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.


  • 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.


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.


  • 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.


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.


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


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).


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.)


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


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.


  • 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.


  • 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


    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?


    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.


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.


  • 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%


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.


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.

Where Does Convexity Come From?

Unless you are a bond or derivatives trader, the term “convexity” usually just makes you want to say “beat it, nerd”.  As if to frustrate us further it has many aliases: “curvature”, “non-linearity”, “gamma”.  It turns out, for such a fancy word “convexity” is quite approachable.

In this post, we will learn:

  • what it looks like
  • common places to find it
  • why you should care

But first, let’s see what convexity isn’t.

Linear P/L

If you buy 1 share of stock for $50 and it goes up 5% you make $2.50. You make 5%. If it goes up 10% you make 10%. Your P/L is linear with respect to what the stock does.

What about leverage?

Leverage does not change the fact that owning a stock has a linear payoff. Leverage may amplify the volatility of the position but not the linearity.  In the chart below, the Levered P/L assumes you buy the same stock but use 50% margin (you borrow $25 out of the $50 it took to buy the stock). So if the stock returns 20% ($50 –> $60), you return 40% ($10 profit on a $25 outlay). In other words, the slope of your P/L with respect to the stock is 2x the unlevered return.

Futures, since they are traded on margin, are levered. But like stocks, they have a linear payoff.  How about a house? If you buy a house with 20% down and it appreciates 20%, you will make 100% on your money. If it appreciates 40% you will make 200%. Your P/L slope is 5x because of the leverage but most importantly it is constant. The rise/run to give you geometry flashbacks is always 5. You can change the slope by paying down the mortgage but changes in the slope are not an intrinsic feature of the investment. Most assets we are used to buying or selling have linear payoffs.

Leverage increases the volatility of the position but leverage does not mean convexity.

Exposures As Deltas

The term “delta” is familiar to many as the option greek which represents a hedge ratio. It answers the question, how much will my option value change per $1 change in stock. I can use that ratio to compute how much stock I would need for an equivalent position.

100 50% delta calls is equivalent to 5,000 shares of stock (100 x .50 delta x 100 share multiplier per option). If I were bullish, I could buy the stock and say I’m long 5,000 deltas. I could also have chosen to buy 100 50% delta calls. I would still say I’m long 5,000 deltas. This allows me to collapse either exposure into the statement “if the stock goes up $1, I will make $5,000”.

We are now using the word “delta” to refer to an exposure. We are agnostic as to whether it’s stock or call options. Either way, we have the same root on — we want the stock to go up, and we know how much we will make when it does.

In the stock example, every dollar change led to a constant P/L change. This is because your exposure is staying constant. We can refer to your exposure as your delta. Instead of referring to the sensitivity of an option to its stock, we are using delta as the slope of your P/L with respect to the stock. This allows us to generalize any exposure into a slope.

Example of a typical call overwriter:

  • You own 3,000 shares of stock = long 3,000 deltas.
  • You are short 30 20% delta calls (equivalent of 600 shares) = short 600 deltas

Net delta: Long 2,400.

This represents the slope of your P/L with respect to the stock. If the stock goes up $1, you make $2,400. If the stock goes down $1, you lose $2,400

In fact, this is how pros combine their positions. They want to know their aggregate delta, or exposure due to changes in the stock price. In fact, pros may even use the correlation between stocks to estimate a net delta with respect to say the SP500. A fund that is trying to stay market neutral might have a basket of alpha-generating longs hedged with a delta equivalent amount of short index futures.

A typical end-of-day report might include the firm’s best estimate of delta with respect to the Nasdaq, delta with respect to SP500, delta with respect to Eurostoxx, and so forth.

It’s worth restating:

Think of a position as a “delta” — the slope of your P/L to a market. 

Not All Deltas Are Created Equal

Let’s compare the deltas of a long stock position vs a long call position. Both positions are long 5,000 deltas

  • Stock position

Instrument: 5,000 shares
P/L: $5,000 per $1 change in stock

  • Call Position

Instrument: 100 50% delta calls
P/L: $5,000 for the first $1 change in stock

Bang! Imagine the stock went to $100. At $100, the stock position is still making $5,000 per $1 change. But the call position is now making $10,000 per $1 change.


Because the 50 strike is so deep in-in-the-money that the calls now have 100% delta. The calls are now equivalent to a 10,000 share position! (100 calls x 1.00 delta x 100 share multiplier). The call owner is long 10,000 deltas now. At $100, your delta per $1 change is $10,000 not $5,000.

Here’s the delta of that 1 year 50 strike call at different stock prices.

Contrast this with the stock owner, who still has just 5,000 deltas. Pure stock is a linear exposure. Linear instruments have constant slopes with respect to the market. They are benign. If you are long 5,000 stock deltas you don’t suddenly find yourself long 10,000 deltas. Stock and futures can be said to be 100% delta instruments. The P/L moves 1 for 1 with the asset price.

But options payoff are non-linear because your delta, your exposure, is actually changing. This is the key to identifying convexity. Payoffs are non-linear not because the asset is volatile. Payoffs are non-linear because the delta of your position is changing. This in turn causes the slope of your P/L with respect to the market’s moves to change.

If you are dealing with an instrument whose exposure changes for the same given change, you are dealing with a non-linear exposure.

Curved P/L

Curvature is a nice term because it reminds us that we are concerned with the shape of a payoff. Curvature indicates that the slope of your payoff changes. Your delta is morphing due to time, changes in volatility, and in these examples, due to changes in the asset price.

If the slope of the payoff is changing then our P/L will curve with respect to changes to a variable or asset price.


Earlier we saw that stock P/L’s are constant. They have a constant slope because their deltas don’t change. Your P/L matches the stock’s changes. It’s just weighted by the size of your position.

Option deltas do change. We know a call delta varies between 0 and 100%. Imagine a stock is trading for $40. You buy a 1 year 50 strike call @ 16% vol. It will start with a 9% delta. Curvature refers to the idea that the delta of that call does not stay constant. As the stock rallies, your delta grows allowing you to make much more than $.09 per $1 change.

When we zoom into the classic hockey stick graph of a call option value with respect to stock we can see the curvature. It’s the difference between assuming a constant $.09 per dollar and what actually happens as the option gains sensitivity to the stock price.

If you drive a car 30mph for 30 minutes you will travel 15 miles. If 10 minutes into the trip you instantly accelerated to 60mph and stayed at the speed for 20 more minutes you will find that you have traveled 25 miles. The difference between 25 miles and your original linear estimate of 15 miles is curvature.

In the analogy, your velocity at any one point in time is your delta. The change in velocity as you went up a gear was a change in delta (and Greekophiles will recognize the acceleration as gamma).

If you want the opportunity to test your understanding go back to the call overwriter example:

  • You own 3,000 shares of stock = long 3,000 deltas.
  • You are short 30 20% delta calls (equivalent of 600 shares) = short 600 deltas

Net delta: Long 2,400

Can you estimate your net delta if the stock rips higher and the call becomes 100% delta? How about if the stock tanks and the calls become 0% delta? Your net delta will tell you how sensitive your P/L is to the change in the stock price. It’s not going to be $2,400 per $1 anymore.


Bonds have naturally curved payoffs with respect to interest rates. Consider the present value of a note with the following terms:

Face value: $1000
Coupon: 5%
Schedule: Semi-Annual
Maturity: 10 years

Suppose you buy the bond when prevailing interest rates are 5%. If interest rates go to 0, you will make a 68% return. If interest rates blow out to 10% you will only lose 32%.

It turns out then as interest rates fall, you actually make money at an increasing rate. As rates rise, you lose money at a decreasing rate. So again, your delta with respect to interest rate changes. In bond world, the equivalent of delta is duration. It’s the answer to the question “how much does my bond change in value for a 1% change in rates?”

So where does the curvature in bond payoff come from? The fact that the bond duration changes as interest rates change. This is reminiscent of how the option call delta changed as the stock price rallied.

The red line shows the bond duration when yields are 10%. But as interest rates fall we can see the bond duration increases, making the bonds even more sensitive to rates decline. The payoff curvature is a product of your position becoming increasingly sensitive to rates. Again, contrast with stocks where your position sensitivity to the price stays constant.


Convexity is a confusing concept. The confusion I think stems from the fact that convex exposures lead to headline-grabbing P/Ls (usually losses). It can be easy to confuse that with huge losses because people were simply levered in highly volatile assets. Confusion is doubly justified when the biggest P/Ls are a combination of leverage plus convexity. Features that often reside in the same instruments such as options.

To keep the idea of convexity straight (you see what I did there), don’t just focus on how a position responds to the market. That only tells you your delta. That’s a snapshot in time. Instantaneous.

Instead, pay extra attention to how the position’s sensitivity to the market changes. Sensitivity, delta, and slope are the same idea. If the slope of your P/L is changing you are playing with convexity. Expect a curved P/L.

Curves are exponential functions. The year is 2020. By now everyone knows how many times you can fold a piece of paper before it reaches the moon. Understanding convexity is par for our times.

Why Option Traders Focus on Vega

Options are derivatives. They derive their value from how the underlying actually moves as well as the market’s perception of how much they will move. So there’s a realized and implied component to the value of an option. When people start using options they are usually attracted to them as an inherently levered way to hedge or speculate on a stock. In other words, they are interested in options as a bet on direction.

In the course of trading options, directional players tend to acquire a solid understanding of delta and gamma.

  • Delta is the option’s sensitivity to the stock price

The more OTM the option is the less sensitive it is to the stock price. Simply the option’s “moneyness” drives it’s delta

  • Gamma is the option delta’s sensitivity to the stock price

Formally, gamma is the second derivative the option’s value with respect to the stock price. Intuitively, it is a measure of how the option’s delta changes as the stock’s moneyness changes. If a call is 20% OTM it may have a low delta. Say 5%. But if the stock suddenly surged 20%, the call would be ATM or 50% delta. It is now highly sensitive to the stock price. That increased sensitivity was the visible effect of gamma.

The movement of stocks is driving the value of options by pushing a giant universe of options in and out of the money at all times. The directional players are getting their fix whether its hedging or punting. This creates a robust ecosystem of buyers and sellers who are able to make very specific bets targeting the size and timing stock moves.

Enter Vol Traders

As the directional traders sling option prices around based on their outlooks for stocks, a much smaller segment of traders will notice a relative lack of attention on the other significant driver of an option’s price — it’s implied volatility. The perception of how much a stock will move in the future sets the price of the option today. In fact, it’s the largest unknown in an option price. With continuous bid/ask prices for stocks, a highly liquid interest rate market, and well-estimated dividends the other inputs into listed option prices are trivial.

So this smaller group of professional traders is actually buying and selling levels of a derived value — implied volatility. This is not strange. A stock trader is dealing in implied levels too — forward earnings. They are converting those to multiples to make comparisons of risk/reward across stocks. Option dealers do the same. They compare levels of volatility to find bargains or to sell overpriced perceptions of future volatility.

So in addition to delta and gamma, option pros are focused on vega — the change in an option price due to changes in implied volatility.

A Basic Example of Vega

Vol traders sometimes think in terms of straddles. To be long a straddle is to own the call and put on the same strike. If you own an ATM straddle you are agnostic on direction, you are just rooting for a large move. Straddles, since they are just the sum of a call and put, can be used to bet on implied volatility changing as well.

If a straddle is worth $10.00 and has a vega of $.30 then we can say for every 1 point move in the vol the straddle will change by $.30. So if vol increases 1 point, the straddle will appreciate to $10.30

This should remind you of delta. If the straddle had a delta of 30% then for a $1 increase in the stock the straddle would increase by $.30, also to $10.30. Delta describes how an option responds to the stock while the vega describes how the option responds to the vol.

Vega As An Exposure

If you are a stock trader you will measure your risk in how many dollars you are long or short. This allows you to answer what your p/l is for say a 1% move. Long $1,000,000 worth of stock and it goes up 5% you make $50,000.

We can do the same with vega. If I own 100 of those $10 straddles then I can say I’m long 3,000 vega:

100 x $.30 x 100 = 3,000

              # of straddles x vega per straddle x option multiplier = 3,000


So if volatility increases by 1 point, the straddles increase to $10.30. Since I owned 100 straddles the value of my position goes from $100,000 1 to $103,000 for a net p/l of $3,000.

If vol started at 20% and actually increased 10 points to 30%, the p/l would have been $30,000:

3000 vega x 10 vol points


Vega Influences Everything

When a directional stock trader sizes their position, the volatility of the stock is a key input. So it follows that one of an option trader’s primary concerns is how volatile the implied volatility is. As we saw above, a shift in perceptions of future volatility can lead to significant gains or losses due to vega exposure.

There’s more.

While vega measures a direct exposure, namely the p/l of an option position due strictly to changes in implied vol, it also acts as an indicator that a position has additional sensitivities to implied volatility. Recall that delta and gamma are driven by moneyness (aka how far in or out of the money) a position is.

But moneyness depends on volatility!

If a stock is trading for $50 we might compute that the 1 year $60 strike call which is 20% OTM is 1 standard deviation away for a given vol. Perhaps it has a 15% delta. What if it’s volatility quadruples?

That strike which is $10 away in fixed dollar space is now much closer is standard deviation space. In fact, it’s delta could now be 50%! 2 So the size of the position’s vega indicates the potential for change in the entire portfolio’s Greeks.

An option portfolio’s dynamic properties can lead to very complex exposures quickly as sensitivity to volatility propagates thru every line item in a portfolio. Professionals will use many more types of Greek’s to measure individual sensitivities. How does your delta change as time passes? How does your delta change as implied vol declines?

You can get as nerdy and esoteric as you want in trying to flatten your exposure to every greek. In practice, nobody does this (well maybe the French), but because portfolios have complex dependencies on implied volatility it’s handy to remember that large gross and net vegas point to the possibility that a portfolio’s risk is quite dynamic under the hood.


Next Step: Vol convexity

The importance of vega warrants further discussion. In the next part, I will cover the “gamma of vol” — how vega itself responds to changes in volatility. Just like a directional trader might use option gamma to acquire convexity with respect to the stock price, the option trader is looking to acquire convexity with respect to volatility.

Pro’s call it volga, but in the spirit of Moontower we will build an intuition without calculus. Stay tuned…



Lessons From The .50 Delta Option

I was chatting with a quant friend who was bouncing an options idea off me. In the course of the conversation, he was surprised I did not assume the .50 delta option was the ATM (at-the-money) option. My friend is much smarter than me on finance stuff but options aren’t his native professional language. So if this idea had him tripped up I realized I had a reason to write a post.

If I do this correctly you will gain a better understanding of:

  1. Delta as a hedge ratio, not a probability
  2. How volatility affects the mean, median, and mode of these returns
  3. The relationship of arithmetic to geometric returns in option theory
  4. What these distributions mean for the value of popular option structures

Some housecleaning:

  • Option math is known for being calculus heavy. If you are a layperson, you are in luck, this tour guide likes to stick to the roads he knows. You won’t find complex equations here. If you are a quant, I suspect you can still benefit from an intuitive approach.
  • We are going to ignore the cost of carry (interests and dividends). While crucial to actual implementation it is just distracting to the intuition.

Delta Is A Hedge Ratio Not a Probability

Often delta and “probability of finishing ITM (in-the-money)” are indistinguishable. But they are not the same thing. The fact that they are not equivalent holds many insights.

Before we go there, let us revisit the most basic definition of delta.

Option delta is the change in option price per $1 change in the underlying

Consider the following example:

Stock is trading for $1. It’s a biotech and tomorrow there is a ruling:

    • 90% of the time the stock goes to zero
    • 10% of the time the stock goes to $10

First take note, the stock is correctly priced at $1 based on expected value (.90 x $0 + .10 x $10). So here are my questions.

  • What is the $5 call worth?

Back to the expected value:

    • 90% of the time the call expires worthless.
    • 10% of the time the call is worth $5

.9 x $0 + .10 x $5 = $.50

The call is worth $.50

  • Now, what is the delta of the $5 call?

$5 strike call =$.50

Delta = (change in option price) / (change in stock price)

    • In the down case, the call goes from $.50 to zero as the stock goes from $1 to zero.

Delta = $.50 / $1.00 = .50

    • In the up case, the call goes from $.50 to $5 while the stock goes from $1 to $10

Delta = $4.50 / $9.00 = .50

The call has a .50 delta

Using The Delta As a Hedge Ratio

Let’s suppose you sell the $5 call to a punter for $.50 and to hedge you buy 50 shares of stock. Each option contract corresponds to a 100 share deliverable.

  • Down scenario P/L:

Short Call P/L = $.50 x 100 = $50

Long Stock P/L = -$1.00 x 50 = -$50

Total P/L = $0

  • Up scenario P/L:

Short Call P/L = -$4.50 x 100 = -$450

Long Stock P/L = $9.00 x 50 = $450

Total P/L = $0

Eureka, it works! If you hedge your option position on a .50 delta your p/l in both cases is zero.

But if you recall, the probability of the $5 call finishing in the money was just 10%. It’s worth restating. In this binary example, the 400% OTM call has a 50% delta despite only having a 10% chance of finishing in the money.

I’ll leave it to you to repeat this example with a balanced distribution. Say a $5 stock that is equally likely to go to zero or $10. You will find the 50% delta call turns out to be ATM. Something you are used to seeing.

The key observation turns out to be:

The more positively skewed the distribution, the further OTM the 50% call will be. If a stock is able to go up 1000% and you sell a 400% OTM call on it you are going to need far more than a token amount of long stock to hedge.

The more positively skewed a distribution, the more the hedge ratio diverges from the “probability of finishing ITM”.


The Effect of Pure Volatility

Not to lead the witness too much, but an obvious feature of the binary example is the biotech stock is very volatile. That’s not a technical definition but a common-sense observation. “This thing is gonna move 100% or 900%!”.

Without math, consider how volatility alone affects a stock’s returns. If the stock price remains unchanged because we do not vary the expected value but instead inject more volatility what is happening?

  1. We are increasing the upside of possible payoffs.

In the biotech example, more volatility can mean the upside is not $10 but $20. 

  1. The counterbalance to the greater upside is a lower probability of rallying

If the stock is still worth $1 then the probability of the up scenario has just halved to 5% (95% x $0 + 5% * $20 = $1, the current price).

If we inject volatility into a price that is bounded by zero, the probability of the stock going down is necessarily increasing.

So volatility alone alters the shape of a stock’s distribution if you keep the stock price unchanged.

Let’s see how this works as we move from binary distributions to more common continuous scenarios.

How Volatility Affects Continuous Distributions

Let’s start with a simulation of a subjectively volatile stock.


  • The stock is $50
  • The annual standard deviation is 80%.

A basic presumption of option models is that returns are normally distributed but this leads to a lognormal distribution of stock prices 1.

Running the simulation:

  • I took a return chosen from a normal distribution with a mean of 0 1 and standard deviation of .80
  • I then ran that return through a simple log process to simulate continuous compounding. 2

S (T1) = S (T0) x e(random generated return)

  • I ran this 10,000 times.

Before we get to the chart note some key observations:

  • You get a positively skewed lognormal distribution bounded by zero. This is expected.
  • The median terminal stock price is $50 corresponding to a median return (aka the geometric mean) close to zero as expected.
  • The mean stock price is $68. corresponding to a mean return of 38%.
  • The modal stock price is $20 corresponding to a modal return of -60%.

Simulation Vs Theory

Let’s compare the simulation to what option theory predicts.

  • Median

As we stated earlier median expected return is 0 from theory and this lines up with the simulation.

  • Mode

The mode in the simulation lines up reasonably 3 with option theory which expects the mode to be:

S x e2 where σ is volatility

Note how volatility pulls your most likely outcomes lower. In this case, the most likely landing spot for the stock is $20 corresponding to a total return of -60%!

Average Arithmetic Returns

Look at the chart again. Note how the average arithmetic mean stock price is $68.89 in this sample. If the median return is 0, the positively skewed distribution has a mean arithmetic return of +37.8%! We don’t want to get excited about this since as investors we care about geometric returns which are zero here, but this 38% OTM strike is still very interesting.

It turns out it corresponds to the strike of the .50 delta option!

The equation for that strike:

S x e2/2)

That strike corresponds to 68.86 which is very close to the simulation result of 68.89.

This is the call that you must hedge with 50% of the underlyer.

The formula will look familiar if you remember that the geometric mean is pulled down from the arithmetic mean in proportion to the variance.

[This strike is special for option traders. This is the strike that has the maximum vega and gamma on the option surface. As implied vol changes the location of this strike can change, but it represents the maximum vega any strike can have for a given spot price. I’ll leave it to the reader to see how this relates to strategies that are convex in vol such as ratio’d vega neutral butterflies.]

Interesting Observations About Options

  • Even in a continuous distribution, the higher the volatility, the more positively skewed the distribution, the further OTM the 50d call strike lives.
  • The cheapest straddle will occur at the median outcome or the ATM4 strike. 
  • The most expensive butterfly will have its “body” near the theoretical mode. This makes sense since a butterfly which is just a spread of 2 vertical spreads is a pure bet on the distribution. If you chart the price of all the butterflies equidistantly across strikes you will have drawn the probability density function implied by the options market!

Enter Black Scholes

In a positively skewed distribution, the probability of finishing in the money for a call was lower than the delta. In the binary example where the stock had only a 10% chance of being worth $10, the probability of the $5 call was much lower than the delta of the $5 call.

What does this have to do with Black Scholes?

In Black Scholes:

  • The term for delta is N(d1).
  • The term for the probability of finishing in the money is N(d2).

What’s the relationship between d2 and d1?

  • d2 = d1 – σ√t

The math defines the relationship we figured out intuitively:

The higher the volatility 5 the more delta and probability will diverge!

Delta and probability are only similar when an option is near expiration or when it’s vol is “low”.

From Theory to the Real World

Markets compensate for Black Schole’s lognormal assumptions by implying a volatility skew. While a biotech stock might have a positive skew on steroids, a typical stock’s distribution looks more normal than positive. By pumping up the implied volatility of the downside puts and lowering the implied vols on the upside calls, the market:

  • Increases the value of all the call spreads.
  • Shifts the implied mode rightward.
  • Shifts the 50d call closer to ATM. Actually, it lowers all call deltas and raises all put deltas. This is important since deltas are the hedge ratios.
  • Fattens the left tail relative to a positive distribution and at least in index options even more than a normal distribution.

These adjustments reconcile the desirability of a simple, easy to compute model like Black Scholes which uses lognormal distributions with empirically consistent asset distributions that we observe in markets.



The next time you hear delta used as probability, remember this is only really useful when options are near-dated. Since most option activity occurs in the front end of the term structure the assumption is typically harmless.

Taking the time to understand why they differ turns out to be a great exercise in building an intuition of investment returns and their distributions. 



Shorting Bimodal Stocks


My friend and former colleague Jason took exception to the viral tweet I referenced last week about how shorting a bi-modal company is like an option. Not because it isn’t but because all equity is an option.

In short, the entire viral tweet is a tautology.

Jason joined Twitter to respond to it. Jason’s gripe was that all equity is effectively a call option struck at zero (you can argue that a positive book value sets the call strike higher but it doesn’t materially change the point).

Jason argues that if the viral tweet pretends it is saying anything beyond “being short stock is like being short an option”, you are mistaken. There are several reasons why, and I’ll use my intro to twitter to go thru them…(Jason’s reply)

Where do I stand?

I liked the original @HedgeDirty tweet because I am just a fan of presenting ideas in different ways. The flaws in the tweet are real and technical but it conveyed a correct impression even if it got there incorrectly. To Jason and @HedgeDirty the riskiness of shorting a bimodal company whose equity is probably worthless is obvious. I appreciated the narrative style which reinforced that point. Even if it’s self-evident to anyone who shorts stocks.

But I also see learning opportunities in deconstructing the flaws in the tweet.

You can read Jason’s reply to see the flaws he found. Especially resonant was the observation that the 0 strike call is a 100 delta call. It has no gamma or theta. The original tweet claimed otherwise.

What would I focus on?

1) Let’s do a little math to find the annualized vol. The first thing to note is how the bi-modality creates a very volatile stock. 125% per year standard deviation.

(Warning: it doesn’t make much sense to use standard deviation based on a normal looking return when we already stated it was bi-modal but the obvious takeaway is “damn, this thing is going to make large moves in return space.” If you know nothing else, you know going short this thing is like barebacking a bucking bull. Size appropriately.)

2) My biggest gripe with the example. The stock doesn’t trade for $185 because it’s hard to be short stocks.

If the stock is trading for $185 the market is implying different distribution. Either the 80% and 20% are not true, the stock’s upside is more than $250 (1.25B EV), or the recovery value is much higher.

It’s not trading for $185 with a theta that pushes towards zero or anything like that. We don’t need to invoke Greeks for an alleged $50 fair stock trading for $185. If it’s trading $185 the market doesn’t agree with the assumptions that compute a $50 fair value. Full stop.


Despite the flaws, I enjoyed the original tweet. You can read plenty of @HedgeDirty threads and see it’s a good account to follow. For example, the Why You Should Never Hold Levered ETFthread is great. I’ve written about the brutal math of levered, especially inverse levered funds before.

Shorting bi-modally distributed stocks is hairy. If it feels like it’s being short an option it’s only because the stock is volatile, equity is an option, and being short options is volatile. Nobody shorting stocks should have needed an education from that tweet. For everyone else, they should be aware of errors in mapping it to option theory even if I think overall the thread was net positively educational.

As for Jason who has been trading options since the late 90s, I imagine he felt that @HedgeDirty borrowed his bass and played it with a pick. Only Paul McCartney gets away with that.