin tosses or dice throwing.

σ= √(.9 x .05552 + .1 x .502)

σ = 16.7% (again, not annualized so we can compare)

Note that your straddle is 10% but your volatility is 16.7%. That ratio is not the 80% we saw in the lognormal world, but instead it is 60%.

Note you cannot repeat the earlier process to find the win rate. You can’t just NORMDIST(.6,0,1,True) because the distribution of returns is not normal. Luckily, with a binary distribution our win rate is easy to see. In this example, if you pay 10% for the straddle you lose 90% of the time.

Even if you paid 6% for the straddle you still lose 90% of the time. However if you bought the straddle that ‘cheap’, your expectancy will be massively positive!

My Own Humble Opinion

When there is a short time to expiration, arbitrarily let’s say a few weeks, my mind’s intuition might latch on to a straddle price. I might think in terms of expected move as one does for earnings in getting a feel for what is the right price. But on longer time frames I prefer to think of implied vol because I am going to be dynamically hedging. Measures of realized vol can be readily compared with implied vol.

If I look at a straddle price for a long period of time, say 1 year, I might fall into a trap thinking “20%? That just sounds high.” I’d rather just compare the implied vol which would be 25% (remember 1.25 x straddle), to realized vol since I am interested in the expectancy of the trades, not the win-rate.

There are all kinds of house of mirrors when looking at vols and straddles and thinking about winning percentages. As Lemon says, it’s “obfuscatory”. Everyone should do what works for them.

If you tend to be long vol, be aware having more losing months than winning months might be completely normal. It’s baked into the math. And the more skewed the distribution, the worse your batting average will be.

But in the long run it’s your slugging percentage that matters.

Recap

  • Straddles as a percent of spot are 80% of the volatility (all annualized)
  • Straddles tell you the average move.
  • Fair straddles have zero expectancy.
  • You lose more often when you win when you are long a straddle.
  • Your win sizes are larger than your losses.
  • Skewed distributions change the relationship between win rates and expectancy. They also change the relationship between straddle prices and standard deviations.

Binary Straddle Example Based On The 2016 Election

This is a dramatization loosely based on the 2016 election.

It may be hard to remember, but leading up to the election the market would sell-off when Trump’s odds increased and vice versa. So let’s make some assumptions.

  • It’s the morning of the election, the SPX index is trading for $100 and the election day straddle is trading for $10.
  • If Donald Trump wins the SPX goes down. If he loses the SPX goes up.
  • The SPX price is completely binary. It will go to either an “up price” or a “down price”.
  • Trump is liquidly trading at 10 cents on the dollar to win the electoral college in betting markets.

If Trump wins the election where does the SPX go?

[This section is blank for your algebra]

If you felt lazy here’s my work:

  • The expected value of the 1 day change in SPX is 0. It’s fairly priced at $100.
  • The probability of the SPX going down is 10% since that’s Trump’s implied probability of winning.

    For both of these statements to be true in a binary situation we know the expected down move which occurs 10% of the time is 9x the expected up move when Trump loses.

    P(up) Stock_up + [1-P(up)] x Stock_down = 0
    .9 x Stock_up + .10 x Stock_down = 0
    .9 x Stock_up = – .10 x Stock_down
    Stock_down / Stock_up = -9 / 1

  • Now let’s bring in the straddle.

    The straddle is trading $10 or 10% of spot. The straddle is the expected absolute value of the change in the SPX.

    P(up) x Size_up + [1-P(up)] x Size_down = Straddle
    .90 x Size_up + .10 x Size_down = 10

    Using the substitution that Size_down = 9 x Size_up:
    .9 x Size_up + .1 (9 x Size_up) = 10
    1.8(Size_up) = 10

    Size_up = $5.55
    So Size_down which is 9x Size _up must be $50

If Trump has a 10% chance to win the election tanking the market AND the straddle is worth $10 then the market was expected to rally 5.55% if he lost. If he won the implied sell-off was 50%!

If that didn’t sound reasonable to you (but you were certain the event was a true binary) then there are relative bets to be made between vertical spreads, outright straddles and election odds depending on what you disagreed with.

To recap:

The exercise here was to turn a binary event with

a) an implied probability

and

b) a straddle

into an implied up and implied down price after the election.

Formulas you can remember based on the above algebra:
Up Move Magnitude = straddle / (2 x P(up))
Down Move Magnitude = Up Move x P(up)/P(down)


A little post-script based on my memory of 2016. At the beginning of the year, there were giant buyers of gold and upside call verticals in gold. Whispers were that it was Drunkenmiller and perhaps a few other macro whales. Well, whoever was buying these call spreads was spot on. Gold had a sharp rally in Q1 of 2016 before settling in somewhere like up 20% in the first half of 2016. A big move for a sub-15% vol asset.

Fast forward to election night. The futures markets were unhinged. In the peak of panic over Trump winning, the SPX was down nearly 10% while gold spiked higher. By the light of the following morning, the market had whipsawed from those points and Drunkenmiller or whoever was leaving footprints in gold had allegedly used the election night headfake to rebalance the long gold position on the highs into an SPX position on the lows.

The 10% straddle seemed to be well-priced, but somehow the GOAT macro trader realized the sign of the Trump move was exactly backward!


Some broker chatter I loosely recall after the election:

Banks that were long Nikkei variance hedged with short US variance allegedly crushed it that night as the Nikkei observation for the variance calc was down over 5% while the US point-to-point return was little changed despite the hellacious path. The Japanese markets closed in the middle of the US night when SPX was at its lows.

There are a number of exotics and bank traders who read this so maybe one of them will fill me in on the color or veracity of that 🙂

DCF As A Lower Bound

Sunblock stock (SUN) makes 10% in a sunny year. Loses 2% in a rainy year.

Umbrella stock (RAIN) loses 2% in a sunny year. Makes 2% in a rainy year.

Assume:

  • The year is 50% to be sunny.
  • The risk-free rate is 0%

A few things to think about

  1. SUN has a higher expected return and Sharpe than RAIN
  2. We can see the stocks have -1 correlation
  3. There is an arbitrage. You can put 50% into each stock and earn 4% in sunny years and 0% in rainy years for an EV of +2% on the portfolio

What can we expect?

The market prices of these stocks will adjust.

Let’s keep it simple and presume:

  1.  SUN’s price stays constant. Its returns characteristics are unchanged.
  2. RAIN’s price is to be bid up so it returns only 1% in a rainy year and loses 3% in a sunny year. Note that RAIN’s expected value is now -1% per year instead of zero.

Why would the market bid that much?

This is the subject of my latest post, You Don’t See The Whole Picture. (Link)

Expect to find:

  • A simple math example to show how the diversification benefits of an asset can benefit a portfolio EVEN if the asset has a negative expected return
  • Examples from the market-making and option trading worlds which describe the “supply chain of edge”. When you see prices that don’t make sense it’s possible you don’t see the info embedded in a higher link in the chain. Whether that’s due to analytical or structural limitations, incentives, or something else is a question you need to consider.

Some Musings I Left Out 

If I felt comfortable larping as an actual businessperson I might have included a few more thoughts in the post:

ComplementsFB can pay up for WhatsApp because they are the most efficient buyer. So the price to a bystander, who can’t see Zuckerburg’s dashboard, looks insane. And in fact, in isolation, the price might be insane. But to the party where its value is highest, it can be a bargain.

Disney paid $4b to buy Star Wars rights. It was a win/win for Lucas and Mickey. The synergies lower the effective price.

Substitutes

Sometimes tech giants scoop up small firms as acqui-hires or to leap-frog R&D time/cost. But I imagine sometimes it’s just defense. Kill Simba before he grows up to inherit the Sahara. Once again, the price looks high in isolation but this “strategic buying” is informed by a wider context.

A Lower Bound

The stand-alone value of a business is the intrinsic value of a call option. But, there is a non-zero chance that some combination makes the asset worth even more. An excessive price is a mix of intrinsic and extrinsic. Going further, is it possible the extrinsic premium increases in proportion to connectivity?

Louis Pasteur wasn’t doing R&D at chocolate chip cookie company, but he would have been paid more at a Nabisco than at his local French universities. But they need to find each other.

In a connected world, awash in capital, the DCF of any business in isolation might be just where the bidding starts.


The most practical implication of these ideas is that you are not paid for diversifiable risks, so you incinerate theoretical money when you don’t diversify. This is true regardless of your actual investment performance.

The Diversification Imperative is a reminder of the only free lunch in investing. (Link)

You Don’t See The Whole Picture

Overpriced Or Just Overpriced In Isolation

☀️Sunblock stock (SUN) makes 10% in sunny year. Loses 2% in rainy year.

☂️Umbrella stock (RAIN) loses 2% in sunny year. Makes 2% in rainy year.

Assume:

  • The year is 50% to be sunny.
  • The risk free rate is 0

A few things to think about:

  1. SUN has a higher expected return and Sharpe than RAIN
  2. We can see the stocks have -1 correlation
  3. There is an arbitrage. You can put 50% into each stock and earn 4% in sunny years and 0% in rainy years for an EV of +2% on the portfolio

What can we expect?

The market prices of these stocks will adjust to there is no arb.

Let’s keep it simple and presume:

  1.  SUN’s price stays constant so its returns characteristics are unchanged.
  2. RAIN’s price is to be bid up so it returns only 1% in a rainy year and loses 3% in a sunny year. Note that RAIN’s expected value is now -1% per year instead of zero.
Why would the market bid that much?

Because there’s still an arb.

You could put 30% of the portfolio into SUN and 70% in RAIN and still earn 50 bps per year with NO risk (remember RFR is 0%)!

What can we generalize?
  • A low or neg correlated asset, even one with a negative expected return, can improve a portfolio.
  • Assets can look appear overpriced in isolation, yet their price is more than justifiable.

When You Don’t Understand The Price You Don’t Understand The Picture

Price is set by the buyer best equipped to underwrite the risk.

If you weren’t willing to bid RAIN up you can bet SUN would have.

This leads to 2 important warnings.

1. You must diversify

Financial theory dictates that you do not get paid for diversifiable risks. To be blunt, you are incinerating money if you don’t diversify. The SUN/RAIN example can show how you would expect to lose money in RAIN in isolation because the market is priced assuming you could buy SUN. I cover this idea more in The Diversification Imperative.

2. You might be a tourist

It’s worth asking yourself, does X look overpriced because I have the wrong perspective? You are looking at RAIN but don’t see what the SUN investor sees.

A Market-Maker Example

If X is willing to pay me a high looking price for a stock or option, what’s the probability they are selling something else to someone else such that they are happy to pay me the “high” price?

Let’s say a call overwriter sees a modest surge in implied vol and is happy to collect some extra premium. Except he’s selling calls to a Citadel market-maker who’s happy to pay the “high” price because her desk is selling index vol. In fact, they are selling index implied correlation at 110%. You might be happy selling the calls for 2% when they are usually worth 1%, but if the person buying them from you knows they are worth 3% at the time you sold them then make no mistake, you are playing a losing game.

However, if your professional edge is in deeply understanding the stock you are selling calls on, then you might be the one capturing the edge in the expensive calls. You are capturing it ultimately from the fact that index volatility is ripping higher and market makers are simply capturing the margin between the weighted option prices of the single stock in proportion to the index volatility. So you, the informed single stock manager, is making edge against the index volatility buyer who set off the chain of events.

The decomposition of the edge between you and the market maker is unclear. But the lesson is you must know where you stand in the pecking order. When a market maker is asked why they are buying Stock A for $100 they respond “because I can sell Stock Z at $110”. There’s always a relative value reason. The more you internalize the SUN/RAIN example and how correlation relates to diversification the more natural this reasoning becomes.

Another example

Let’s consider another option relative value trade.  If volatility surges in A but not in B and they are tightly correlated let’s look at how 2 different market participants might react.

Naive

The naive investor isn’t aware of what is not monitoring the universe of names. They do not think cross-sectionally. They see a surge in A and decide to sell it. It may or may not work out. It’s a risky trade with commensurate reward potential.

Sophisticated

The sophisticated trader recognizes they can sell A and buy B whose option prices are still stale (perhaps there has been a systematic seller in B who has been price insensitive. Maybe from the same class of investor our friend “naive” came from. They don’t look at the market broadly and realize the thing they are selling is starting to “stick out” as cheap to all the sharps).

Here’s the key: the sophisticated trader will do the same trade as the naive one but by hedging the vol with B, they can do the whole package bigger than if they simply sold A naked.

The sophisticated traders are the ones who see lots of flow. They “know where everything is”. While in this example, sophisticated and naive both sold A there will be times when sophisticated is lifting naive’s offer. Sophisticated has sorted the entire market and is optimizing buys and sells cross-sectionally.

Are you the fish at the table?

Flow traders and market makers are always wondering if their counterparty is legging a portfolio that they’d like to leg themselves if they saw the whole picture.

Sometimes it’s not possible because of structural reasons. For example, the risk that banks exhaust from structured product issuance or facilitating commodity hedges for corporations originates from a relationship nobody else can access.

A bank charter means some captive audiences. But that exhaust risk is recycled through the market much like a good flows through a vertical supply chain from wholesaler to retailer, with a markup being tacked on incrementally until its sold to a Robinhood client.

The markups are not explicitly in dollars but in the currency that lubricates financial markets — risk/reward. Mathematical expectancy, like a house’s edge, is priced by its most efficient holder.

If prices are always being set by the party who most efficiently underwrites/hedges/prices the risk and you know you are not one of those parties then you should wonder…

am I being arbed?

Lessons From The Layup – Corner 3 Spread

During an interview with Ted Seides, investor Andrew Tsai recounts an internship at the well-known trading firm Susquehanna in the mid-90s (disclosure: I worked there for 8 years after college). In particular, he remembers a company outing to a dog track that summer:

I’m sitting next to one of the partners and I’m looking at the sheet of all the races, and he’s like “How are you gonna bet?” I respond, “Well, I’ve never really done this before but this dog looks like he’s got a good track record and he’s been running strong lately.”

The guy looked at me like I was a complete idiot.

He’s like, “What are you talking about, ‘How is this dog doing?'”

Andrew is perplexed. Well, isn’t that kind of what we’re talking about.

The partner starts to explain, Look at the relative value of this dog and that dog.

The lightbulb went on for Andrew.

“We started talking about spread trading and trying to capture that basis and I’m like ‘These are my guys’. It was really this culture of dissection that I loved.”

Relative Value Goggles

One of my favorite Twitter follows is the anonymous account @econompic. He’s in my top 5 and you should follow him too (only about 15% of my followers follow him which is basically as stupid as a butterfly trading for a credit). Go for the finance stuff and stay for takes on breakfast cereal, Weezer, and the NBA. Oh and the polls. You see, Jake’s polls act like the Susquehanna partner while Andrew is the rest of #fintwit. They are cleverly designed to surface mispricings in how people think about risk or relative value.

His relative value instincts are well-tuned. It’s like he has goggles that allow him to filter the world through prices. It’s a lens that’s critical for trading. One of his recent tweets is a great example of this. I’ll withhold the full tweet for now since it has spoilers. Let’s start with this screenshot:

So which shot do you take?

(take note of your answer and reasoning before continuing)

Spread Perception

The first thing that should leap off the screen is the gap between the free throw and the top-of-the-key 3. Using NBA dimensions, that’s a 15′ shot vs a 23’9″ shot. And you are rewarded 5X for it from the benevolent genie offering this bet. The reflex you need to hone is that:

Prices imply probabilities

Why?

Because of expected value. Expected value is the probability of payoff times its magnitude. Would you pay Best Buy $50/yr to insure a $1,000 TV? If there’s more than a 5% chance that it fails you might. If there was a $500 deductible then the benefit is cut by half and you need to think there’s at least a 10% chance the TV fails. And if you think you get more TV per buck every year thanks to innovation then purchasing insurance implies an even greater defect rate.

So when you weigh the cost of different choices (insure vs not insure, fix vs replace, cheaper product vs more durable product) you are implicitly weighing probabilities. Making that explicit can expose mispricings.

Let’s go back to basketball.
Dissecting the basketball shot.

Just to get a hang for the reasoning let’s start with a simplifying assumption. You are 100% to make the layup.

  • Free Throws
    How confident do you need to be from the free-throw line to forgo the certain $50,000 you’d make from a layup? At least 50% confident. If you can shoot a free throw with a better percentage than a coin flip the free throw “has more equity”. If you are a 60% free throw shooter than that option is worth $60,000.
  • Top-of-the-key 3
    $500k to make this shot. You only need to be 10% confident to justify forgoing the layup for a chance at some big money.

    Ok, here is where the probabilities should really get your senses tingling. The free throw implies a 50% probability and the top-of-the-key 3 implies 10%. Are you 5x more likely to make a free throw than this 3-pointer?

    Unless you are 7 and literally can’t heave a ball from the 3-point line, it’s hard to imagine your chance of making these shots to be so far apart. In fact, if the 7-year-old can’t reach the rim at all from long range, I have my doubts they can shoot consistently shoot 50% from the stripe in the first place. But I’m willing to concede that possibility. For an adult, that spread is too wide. You either can’t hit free throws with a .500 percentage or your chance of making a top-of-the-key 3 is greater than 10%.

    To take an outside view, consider NBA players. Guys who shoot about 40% in games, can shoot between 65-75% in practice. HS coaches can tell you that a 30% 3-pt shooter can make about half their shots in practice. Since free throw percentages are bounded by 100% you are talking about no more than a spread of 2x between free throw and 3-pt percentage. Your margin of error on the spread could be 100% and you’d still only have a spread of 4x. These shots are priced at 5x!

    An exactly 50% free throw shooter be a 12.5% 3-point-shooter using the most conservative estimates and this top-of-the key 3 is still “too cheap”. And remember, there is a conditional probability aspect to this since we are dealing in relative pricing. If you are certain you need a miracle to hit an uncontested 3-pointer there is almost no chance you are truly a 50% free throw shooter.

  • The rest of the table

To amateurs the corner-3, without a view of the backboard or the chance for a lucky bank shot, is daunting. But are you really half as likely to hit a corner-3 vs the key-3? As we get into the low probability shots it’s reasonable for a person who really knows their habits to potentially parse these odds but it takes quite a bit of experience to know that you are really 100% better at top-of-the-key 3s then corner 3s. Without that conviction, I’d take the better implied odds in the corner-3.

The entire payoff schedule suggests that you should either take a layup or a corner 3 as you are being offered very cheap relative pricing on those options. You can check out the rest of the tweet for the comments and replies. (Link)

What If You’re Broke?

If you read the thread there’s mention about how being broke can push you towards the layup even if the expected value of another choice is higher. This is a great opportunity to bring ideas like “risk aversion” or “diminishing marginal utility of wealth” into practical consideration.

The expected value framework above is an optimal case. It assumes every dollar has equivalent value to the player. The fancy term for this is “risk neutral”. If you have $5,000 and making another $5,000 has a “happiness value” that is equal and opposite to the “sadness value” that you experience if you lose $5,000 then you are risk-neutral. Since you are not a robot and need to eat, you are not risk-neutral. You would not bet all your money on a 50/50 coin flip. And you probably wouldn’t do it if you had a 60% of winning the flip. You are “risk-averse”.

A related concept is the diminishing value of additional wealth. This is pretty obvious. Jeff Bezos’ first million probably felt good. Today, it would be an imperceptible amount on his Mint dashboard.

Without knowing the lingo we all understand the intuition. If you are a broke college kid you might always opt for the layup. A sure $50k might mean getting out from under that 15% credit card APR, while $100k is ‘nice to have’, not ‘need to have’. That first $50k can be life-changing by getting you off the wrong path.

Likewise, the rich gal with a vacation house in Malibu is not so constrained. She can rely on the optimal pure expected value prescription. Just as a trading firm with a huge bankroll is willing to bet large sums on small edges. They will optimize for EV when the bet sizes are small relative to capital.

Our intuition moves us in the right direction. It tells us that the college student will be more conservative in choosing which shot to take. By mixing in a simple concept like “utility of wealth”, we can actually re-price all the probabilities implied by the shot payoffs.

Adjusting Probabilities For Risk Aversion

Linear vs Concave Utility

  • Risk-neutral utility curves are linear.
    If you are the risk-neutral robot every dollar you make is worth exactly the same to you. Your second million is as sweet as the first. That’s a linear utility function. Those are the curves embedded in any expected value proposition which simply spits out “pick the highest one”. I presumed such a framework in the prior table that said: “Min Probability To Accept Shot”.
  • Risk-averse utility curves are concave
    If you are risk-averse, every additional dollar is not worth quite as much as the one before it. And every extra dollar you lose hurts just a tad more than the one before it. Losing your rent money hurts more than losing your Ferrari money. So instead of a linear function, we need a function that:

    1. Is always increasing to reflect that more money is always better than less money (‘Mo Problems and other first-world complaints notwithstanding).

    2. Slope starts out faster than the linear model then flattens as we make more money.

    Luckily, there is a simple function that does exactly that. The log or natural log function. People who study “risk-aversion” and diminishing marginal utility of wealth don’t think about it linearly. They don’t presume $5,000,000 is twice as “useful” as $2,500,000. They might say it’s only 1.75 as “useful” ( ln 5 / ln 2.5 = 1.75).

    Visually,

Re-computing Minimum Probabilities As A Function Of Starting Wealth

  • 25 year old with $10,000 to his name.
    The guaranteed layup increases his wealth by 6x and log wealth by 2.8x.
    The free throw increases his wealth by 11x but his log wealth by only 3.4x!

    Look how much it raises the minimum probabilities for him to accept various shots if he has a log wealth utility preference. He needs to shoot 3s as well as a good [contested] NBA shooter to gamble on the big money instead of the layup!

  • Give that guy a $10,000,000 bank account, and he’ll choose according to Spock-like expected value prescriptions.
  • Finally, check out the implied minimum shot probabilities for various levels of wealth. The larger your bankroll the more you can rely on probabilities imputed simply by expected value. If you are fabulously rich, you aren’t paying up for life insurance, home insurance, and so forth. You’ll deal with those bills as they come. For most of us, calamities mean financial ruin.
    How we decide depends not just on the expected value but on our own situations. The more secure we are (on the flatter section of the log wealth curve) the more we can afford to act optimally.

    (There is quite a bit of fuel for liberal policymakers here. They will realize that this is another example of Matthew effect or accumulated advantage. Richer people can avoid negative EV trades like insurance. Another thought. The inflection point on the so-called Laffer curve is probably much further to the right if we re-scale the axis in terms of log wealth suggesting we may tolerate much steeper graduated tax brackets. I’m not making a political opinion so don’t @ me. I’m just observing things that I’m sure have been discussed elsewhere.)

Conclusion

Prices impute probabilities. By taking the extra effort to make this explicit we can de-fog our relative value goggles. This improves our decision making in trading and life.

Since we are not “risk-neutral” robots the correct decisions are often theoretical. Translating the prescription to your own situation is an extra step that we typically leave to our intuition. This is quite reasonable. At the end of the day, we aren’t going to define our own wealth functions in Excel (log wealth is just one example of a non-linear function that seems to accommodate our intuition but the actual slopes and smoothness can vary quite a bit from person to person).

I recommend following Jake. His polls will help you tune your intuition.

You Can Mock Trade With A Deck Of Cards

Here’s a mock trading game I learned as a trainee to simulate futures and options market making. This game was commonly used as a day 1 exercise in trading class or when interviewing cohorts of college grads during recruiting “combines”.


The Futures Game

What you need:

  1. A deck of cards
  2. Nerdy friends (the more the better)
  3. A paper and pen per person to use as a tradelog

Setup:

You want to deal out enough cards to players (these are the market makers) so that there is about 25 remaining in the deck. There’s some leeway here.

Example:

  • You have 6 players. So deal them each 4 cards leaving 28 cards undealt.
  • Market makers may look at their hands but don’t share info.
  • The undealt cards are known as the “public pile”. They should be evenly divided into 4 or 5 sub-piles ideally (again there’s leeway depending on how many cards there are).
  • The sub-piles are going to represent “trading days”.
  • The cards themselves are news flow which will move the futures prices.

Description of futures prices:

  • The futures are the 4 suits. There’s a club’s market, a spades market, etc.
  • The final settlement price of the futures will be the sum of the ranks of cards in the public pile. (Ace =1 thru King = 13). So the maximum any future can be worth is 911

    It’s best to define the tradeable universe to keep the liquidity centralized.

    So you could have a diamond market, a spades market, and a “reds” market (which would be an index settling to the sum of diamonds and hearts).


    How To Play


    The first trading day

    • Reveal the cards in the first public sub-pile.
    • Market makers make bids and offers for the various markets. Tight 2 sided markets should be encouraged/required. For example:John: “I’m 65 bid for Hearts and offered at 68”

      Jen: “I’ll pay 67 for 5 Hearts contracts” (perhaps Jen is holding no Hearts in her hand)

      John: “Sold you 5 at 67” (John is holding 16 points of Hearts in his hand)

    • Record all your trades on your own pad or paper:1. Which contract you bought/sold
      2. Quantity of contracts
      3. Price of contracts
      4. Counterparty

    So for example, if I paid 51 for 4 “clubs contracts” from Mary I would record that information on my paper. Mary would record her sale of the 4 contracts at 51 on her card with me as the counterparty.

    • The trading is open outcry. There are no turns.

    Settling the trading day

    1. When the trading peters out for that “day” everyone should check their trades against their counterparties to make sure there are no so-called breaks or “outtrades”.
    2. On a central eraseboard or paper the “closing price” of each market can be recorded. So if the King of clubs and 3 of clubs were revealed from the sub-pile, then clubs “settled at 15”. Clubs might have traded 53 last in the expectation that more clubs will be revealed on subsequent days.
    3. Repeat this process for all remaining tradings days

    The last settlement

    • Compute “P/L” for all trades.

    If I bought 4 clubs contracts for $51 and clubs final settlement was $63 then I made a profit of $12 x 4 or $48. Mary’s loss would match that amount for that trade.

    The total P/L of all traders should sum to zero at the end of the game.

    Options Variant

    • Either the same group or a different group of people could choose to trade calls and puts on the final settlement price of the futures.

    So if I paid 3 for Clubs 55 calls and the final settlement was $63 then I profit the difference between the $63 and the strike ($55) minus the premium I outlayed:

    $63-$55 – $3 = $5

    • You could even get fancy and trade “vol”. You could sell say 10 clubs calls and buy 5 clubs futures to hedge the delta.
    • This game is played the same way the futures game is played or in conjunction. Repeat the process for all trading days then compute P/Ls at the end. Again if there are no errors the game should be zero-sum.


Mock Trading Options With Market Makers

I got into options trading straight out of college. In 2000, the option exchanges were bustling. The Amex in NYC (where I was based), the PHLX, the P-Coast, and of course the CBOE. As a trainee, your day consisted of assisting the option market makers and specialists. Building spreadsheets, running risk reports (hitting a macro, then killing some trees), and the worst part of the job — the pre-open routine of reconciling positions and breaks. Hopefully, you’d finish before your trader sauntered into the office hungover.

During the actual trading day, your duties were pretty limited. I remember going to Cafe World at the corner of Trinity and Rector with a diagram of where my trader wanted each dish from the buffet arranged on his plate. Although you aren’t paid much you are still a liability for your first 6-12 months.

Mock Trading

Your main purpose in the cocoon phase is to learn. After the market close, you’d attend “mock” which was short for mock-trading. Mock would be led by senior traders. “Senior” basically meant a market-maker that was now “on a badge” the credential you needed to trade on the floor. You were getting taught by people that ranged from 1 to 5 years older than you which should tell you a) how start-uppy the culture and b) how much every day’s hundreds of trades added up to valuable experience quickly.

At my firm, mock was basically hunger games. You’d stand around for an hour shoulder to shoulder with 15 guys (yes it was mostly guys) in front of a dry erase board as 3 or 4 senior traders posed as brokers barking out orders and moving the stock and option prices around setting up opportunities for the trainees to spot arbitrages.

You’d have to hedge your trades (nothing like selling one of your teacher’s some puts as another teacher announced the stock bid was now 25 cents lower), lean markets based on what prevailing bids or offers were “resting” on the exchange book, read body language, remember all the verbally announced orders that might have been announced but were not in play until the stock moved. Memory, pattern recognition, and extremely fast mental math. In fact, everyone in the room would play a timed put/call parity computer game during the day to prepare (I actually trained during the tail end of the fractions era).

So for fun, I thought I’d share an example of what mock trading would be like.

Spot The Edge

Requirements and assumptions:

  • Stay delta-neutral. If you want to buy or sell the stock you must cross the spread.
  • Options markets are all 500 up, meaning the bids and offers have 500 contracts on them.
  • Cost of carry = 0%
  • 90 days until expiration
  • You will need to know Put/Call Parity

    Call = (Stock Price – Strike Price) + Put + Cost of Carry

    Since there’s no cost of carry let’s restate this more simply:

    Call = Intrinsic + Put

Ok, here’s the option’s board:

A broker walks into the pit and announces:

I have 200 XYZ 55 straddles offered at $4.15!

I’ll get you started with a hint. Be the first person to yell: “Buy em!”

Now go figure out why.

Here are the exercises you can do with the information above.

  1. Compute the implied volatility.
  2. Find the arbitrages or best series of trades in conjunction with the broker orders that are being shouted into the pit.
  3. Report your remaining position and at what average price it was established.

    Extra credit: Compute your P/L. You may reference an option model after the mock trading session ends.

It’s all spoilers ahead so if you actually want to do this, don’t scroll further until you are done.

Answers

  • Compute the implied volatility

The approximation for the ATM straddle is given by the expression1 :

Straddle = .8Sσ√T

where S = stock price
σ = implied volatility
T = time to expiry (in years)

Let’s use mid-market of the 55 put and put/call parity to get the call price.

C = Intrinsic + P

C = 0 + $2.10 = $2.10

Since the straddle is just C + P we get $4.20 for the straddle. Plugging into the approximation:

$4.20 = .8 x $55 x σ x √.25

Solving for σ we get an implied volatility of 19%

  • What series of trades do we do?

    1. Buy 200 55 straddles for $4.15
    2. Sell 200 55 calls at $2.15
    3. Sell 400 60 calls at $1.05
    4. Buy 400 65 calls for $.05
    5. Sell 200 65 puts at $10.10
    6. Sell 3,000 shares of stock for $54.95

    Whoa. That’s a lot of trading. Because of put/call parity, traders can collapse their thinking and position by strike. A call is a put and a put is a call. You can always convert one into the other by taking the opposing delta in the underlying.

    Let’s summarize these trades by strike.

    65 Strike

    Buy 400 65 calls for $.05
    Sell 200 65 puts at $10.10

    Re-factoring:

    1. Buy 200 65 calls for $.05 and sell 200 65 puts at $10.10. Buying a call and selling a put on the same strike is known as a combo. It is the same thing as synthetically buying the stock. Why? Think about it, no matter what happens you will be buying the stock for $65 at expiration. You’ll either exercise the call or be assigned on the put. But you collected $10.05 today to make that commitment so you effectively bought the stock for $65 – $10.05 today or $54.95. Sweet.

    So this can be summarized simply as buying 20,000 shares for $54.95

    2. You also bought 200 extra 65 calls for .05

    60 Strike

    Sell 400 60 calls at $1.05

    55 Strike

    Buy 200 55 straddles for $4.15
    Sell 200 55 calls at $2.15

    Since you bought 200 straddles, you bought 200 calls and 200 puts. The calls cancel out and you are left long 200 puts at a net price of $2.00 (spent $4.15 200x in straddle premia and collected $2.15 200x in call premia).

    Now remember we synthetically bought 20,000 shares for $54.95 via the 65 strike combos.

    Back to put/call parity.

    C = (S-K) + P
    C = ($54.95 – $55) + $2.00
    C = $1.95

    So the combo plus these 200 55 puts means you legged buying 200 55 calls for $1.95

  • What is our residual position and at what average price?

    Let’s do what option traders do and show the net position by strike. That’s how we see what we actually have on. It allows us to make sense of the complexity at a glance.

  • 1. First, we can see the 200/-400/200 pattern on equidistant strikes (ie they are each $5 apart). That is a butterfly. A relatively low-risk distributional trade that has very little vega, gamma, and theta with several months until expiration.

    What price did we leg it for?

    Recall:

    1. We bought 55 strike call synthetically for $1.95
    2. We sold 2x as many 60 calls at $1.05
    3. We bought the 65 calls for $.05

    Adding up, $1.95 + (2 x -$1.05) + $.05 = -$.10

    Negative 10 cents?

    Correct. You just legged buying a structure that can never be worth less than zero for a credit. Arbitrage.

    What is the delta of our total position?

    Option traders want to stay delta-neutral. So estimating the deltas (or having Black Scholes spit them out) we compute the delta contribution of each strike and find we must sell or short 3,000 shares to be delta neutral.

  • Extra Credit: What’s the P/L?

    Butterfly P/L

    Using a flat 19% implied vol I get a Black Scholes value of $.93 for the butterfly. We actually got paid $.10 to own it. So our theoretical profit or edge is $1.03 times 200 contracts.

    $1.03 x 200 contracts x 100 multiplier = $20,600 profit

    Combo or Synthetic Stock P/L

    We bought 20,000 shares of stock synthetically for $54.95 via the 200 65 strike combos. If the stock is marked at mid or $55.025 then we made $.075 on 20,000 shares or $1,500.

    Stock P/L

    We did need to sell 3,000 shares at $55.00 (the bid) to hedge 3,000 shares or deltas. If the stock is marked at mid or $55.025 then we lost $.025 on 3,000 shares or $75

    Total profit: $20,600 + $1,500 – $75 or $22,025!

Wrapping Up

Back in those olden days, we’d play this game after market hours but you can imagine multiple brokers shouting orders at the same time and more months than just a single expiry. We studied many different types of arbitrage relationships so we could spot mispricings from many angles.

You’d take what you learned from these games and apply it during the trading day. You’d watch how market makers and brokers in the pits reacted to different orders as you start to piece the matrix together. At my firm, the people who performed best were sent to a Philly suburb for 3 months. This was known as “class” and it was held 4 times a year. “Class” was theory and option nerd stuff until lunch then mock for the rest of the afternoon. Mock had a simulation environment with electronic overhead screens just like the exchanges and everyone held a tablet PC with stock trading software and a proper option model. This is where you started going beyond mock and getting into more game theory and real-life trading scenarios.

The faster you got into a “class” cohort the faster you got your own “badge”, P/L, and risk budget (not to mention enough comp to rent a 400 sq ft studio without a roommate).

Times have changed. The game isn’t about mental math and yelling loud and having the best memory. But this was how my intuition was built up and the lessons still permeate how I think about trading today.

Finding Vol Convexity

In this post, we will learn what it means for a position to be convex with respect to volatility.

In preparation for this post, you may want a refresher.

  • Vega is the sensitivity of a P/L to changes in volatility. This is the exposure volatility traders are taking active views on. It requires tremendous attention since changes in vol directly affect P/L via vega but also impacts or distorts the “moneyness” of all options in a portfolio. In that way, large vega exposures are signs that the risks under the hood of a portfolio are especially dynamic.

    Refresher Post: Why Option Traders Focus On Vega (Link)

  • Convexity is the idea that there are non-linear P/L sensitivities within a portfolio. The curvature of the P/L derives from the fact that the exposures change as the market moves. Option deltas are not constant. That means deltas derived from options, as opposed to deltas derived from so-called “delta one” instruments like common stock or futures, are subject to change as the market moves. Vol convexity is the same phenomenon. Instead of applying to a delta, it applies to vega.

    Refresher Post: Where Does Convexity Come From? (Link)

In Moontower style we will do this without anything more than middle school math. This 80/20 approach provides the intuition without the brain damage that only a relative handful of people need to know.

Mapping Directional Trading To Volatility Trading


Directional Traders

Most investors are looking to profit from the direction of stocks.  Stated another way, most investors are taking active delta exposures. The size of their delta determines the slope of their P/L with respect to the market’s movement.

Directional Convexity

Some of these investors use options to make directional bets. This gives their positions convexity with respect to the changes in stock (also known as gamma). The convexity derives from the fact that their delta or P/L slope changes as the stock moves.

Volatility Traders

Now consider another, much small, class of investor. The option traders who try to keep delta-neutral portfolios. They are not seeking active delta exposure. They have no alpha in that game. Instead, they are taking active vega exposures. The size of their vega determines the slope of their P/L with respect to changes in implied volatility.

Volatility Convexity

Like the directional traders who use options, vol traders maintain convex exposures with respect to changes in the stock. Again, that’s gamma. But vol traders are much more focused on vol convexity. The reason vol traders focus on this more than directional traders is that vol traders typically run large portfolios of options across names, strikes, and tenors. These portfolios can include exotic and vanilla options. The presence of vol convexity means vol changes propagate through the entire portfolio in uneven ways. Risk managers model how vega exposures morph with vol changes.

For directional traders with just a few line items of options on their books, vol convexity is going to be much further down on the list of concerns. Somewhere in between “What’s for lunch?” and getting flamed by intern on Glassdoor.

Maximum Vega

Vol traders often think in terms of straddles. In fact, in many markets, brokers publish “straddle runs” every few hours. This is just a list of straddle prices and their implied vol per expiration.

At-The-Money Vega

A handy formula every novice trader learns is the at-the-money straddle approximation1:

Straddle = .8Sσ√T

where S = stock price
                        σ = implied volatility
                                     T = time to expiry (in years)

So if there is 1 year until expiration, the 1 year ATM straddle on a 16% vol, $50 stock is $6.40 (.8 x 50 x .16).

So if implied volatility goes up 1 point to 17% how much does the straddle change?

.8 x 50 x.17 = $6.80

So the straddle increased by $.40 for a 1 point increase in vol. Recall that vega is the sensitivity of the option price with respect to vol. Voila, the straddle vega is $.40

More generally this can be seen from re-arranging the approximation formula.

Vega = Straddle/σ = .8S√T

Ok, so we have quickly found the ATM straddle price and ATM straddle vega. Look again at the expression for the straddle vega.

 .8S√T

There are 2 big insights here. The first can be seen from the expression. The second cannot.

  1. The vega of the ATM straddle does not depend on the level of implied vol.

    The vega only cares about the stock price and time to expiration. So whether you are talking about a $50 crazy biotech stock or a $50 bond ETF the 1-year vega is exactly the same even if the straddle prices will vary according to the implied vols.

  2. The vega of the ATM option is the maximum vega of any option in that expiry.

    This statement implies that the vega of an option varies by strike. All of the other strikes have a lower vega. They are less sensitive to vol than this one. That makes sense. This option has the greatest extrinsic value.

    (I have a confession. The maximum vega actually occurs at the 50% delta option strike, not the at-the-money or at-the-forward. I used ATM because it is more intuitive. The hand-waving should not trouble you. Going forward I will use the .50 delta option for the charts. If you need a refresher see my post Lessons From The .50 Delta OptionDon’t worry, the intuition is not going to change if you fail to appreciate the difference)

Vega Across Strikes

While we were able to compute the vega for the ATM straddle to be $.40 from the straddle approximation, how about the rest of the strikes?

For those, we need to rely on Black Scholes. You can find the formula for vega anywhere online. Let’s feed in a $50 stock, 0 carry, 16% vol, a 1-year tenor, and a strike into a vega formula. We will do this for a range of strikes.

Here’s the curve we get:

This chart assumes a single option per strike which is why the vega of the .50 delta strike is $.20 (not $.40 like the straddle vega).

The big takeaways:

  1. The vega of a non-.50d option does depend on the level of vol.
  2. There is a maximum vega any option can have and it occurs at the .50d option


The Source of Convexity

If option traders’ profits are a function of vol changes, then their vega positions represent the slope of that exposure. If the vega of the position can change as vol moves around then their position sizes are changing as vol moves around. The changes in exposure or vega due to vol changes create a curved P/L.

Let’s see how changes in volatility affect vegas.

When Vol Increases All Strikes Become Closer To .50 Delta

Here’s the vega by strike chart the same stock. The blue line assumes 16% vol across all strikes. The red line is 32% vol across all strikes.

In fact, imagine overnight, the stock’s vol doubled from 16% to 32%.

The maximum vega at any strike is still fixed at $.20, it just occurs at the new .50 delta strike. The .50 delta strike moved up $2 or about 4% but look how the vega of the options at nearly every strike increased. This is intuitive. If you double the vol then a strike that used to be 1 standard deviation away is now 1/2 a standard deviation away. All the OTM deltas are creeping closer to .50 while of course, the .50 delta option remains .50 delta.

Watch How Your Position Changes

You can start to see the reason why a position can be convex with respect to changes in vol. Imagine you were long the .50 delta option and short the way OTM 90 strike call.

  • At 16% vol the call you are long has $.20 of vega and the call you are short has 0 vega. You are unequivocally long volatility. Even if you are long 1 .50 delta call and short 10 90 strike calls you are long vol (1 x $.20 + (-10) x $0). Your portfolio’s net vega is long $.20 of vega

  • At 32% vol, the call you are long has slightly less than $.20 of vega since the .50 delta option has shifted to the right. Let’s still use $.20 to make the point. The calls you are short now possess $.05 of vega. Your new position vega computed as (1 x $.20 + (-10) x $.05) or -$.30 of vega. You are now short vol!

Your vega which represents your slope of P/L with respect to vol has changed simply by the vol changing. The higher the vol goes, the short vol you become.

Observe:

  • The strikes near the meat of the distribution can only gain so much vega. Remember, maximum strike vega is only a function of spot price and time to expiry.
  • Further OTM options become “closer” to 50d. This pushes their vega up relative to the ATM option.

This chart shows vega profiles across strikes over a wider range of vols. At extreme vols lots of strikes look like .50d options!

Trade Examples

Long ATM option, short OTM option. (Long vega, short “vol of vol”)

Starting conditions:

Stock price =  $50
Implied vol = 16%

Portfolio

Long leg

        • 1 .50 delta call @$50.64 strike (approximately ATM)
        • Vega =$.20
        • Premium = $2.90

Short leg

        • Short 1 .14 delta call @$60 strike (approximately 20% OTM)
        • Vega = $.11
        • Premium = $.55

Summed as a vertical call spread

        • Premium = $2.90 – $.55 = $2.35
        • Vega = $.20 – $.11 = $.09
        • Note the position is long volatility


Now let’s change implied vol up and down.


It’s a busy picture. Let’s walk through the scenarios:

We start at 16% vol and increase vol

        • Call spread value increases (solid green line) because the position is long vega. Your P/L is rising since this is the position you are long.
        • However, the ATM call vega (blue dash line) stays relatively fixed while OTM call vega increases (red dashed line) causing the call spread vega (green dashed line) to decline from its initial value.
        • As vol increases your vol length is decreasing. Whoa, this looks like negative gamma with respect to vol!

We start at 16% vol and decrease vol

        • Call spread value decreases (solid green line) because the position is long vega. Your P/L is falling since this is the position you are long.
        • However, the ATM call vega (blue dash line) stays relatively fixed while OTM call vega decreases (red dashed line) causing the call spread vega (green dashed line) to increase from its initial value.
        • As vol declines your vol length is increasing. Whoa again, this looks like negative gamma with respect to vol!

Ratio Trade. Short ATM option, long extra OTM options. (Vega neutral, long “vol of vol”)

Same starting conditions:

Stock price =  $50
Implied vol = 16%

We are targeting a vega-neutral portfolio

New Portfolio

Long leg

        • Some amount of .14 delta calls @$60 strike (approximately 20% OTM)
        • Vega per option =$.11

Since the short leg has $.20 of vega and our long leg has $.11 of vega we need to buy 1.75 of the 60 strike OTM calls ($.20 / $.11) to have a net flat vega position.

        • Total vega for the long leg: $.20
        • Premium per option = $.55
        • Total premium = $.9625 ($.55 x 1.75 contracts)

Short leg

        • Short 1 .50 delta call @$50.64 strike (approximately ATM)
        • Vega = $.20
        • Premium = $2.90

Summed as a ratioed vertical call spread

        • Premium = $2.90 – $.9625 = $1.9375
        • Vega = $.20 – $.20 = $0
        • The position is flat volatility


You know what’s coming. Let’s change the implied volatility and look at the structure price. Remember you shorted the ATM option at $2.90 and bought 1.75 OTM calls for a total premium of $.9625.  

In other words, you shorted this structure for an upfront premium of $1.9375. Watch what happens to its value when you raise or lower the implied vol.

To understand why the structure behaves like this, look at the scenarios.

We start at 16% vol and increase vol

        • While both your longs and shorts increase in value, your longs pick up extra vega, while your shorts are already as sensitive as they will ever be to vega. So every uptick in vol causes the structures net vega to become long vol.
        • By being short this structure you are getting longer vol as vol increase. If your vol exposure gets longer as vol increases your exposure is convex with respect to vol.

We start at 16% vol and decrease vol

        • The opposite scenario occurs. As vol declines your short option leg has a fixed vega while your long vol leg that is OTM “goes away” as it’s vol declines. If vol is very low that call is extremely far OTM in standard deviation space. Imagine the extreme downward vol shock scenario — the stock is taken over at its current $50 for cash. All the options go to zero, the structure goes to zero, and you simply collect the premium you sold the structure at.
        • By being short this structure you are getting shorter vol as vol declines. If your vol exposure gets more short as vol falls your exposure is convex with respect to vol.

One last chart to drive it home. The green line is your P/L as vol changes. Notice that your max P/L in the vol declining scenario is $1.9375, the entire value of the structure. It is unbounded on the upside. It looks like the more familiar picture of being long a straddle! The fact that the P/L chart is curved and not linear is convexity and as we know, results from the size of exposure changing with respect to vol.

The blue line shows exactly how that vega exposure changes with respect to vol. You started vega-neutral. As vol increased you got longer. As vol fell you got shorter.


Caveats And FYIs

This was intended to be an introduction. But here’s a non-exhaustive list of “gotchas”:

  • Option surfaces usually have a skew. OTM options often trade at a premium volatility to ATM options which reduces the spread of vegas between the options. Less room for relative narrowing.
  • We haven’t talked about the cost of those premium vols. Short gamma, paying theta anyone?
  • In these examples, we shocked the vols up and down uniformly across the strikes. I’ll leave it to you to consider what adding a fixed amount of variance per strike would do to a vol surface.
  • I completely ignored the fact that as you change the vols you are changing the location of the .50 delta option, or for that matter the delta of every option. In other words, I showed fixed strike vol behavior assuming a uniform shock. Adjusting for that is recursive and frankly unneeded for the intuition.
  • Volga is the term for the sensitivity of an option’s vega with respect to vol. Volga itself changes as vol changes. That .50 delta option has starts with little sensitivity to vol. But if we crank vol higher that option moves further from .50 delta as the new .50 delta strike has moved somewhere to the right. So it follows that the old .50 delta is picking up volga, or sensitivity to vol. Again, no need to go full Christopher Nolan I’m just leaving breadcrumbs for the committed.

The main intuition I want you to get is that OTM options are sensitive to the vol of vol because their vegas can bounce around between 0 and the maximum vega. ATM options are already at their maximum vega. So structures that own extra options relative to be being short the ATM are convex in vol.

Conclusion

Vol convexity is important because changes in vol influence much of the greeks. Understanding the concept can be used for defense and offense. Vol directly impacts option prices according to their vega. But it also changes their vega.

Who should care?

  • Anyone who wonders how a nickel option can go to $20.

When you combine the convexity of options with respect to vol (volga) with the convexity of options due to changes in the stock price (gamma) you get nitroglycerine.

  • That segment of the market who takes active views on volatility, not direction.

Remember even when dealing with non-linear instruments, a snapshot of a portfolio at a single point in time might show it to be vega-neutral. But a photo of a car can make it look parked. Only the video can show how fast the car can move.