Most people who get into options are seduced by levered returns, but for the relationship to go from a fling to the real thing, they commit to learning about “vol”: implied vol, realized vol, vol surfaces. I’ve declared that options are ALWAYS about vol.

This is snobbery to the same degree as reserving “champagne” for sparkling wine that originates from a particular region of France. I resort to such snobbery on options to make the distinction between an option trade working for directional reasons vs vol reasons (if it works for the former and not the latter you were probably better to just trade the stock). But as all strong pronouncements go, they obscure truth. Sometimes to deceive, but other times, like in my example, it’s to move the emphasis to what matters without heat loss from caveats and equivocations.

In this post, we discuss the full truth. Options are always about vol unless they are about funding.

Funding is boring, bean-counting stuff. We want sexy. Think about it — what do you hear more about, rho or vanna? The opposite of love is not hate, it’s apathy. I get it, so many things vying for our attention, the investor neglect of “cost of carry” seems like a weird thing to wear a ribbon for — except for the fact that understanding cost of carry is both the easiest and most widely applicable “win-win” in the options world. Its neglect is quite tragic.

We will fix this over the next 2 posts. These are foundational posts that might well represent the largest gap between what people know and what they should know. It’s the basic blocking and tackling of options that every professional training program starts with whether you are going near options as a trader, quant, stock loan, ops or broker.

I’ll add that given the rise of heavily borrowed, speculative “meme” stocks, in a landscape where interest rates are not pinned to zero, this topic has never been so timely.

Today, we:

1. Start with a puzzle
2. Show how the solution informs arbitrage theory

Next week, we go from theory to practice:

1. We’ll show just how much money you could be leaving on the table in both longs or shorts by not letting the options market finance your position. This is relevant to any investor.
2. For those who are more vol-inclined we see how this work forms the foundation of modeling option surfaces. We’ll conclude with related considerations that are out of scope.

Onwards…

A market puzzle

Stock price: $100

1-year 100 strike call: $8.00

1-year 100 strike put: $7.00

The risk-free rate proxied by SOFR: 4% (assume this stays constant)

Dividends: $0…the company is not expected to pay a dividend

Your objective: Capture the return of owning the stock for the next year. Ignore taxes.

What’s the best way to do this assuming God confirmed the SOFR and dividend assumptions?

The first things that come to mind:

1) buy calls

Owning the return of a stock looks like a straight line. If the stock goes up 5%, you make 5% and vice versa. We know that outright option position payoffs look like hockey stick diagrams. If you buy the call and the stock only goes up 4% over the course of the year, you lose 50% of your premium vs earning 4% on your invested capital. Rule this out.

2) buy the stock

This works. It answers the question faithfully.

But there’s a problem.

This is exactly the answer an investor who doesn’t understand options would choose.

It turns out this investor is about to:

a) underperform someone who understands options. If this is a professional investor who has a zero-sum mandate of “get that alpha” will soon find themselves with no mandate

or

b) lose money

Countless investors who do not understand options make the mistake of buying a stock when they should have ordered off-menu— they should have bought synthetic stock.

To understand why, we start by breaking down why buying the stock in this setup is either a recipe for underperformance or worse, a losing proposition.

The problem with just buying the stock

The “underperformance” case

First, even if you don’t care about relative performance (an acceptable and even healthy posture for retail or non-professional investors), this is still important because “you could have done better” with this knowledge.

This is not something that you only learn in hindsight. Before doing the trade you can know if buying the stock is inferior!

Inferior to what?

Buying the synthetic future via options!

💡A synthetic future involves buying the call and selling the put on the same strike. Old school traders also call this a “combo”. The easiest way to see this is to just consider the scenarios. Suppose you bought the 100 call and shorted the 100 put. If the stock expires greater than $100, you will exercise and buy the stock for $100. If the stock expires below $100, your short put will be assigned and you will be forced to buy the stock for $100. Either way you are buying the stock for $100. If at some point in the future you are guaranteed to buy the stock for $100, then you are long that exposure right now that moves dollar for dollar with the stock. This video explains it with live data. This video is an ELI5 approach.

In the puzzle, the synthetic future is cheaper than its fair value. Or you can say the stock price is overpriced relative to its synthetic future.

To understand why, we can use our puzzle to step through the cash flows. The logic of the cash flows bridges the theoretical fair value of the synthetic future to the stock price.

Suppose the stock goes up 10% in a year. The “normie” investor who bought the stock makes $10. But what about the option-pilled investor who bought the 1-year synthetic future instead?

The option-pilled investor spends $1 today buying the 100-strike synthetic. They spent $8 on the call but collected $7 for the put. At expiry, the stock is $110 so the 100 call is worth $10 and the put is $0. The position they spent $1 for is worth $10. The total profit is $9 while the regular investor made $10.

The option-pilled investor made $1 less than the stock investor for an equivalent exposure. This makes sense — if you pay $1 for 100-strike combo you have synthetically paid $101 for the stock not $100.

But what are we ignoring?

I’ll start with a hint. This is not a percent return thing. Someone is jumping up and down that you 10x your money with the options. But that’s not fair. To honestly compare returns you also need to fairly compare risk so even though you only laid out $1 you still needed to keep the rest of the cash in reserve in case of margin calls. After all, you are still long $100 worth of stock.

Hopefully, the hint was actually a hint and not just a clarification.

The option-pilled investor acquires the same exposure for $1, but while they must keep the other $99 in a margin account, they do earn interest on that. In 1 year, they make $9 on the shares + $3.96 in interest ($99 * 4%) for a total profit of $12.96 instead of just $10.

Towards theory

We used a simple investing example to demonstrate how the same exposure expressed in 2 different ways led to 2 different cash flows. And one of them simply dominates the other. This is not a “frontier” thing where the p/l is different but the trade-offs varied. This is arbitrage. If the same exposure yields 2 different profits with the same risk then one set of cash flows is mispriced today.

You could buy the synthetic future and short the stock and earn ~$3 (about $4 in interest minus $1 premium for the synthetic future)

Again, somebody reading this is jumping up and down:

“Who cares about earning 3% when SOFR is 4%?”

I didn’t say 3%. I said $3. You can do this with no starting capital in theory. You borrow shares, short them, and collect $100 in the account today. We’ll be conservative and say the collateral you hold against the short is half the proceeds of the short (you still earn interest on collateral) and you spend $1 of the proceeds on the synthetic future while earning $3.96 in interest (4% on $99) for a total profit of $2.96.

You made $2.96 on zero starting capital. Infinite return. Pure arbitrage.

While markets are not perfectly efficient, if you can use a calculator, you can be sure Ken Griffin can too. With almost $3 extra dollars sitting on the sidewalk, the synthetic is too cheap at $1. Ken is going to bid the synthetic higher until he is indifferent between owning the combo vs the stock. As you might guess, that price, the non-arbitrage fair price, must be closer to ~$4.

Assuming we are indifferent or “risk-neutral” between 2 cash flows, the present value of those cash flows must trade for the same price today in a world where Ken Griffins hunt for free-money glitches 24/7. This is derivatives and arbitrage-pricing theory in a sentence.

We must turn this logic into a formula.

• When we buy the synthetic future, we commit to buying the stock for $100 in 1 year.
• With a choice between that commitment vs buying the stock today for $100 we prefer the synthetic because we have the same exposure but only need to set aside the present value of the $100 we need to buy the stock in 1 year.

The difference between the 100 strike and the present value of the strike is the cost of carry. The buyer of the synthetic must pay the carry today to be indifferent between buying the stock or the future.

This leads to 2 important formulas.

“Reversal/Conversion”

The cost of carry is referred to as the “reversal/conversion”. That’s a mouthful, so it’s often shortened to “rev/con”.

R/C = Cost of carry = K - Ke⁻ʳᵗ

where:

K = strike
r = risk-free rate
t = fraction of a year

Using our example:

K = 100
r = .04
t = 1.0

The origin of the term reversal/conversion is worth a mention.

It is actually a quoted value in the broker market as it acts like an EFP or ‘exchange for physical’.

• If you “reverse”, you are doing a package of buying a synthetic future and selling or shorting the underlying stock in equal proportion net of the multiplier (ie for every synthetic you buy, you short 100 shares). In this example, the fair price to pay for the reversal is $3.92. If you are long shares and want to flip into synthetic futures instead you should have to pay $3.92.
• You can also “convert”. If instead you were short shares and wanted to sell the synthetic and buy the stock to exchange your short from physical to options then you should require a payment of $3.92 to be kept whole on the fact that you need to wait a year to receive $100 for selling the stock at expiry. The conversion package is “short the synthetic future, buy the stock”.

The cost of carry or “rev/con” looks similar to the interest on a zero-coupon bond with a face value of the strike.

💡For this post, we are limiting the discussion to European-style options that do not pay dividends…the same type of options the original Black-Scholes equations were derived for.

Fair value of the synthetic future

The buyer of the synthetic must pay cost of carry of the strike up front for there to be no arbitrage between this otherwise costless position as compared to buying the stock.

They should also have to pay the intrinsic value or difference between the stock price and strike price. In this example, if the stock is $100 the 100-strike synthetic future costs $3.92. But what about the 99-strike synthetic future?

The commitment to buy the stock is already $1 in-the-money and you must pay the present value of $99 to account for the carry on the strike.

Synthetic future = Intrinsic + R/C

Synthetic future = (S-K) + R/C

Bonus: Put/Call parity from the fair value of the synthetic future

💡For the algebraically inclined, you can see how this re-arranges to the formal put-call parity formula. Remember the synthetic future involves buying a call and selling a put on the same strike: C-P

Synthetic future = Intrinsic + R/C

C - P = (S-K) + R/C

C = (S-K) + P + R/C

In words,

Call = Intrinsic + Put + cost of carry

All the heuristics are right in the identity:

• The call includes the value of the put on the same strike
• An option must include intrinsic
• The call saves you from funding the stock today so the cost of carry must be added to its value to prevent arbitrage. If interest rates rise, call values increase as the value of not having to spend the cash today is higher!

Re-arrange for the put:

P = (K-S) + C - R/C

In words,

Put = Intrinsic + Call – cost of carry

• The put includes the value of the call on the same strike
• An option must include intrinsic
• Being short via a put option doesn’t give you interest on the proceeds of cash from the short, so the put must be discounted by the cost of carry to prevent arbitrage. If interest rates rise, put values fall as there is more interest to be earned from being short actual shares.

Circling back to the problem with buying the stock

In our puzzle, we contrived a situation where the synthetic was offered too cheap relative to the stock price, assuming God decreed that SOFR is 4%.

We derived the no-arbitrage price for the synthetic by finding our indifference point between the cash flows of owning the stock or the synthetic.

In practice, if the synthetic appears too cheap compared to a SOFR rate, I can assure you there’s no free money on the sidewalk. You should check your assumptions. I’ll check for you — the market is saying you can’t collect SOFR on the proceeds of these short shares. In fact, if the synthetic is extremely cheap relative to the stock price and you try to pick up the free money by buying the synthetic and shorting the shares (that “reversal” trade we talked about), you might find that instead of paying for the package, the market pays you! In other words, the reversal is trading for a credit (the synthetic future is trading cheaper than the stock price). You think you should be delighted…until your broker sends a bill for borrowing the share you shorted instead of you receiving interest on cash proceeds in the account.

If you were bullish on the stock as we stipulated in the puzzle and bought it, you just bought shares in something heavily shorted. Instead, you should have bought the synthetic future for a lower price. If you are bullish and going to get long the stock wouldn’t you rather at least buy it for the lowest price available? That price will be in the options market via synthetics — not the stock market. That’s why I say that even stock traders, ones who don’t care about vol, still need to understand options. You incinerate money buying the stock when you should have just bought the synthetic. “Not incinerating money” is the easiest win in investing.

Next week:

• short stock rebate
• learn to measure the term structure of synthetic stock futures, effectively creating a menu of stock prices at any point in time according to the funding rates until each expiration. This offers another easy win — it’s an option to refinance our positions when the market pricing differs from our prime broker (rare example of a win-win where pros even trade with each other via rev/cons or box trades)
• understand how funding and put/call parity sit at the foundation of surface modeling

Appendix: Dividends and Rev/Con Markets

So far, we assumed no dividends. Real stocks usually pay them, and this changes the fair value of the synthetic future.

where:

q = continuous dividend yield

Dividends lower the forward price because the holder of the synthetic future doesn’t collect them.

2. Intuition

• If dividends = 0, this collapses to the formulas in the main text.
• If dividends are high, the synthetic trades cheaper relative to spot, because you’re forgoing dividend income by not owning the physical shares
• If dividends exceed interest rates, the forward price can even trade below spot

3. Example with dividends
Suppose:

• Spot stock price = $100
• Risk-free rate = 4%
• Dividend yield = 2%
• Time horizon = 1 year

Then:

The implied future is $101.98

Without dividends, the implied future would have been $103.92 . The 2% yield shaved nearly $2 off the forward price.

You can adjust this formula for discrete dividends by deducting the present value of each expected dividend from the strike.

You can see that if the risk-free rate is 0, the R/C is negative or “trades for a credit”. In this case, you would pay to “convert” since being long the stock pays you the dividend. You would need to be paid to “reverse” as you forgo the dividend being long the synthetic future instead of physical shares.

Rev/cons are heavily traded as the funding market through the options can be much tighter than prime broker rates. It’s also a transparent market, whereas stock loan can be opaque beyond your prime broker.

Rev/con markets are the home for price discovery on expected dividends. If you had a divergent view from the market on a future dividend, this is where you go to pick someone off. Rev/cons are clean trades because they have 0 delta (you are offsetting a synthetic future vs shares as a single package — or you can say that you are trading the synthetic future delta neutral. They have no market impact and can often trade in size so for those services that try to tabulate volume to say what market makers are holding rev/cons are a nuisance. If they fail to notice that the option trades are matched with a corresponding stock print, they will attribute greeks when they shouldn’t.

🔗Further reading

Ari wrote You Don’t Use Your Instagram Self to Trade which is a great demonstration of how tricky the details can be. He also uses an equivalent but different representation of the put/call parity equation. I think of his version as combining cost of carry and intrinsic terms to become “intrinsic to the discounted strike”.

Yet another vibe-code project. This one went viral because…it’s a game!

It’s a replica of the one we trained on an eon ago at SIG. It’s a put-call parity game.

The formula for put/call parity is:

C = (S - K) + P + RC

where:

C = call value

P = put value

S = stock price

K = strike price

RC = cost of carry til expiry (ie “reversal/conversion” value)

In the game you are given the strike price and 3 out of 4 of the remaining variables. Solve for the 4th. The game is timed.

 

Try it for yourself:

🕹️Put-Call Parity Trading Game

 

Contextualizing the formula

You don’t want to just raw-dog the formula. You get faster if you can contextualize it because with practice your mind collapses multiple operations into just one or two. You need to try it for ahwile to appreciate what I mean.

But let me give the context.

1. First, note that (S-K) is just “intrinsic value”. If positive, the call is in-the-money, if negative, the put is.
2. The extrinsic portion of the ITM [call/put] is the value of the OTM [put/call]
3. For calls, we add the rev/con (ie cost of carry) for puts we subtract it

Contextualizing the game

The game spread quickly when I shared the link on X. I even had some more recent alum of SIG and one from another MM tell me even in recent years they use a game like this in training.

When I started in 2000, this game was actually in fractions (“steenths”) but decimalization pilots were happening in my clerking months. Even though I started on fractions, I was doing decimals about halfway through assistant year. As clerks we were supposed to play this game when things were slow so you could be fast in mock-trading in class after work so you could actually get selected for the bootcamp (which alone got you a raise) and get on with a trading account.

Speedy mental math was more important back then. Different era obviously, but interesting that they still find use in it. I could speculate as to why but maybe someone reading this will give me the official reason. I would admit that even when I’d get a broker look at an outright option when I was at Parallax I’d automatically do the calc in my head to compare to the same strike option on the chain. Being facile with the calculation was also a requirement on the rare occasion that a broker asked for a synthetic (a topic I discussed in the art of paranoia as well).

It only took about 20 minutes of iteration to make the game with Claude. Which is not much longer than it takes me to play 🙁

Today is a continuation of calendar spreads through the eyes of a vol trader.

Recap

That post is a response to a conundrum that regularly presents itself to vol traders. These scenarios will feel familiar:

Harvesting VRP: “buying a time spread to harvest the front-month VRP” – selling expensive implied against cheaper realized while hedging with back-month options

Buying cheap vol: Vol screens low across the board, but front months are cheaper than back months. Do I buy the cheapest or pay up for duration?

Despite these situations being as common as dust, they don’t have an obvious playbook.

[Which is good because the moontower app has a point of view on this — because this is exactly the type of question you wrangle with when you run a vol book.]

My favorite approach to problems like this is not a backtest but a simulation. A sim is a controlled environment where you can fix assumptions, push a random variable, and get a platonic result that says “this is the shape of the p/l if the assumptions hold”. That might sound simplistic, but if you can’t forecast the output of the platonic case then you can benefit tremendously from some calibration. I predict you’ll benefit.

Understanding the simulation

The simulation approach I introduced uses a strike-resetting model to isolate vol realized vol’s contribution to the p/l — the variable you are betting on when you trade VRP. We initiate the stock at $100 then draw a return from a random walk of X vol. We compute the daily p/l of a portfolio comprised of:

a) the 100-strike calendar call spread (notice it is at-the-money)

b) a share position so that you start each day delta-neutral

So if the draw is +2% then we compute the p/l of the portfolio based on a stock price of $102 and time elapsing one trading day. We then reset the stock to $100 and repeat until M1 expires. We do this to minimize the noise of p/l path dependence that can occur if the stock gets far from the strike, choking off the dollar gamma in the process. In our sim, the dollar gamma starting each day follows the predictable glide path determined only by the DTE falling.

🔢Simulation Parameters

• DTE for M1 and M2
• IV for M1 and M2
• Realized vol to sample daily changes from

In the earlier post, I stepped through 2 examples of buying the calendar spread for a flat IV (ie M1 IV and M2 IV are equal) and the IV is greater than the realized vol (ie positive VRP).

You expect to win in this scenario because you are short gamma and collecting theta while the realized moves are not large enough to punish the seller. The rent or “cost of gamma” was too high for the counterparty who owns M1. For the calendar spread owner, they lose on being long M2 but not as much as they gain on being short M1.

Now I gave 2 examples to highlight that the trade is indeed noisy because there is so much gamma on the last day before expiry that it can make or break the entire p/l.

Today’s post will not only address the noise but the starting approach to the question:

How do we evaluate the term structure premium when either harvesting VRP or getting long cheap vol?

🤖Included in the post is a webapp to let you run a single or thousands of simulations and step thru any single trial day by day to understand exactly how the p/l develops as well as the p/l distributon for the entire batch! You can even clone the app to modify it as you want.

Let’s start hacking away on the questions of whether we should be buying calendar spreads to collect the VRP.

The day after I published the original post I sent out the simulation webapp I vibe-coded with Gemini. It allowed you to put the IV for M1 and M2, input a realized vol, and step through the daily hedged p/l until M1 expires.

Tinkering with that is useful because it gives you a sense of the noise in harvesting VRP. But it’s just a single trial of “hey I put this calendar spread on and hedged it until M1 went away, what happened?” (again, assuming spot resets to $100 daily)

What you’ll find is if you buy the calendar spread for flat vol and the realized is less than the implied, you usually win. If you pay a higher IV in M2, you win less or if you pay too much you actually lose.

Why?

Because the simulation embeds IV “rolldown”:

If M2 differs from M1 we assume a linear glide path for M2 to approach M1 by the time M1 expires. For example, if M1 has 20 dte and is 20% while M2 is 22%, then M2 vol will fall by .10 per day (2 vol points / 20 days).

This is a good place to stop to consider the forces driving a long calendar spread p/l:

• You make more money the fatter the VRP — the theta rent you collect is more than compensating you for short gamma. To be more specific, you lose to your long option theta but more than make it up on your option theta. The theta and gamma of the longer dated option are both smaller than those of the nearer dated.
• If you pay a premium IV for M2 and it “rolls down” to IV of M1, then you lose to vega a little bit each day by the quantity vega * vol change.

Your p/l performance tradeoff looks like this:

For a given VRP, you make more money the cheaper the calendar spread is. This is highly stylized, it’s basically a scribble— the true tradeoff curves may not even be lines.

“Shadow” Theta

In our simulation, M2 IV rolls down to M1 IV which is held constant. We can allocate the vega p/l loss to a descriptive term: “shadow” theta.

[I first heard of shadow greeks in Taleb’s Dynamic Hedging. I’m not referring to his definition of shadow theta, in fact I don’t remember if he had that one, but borrowing the nomenclature “shadow” which did get traction as a way to informally describe p/l sensitivities not already covered by the proper greeks.]

In the simple accounting for a calendar spread, M2 has less theta than M1. But if we expect M2 IV to roll down, then its effective theta will be its Black Scholes theta + shadow theta.

If M2 is a large enough premium to M1, M2 will actually have more theta than M1 until M1 gets close til expiry. You can still win being long such a time spread because if the stock only makes very small moves compared to its IV as M1 expiry approaches you will collect the larger M1 theta just as it hits the steeper section of its decay.

Equipped with shadow theta to account for roll down (or roll up in the event of discounted M2 IVs) in calendar spreads where M2 vol does not equal M1, we can interpret the latest simulation tool.

Built with Claude the new app runs the same delta-hedged process I showed earlier using Gemini, except now we have wrapped that simulation in a loop where we can run N trials of different random walks sampled from the same volatility.

The output allows provides 2 key views:

1. the distribution of p/l’s for the whole batch
2. a drill down into the path of any single trial

I recommend tinkering with it yourself, but let me offer a few examples of scenarios to run.

Run your own time spread simulations

Try it here:

🤖Monte Carlo Delta Hedged Simulator

Allow me to get you started with an example and some interpretation.

Typical market environment

# of trials: 5000

M1 position type: short call
realized vol: .16
M1 DTE: 20
M1 VRP as Percent Premium:15


M2 position type: long call
M2 DTE: 40
Vol Steepness (as % premium/discount to M1 IV): 10

Let’s translate the meaning of these value.

• The position types are describing a short ATM call calendar spread. The stock and strike price are both $100. The rfr is assumed to be 0.
• The 20 DTE and 40 DTE terms are chosen because they act like a 1-month/2-month spread (assuming trading days)
• The stock’s returns are being randomly chosen from a process assuming 0 drift and 16% annual vol, corresponding to ~1% daily vol.
• The M1 IV will be the VRP percent premium tacked on to the realized vol. In this case, IV = .16 * 1.15 = 18.4%
• The M2 IV comes from applying a “vol steepness” to M1 IV. In this case , IV2 = 18.4% * 1.10 = 20.2%

The simulation will then produce a return stream for 20 days (until M1 expires) and hedge back to delta-neutral daily as M2 vol “rolls down” linearly over those 20 days from 20.2% to 18.4%

That is 1 trial path. The output for that trial will show the cumulative p/l for the trade but you can examine any single day in the path (very educational to step thru btw).

When you run the simulation it will actually run 5000 trials and provide summary stats of the p/l distribution.

Let’s run this sim and go through the output.

Setup:

5000 trials…the trade has a mean p/l of -.09 with a st dev of .35 (so -.26 sharpe). It wins 44% of the time and a skewed left tail. It would perform better presumably if the VRP was fatter or if it didn’t have to pay a 2 vol point premium for M2. This lays the groundwork for the next sim a tinkerer might want to run — “What M2 and M1 were the same vol so there was no IV rolldown, how does that look?”

Let’s do it. New sim parameters:

Positive VRP, flat term structure scenario

# of trials: 5000

M1 position type: short call
realized vol: .16
M1 DTE: 20
M1 VRP as Percent Premium:15


M2 position type: long call
M2 DTE: 40
Vol Steepness (as %): 0 ←—— We changed this from 10 to 0!

Now we’re talkin! We win 3/4 of the time, .66 sharpe and while the left tail extends to a larger magnitude, the right tail has more mass.

It’s instructive to look at the details of any single trial out of the 5000 runs.

The table showing each day is larger than I can show in the screenshot but includes “shadow theta” and every other variable that goes into the computation so you can learn it yourself. (You can even make a copy of the app right from the UI if you wanted to customize it).

In the path summary, you can see that even though we draw returns from a 16% vol distribution, this particular trial of 20 trading days realized 17.94% vol, about 1/2 vol point realized VRP — however the path resulted in the trade earning 2.3 vol points. You can see that M1p/l (purple line) and therefore the combined p/l (green line) spiked at the end creating a sizeable total win.

If we look at the stock price path we’ll know why:

The return on the last 3 days, especially the last day, when gamma AND theta are maximized were smaller than implied daily moves of 18.4%/ 16 or 1.15%

Again, this highlights the insane noise of realized vol p/l at expiry. If you were long those front-month options, you were doing pretty well until the last 3 days, and 2 vol points worse than what the plain 1/2 vol point VRP would have predicted based on the stock realizing a healthy 17.9% vol for the full period. The short got bailed out.

You can imagine the opposite scenario as well, where the short was doing well all month, only to get crushed on the last day. That’s the beauty of the sim…you can just go look at any of the thousands of losing trials to find paths that unfolded so painfully.

Wrapping up

The tradeoff between choosing M1 or week 1 vs a longer dated option is a conundrum whether you are looking to buy or sell a single option and need to choose a maturity or if your strategy rests on discerning between months.

Wait.

I’m sorry but this aside really belongs here so bear with me:

Near-dated option p/ls are driven by the gamma/theta tug of war, long-dated options are driven by IV and therefore vega.

This is also why near-dated you think in straddles and long-dated in vol. It’s a habit that comes out of intution for what factors drive the p/l. It’s also a built-in defense against “option illusions”:

1. Every long dated straddle looks optically expensive, because they grow in price monotonically by something that looks like √dte meanwhile you are comparing that growing number to the same spot price visually. But we price the straddles by the sum of all the delta-hedging we can do over the course of its life so its value really depends on like all these little scalps. The most coherent way to bascially amortize the scalps into something of meaning is an IV because we know to compare that to measures of realized for context.
2. On the near-dated option side we know that IV is hyper-sensitive to the DTE we use in our model. And DTE is not an agreed upon number. Trading day to expiry? Calendar days to expiry? If it’s 10am in NY, how much of the day do you think has elapsed? How about at 3pm? These questions don’t have straightforward answers so with little time to expiry the notion of IV really falls apart. But you know what doesn’t? Straddle prices. “Bruh, what’s your market on how much the stock can move in the next 4 hours?” We can actually think about that much more easily than the minute to minute gyrations or in terms of “vol”.

I think most option traders understand these 2 distinctions in their bones.

Ok, back to the concluding remarks.

We’ll finish with a few real-world considerations and thoughts:

• We totally ignored trading costs. Trading costs, both direct and in the form of slippage, are a tax on any delta-hedger whether you are long or short options. In other words, regardless of which way you sim the strategy, there’s no bias in excluding costs — all the results will be worse.
• Correlation of IV to RV…if realized vol increases both M1 and M2 can respond. If it declines, you might make more on your short M1 but you are losing more on M2 as well. If the IV selloff happens closer to M1 expiry it will likely hurt more as you will be longer vega.
• Dollar gamma is so high near expiry that the move on the last day can have a disproportionate impact on the entire strategy’s p/l. This observation should also tell you quite a bit about how much 0dte trading is akin to gambling. As a market maker, going into expiration day with a lot of risk always felt unproductive because that day’s results would swamp all wood we chopped all month. And yet people sign up for this every day? I’d think this requires a highly devoted strategy and focus and is a narrow subset of the general business I’d describe as “vol trading” (although one that can absorb lots of intraday capital).
• A professional vol trader’s surface will, to varying extents, be cleaned for events. That process means that term structures that look descending might actually be ascending and vice versa. Rigorously filling in an event calendar to clean vols is no easy task and is a joint effort between a trading firms’ fundamental research and QR groups, but I suspect its ROI is well worth it for the scaled prop firms and HFs. 15-20 years ago a basic understanding of this was extremely profitable and probably the single most important area of analytical focus in my own trading.

 

Let’s leave it there.

I used a pattern to explain it to my 12-year-old on our car ride on Monday.

Start with:

8*8 = 64

Let’s call that a * b

It feels like if we subtract 1 from a and add 1 to b multipy it should be close to 64

7*9 = 63

Close but a tad lower.

What if we keep the 8 average between the numbers but widen the dispersion between them more:

6*10 = 60

Lower still.

When the deviation from the mean was 1, the product of a * b was just 1 lower. (63 vs 64)

When the deviation from the mean was 2, the product of a * b was 4 lower. (60 vs 64).

Hmm, I have hunch what’s gonna happen here.

5*11 = 55

When the deviation from the mean was 3, the product of a * b was 9 lower. (55 vs 64).

One more to solidify this…

4 * 12 = 48

When the deviation from the mean was 4, the product of a * b was 16 lower. (48 vs 64).

We got the pattern.

For 2 numbers, a and b:

a * b = Mean² – MAD²

where MAD = mean absolute deviation

As soon as the numbers deviate from the mean, their product is dragged down even if the mean is unchanged.

More deviation, more drag.

💡Note that MAD² is just variance when there are only 2 points because the mean is the midpoint, the 2 deviations must be equal.

If there are more points then MAD² < variance. We can see this from simply remembering that MAD ~ .8 * SD therefore MAD² ~ .64 * Variance

In investing, we compound or multiply returns so even if the mean of two returns is the same, the dispersion matters.

The mean of 1.1 and .9 is 1, but the geometric mean is less than 1 (ie when you multiply them together). The amount less than 1 is a function of the deviation of the 2 numbers from the mean of 1.

.8 and 1.2 have a mean of 1, but a geometric mean less than the geometric mean of .9 and 1.1.

.5 and 1.5 have a mean of 1, but a geometric mean less than the geometric mean of .8 and 1.2.

The drag is a function of squared deviation. And no deviation, no drag — the arithmetic and geometric mean are the same in that case.

Summarizing:

Then with numbers that look like returns:

Notice how the difference between the arithmetic and geometric mean is approximately half the variance.

You’ve seen this before.

r−1/2 * σ²

The median expected return (ie geometric return).

AKA the risk-neutral drift from Black-Scholes.

AKA the “volatility drain”.

Paid subs will recall my story of Doug teaching Black-Scholes to my cohort at SIG back in 2001. Four hours in one day to explain the assumptions and four hours the next day to derive the equation. I tried to keep up but dropped off embarrassingly quickly.

I did that webinar to explain how I eventually came to understand the formula. The recording is paywalled but these are the slides for the talk.

Here’s the distilled version:

Start with what we know.

At expiry, a call option is worth the stock price minus the strike price (or zero if the call is “out-of-the-money”)

So today, the call price equals

“the current expected value of the stock given the call is exercised”

minus

“the discounted strike price”

[The strike price gets discounted for both the time value of money AND the probability of exercise.]

Let’s work through this with common sense.

You’re looking at a 1-year $50 strike call. The stock trades at $50 today, risk-free rate is 5%.

Say the call has a 50% chance of being in-the-money.

Let’s also assert that in the state of the world where the call gets exercised, the stock is on average $58*. That happens 50% of the time, so the expected value is 0.50 × $58 = $29.

*Think of this like rolling a die: given that you roll greater than 3, what’s the expected value? It’s 5 (the average of 4, 5, 6).

What about the discounted strike price?

The $50 strike discounted to present value is $50 × e^(-0.05) = $47.56. With a 50% exercise probability: 0.50 × $47.56 = $23.78.

The call value from our definition

“the current expected value of the stock given the call is exercised”

minus

“the discounted strike price”

maps to

$29 – $23.78 = $5.22.

The key insight: we can replicate a call option with a portfolio of stock and cash

You can replicate a call’s payoff by owning some amount of stock. This amount is more commonly referred to as the “delta” (or hedge ratio).

This delta changes as the stock becomes more or less likely to finish in-the-money. As the stock rises, you buy more shares to replicate the call’s potential payoff. As it falls, you sell shares since exercise becomes less likely. You’re buying high and selling low—creating negative cash flows. That sum of negative P&L should is what the option is worth.

You can either buy the option (pre-paying these cash flows) or manufacture it yourself through this delta hedging strategy.

In an arbitrage-free world, the option price must equal the present value of these replicating cash flows. If the option were priced with higher volatility than actual, you could short it, hedge with shares, and pocket the difference.

The self-financing part is elegant.

To replicate the call, you need to buy the “delta” quantity of shares. With what cash? You borrow it—specifically, you borrow $23.78 and use that cash to buy the shares today. This is why the strategy is self-financing: we’re simply borrowing against a future cash flow.

Why does this work?

At expiration, if the call gets exercised, you sell your stock at $50 to the call owner. With 50% exercise probability, your mathematical expectation is to receive $25 in one year. So you can borrow the present value of $25 today (ie $23.78), use that borrowed money to buy the shares, knowing you can repay the loan at expiry with the proceeds from selling those shares.

Notice why call values increase with interest rate:

a call is ultimately the difference in value between the number of shares you need to buy (delta shares) and the number of shares you can afford to buy via the loan. The higher the interest rate, the less you can borrow, the fewer shares you can buy, so the call value—which bridges that gap—increases.

In a sentence…

a call value represents the difference between how much stock you need to buy and how much you can afford to buy to achieve that hockey stick payoff.

Fwiw…

One of the webinar attendees says this diagram made it click. I’ve never seen it anywhere else and came up with it when I wrote A Visual Appreciation for Black-Scholes Delta

 

I can’t remember which of the 3 Todd Simkin interviews on my blog I summarized where he mentions it but Todd was asked if SIG’s secretiveness has been an advantage. He said in trading, it’s been good, but when it comes to recruiting technologists or researchers, it’s been a hindrance. The FAANG companies are household names and since trading firms compete for some of the same talent, you’d want more people to know what SIG is.

I figure this recognition is behind their increased public outreach. Like this awesome video that recently dropped from the lecture series where Professor Costa teaches their trainees about the GFC.

It starts assuming you don’t even know what a bond is and proceeds to cover an unbelievable amount of distance in one hour. The narrative and history going back to the 80s is fantastic and I even learned (or reviewed) a lot of basic market knowledge.

#teaching_goals

While this video is loaded, here’s 5 bits that stood out for me. There’s also a very SIG-esque lesson in there about anchoring bias.

1. Diversification has literal monetary value – Great demonstration of how portfolio theory translates directly into pricing and risk management
2. Reflexivity in credit markets – Default rates weren’t actuarial constants but depended on loan originators’ incentives. Once originators became divorced from risk while retaining pricing/underwriting control, the system became unstable. A systems thinker would have spotted this disconnect.
3. Misaligned incentives drove market distortion – Traders focused narrowly on derivatives markets where the CDS market dwarfed the underlying bond market. Unlike bond issuance (limited by actual capital formation needs), derivative trading appetite was essentially unlimited.
4. Good ideas taken too far become dangerous – Diversification through low correlation assets is sound in principle, but this conceptual acceptance prevented people from asking the critical follow-up: “To what degree is this still safe?” (The opposite is hormesis – sometimes a little of a bad thing is actually beneficial. As the old saying goes “the posion is in the dose”.)
5. “This would turn out to be a fateful decision” – The final section on implied correlation reveals how trading desks completely inverted their hedge ratios between tranches, fundamentally misunderstanding how correlation affects different credits.

I got long IBIT (BTC etf) when my April calls expired ITM. IBIT continued rallying, I was happy to stay long. However, the vol screened cheap.

I decided to “replace” — I sold my long shares and bought calls.

Deciding which calls to buy was tricky. So tricky it inspired this post.

The heart of the conundrum lives in this snapshot from 4/23/25:

That is the constant maturity IV term structure, visually depicting a steep term structure. The implied forwards table below it gives a more granular view.

You can see the problem.

I think vol is cheap…but that was confined to the front of the curve. If we just look at the first 2 monthly expirations, May 16th and June, the vols are 45.8% and 51.1%.

The premium from May to June is so steep that the May/June forward vol was 54.3%, a full 10 points over the weekly IV!

Vol might be cheap but you can only “lock” it in for about a month. Then it doesn’t look so cheap anymore. No easy trades, right? The market is pricing mean reversion in the vol.

I chose to go with what the market gave me — the cheap vol in the front. I sold my shares and bought the May 60 calls.

Well, the stock went up about 15% that month…but expired just below my long strike.

A 15% move in a month corresponds to a 52% vol move (.15 x √12) but I picked the wrong strike. Path is Anton Chigurh’s coin.

I chose the shorter expiry because that’s where vol was cheap. June was 5 vol points richer, more than 10% premium over a single month. Is that too much to pay? Does that nullify the “vol is cheap” thrust behind the trade idea?

There’s no straight answer to such a question.

The term structure balances a tension between the supply and demand in each month. Those flows are made up of both natural needs — directional speculation, hedging — and vol trader positoning insofar as they have opinions on the relative pricing.

The term structure is the jurors’ debate of the evidence. The truth behind the case, just like any trial is not perfectly known. In the market’s case, the ensuing price action, represents the truth.

Discerning between months and trading time spreads is a significant portion of vol trading. Understanding events is a mix of computation and research labor. There’s room for alpha in the act of normalizing term structures to account for idiosyncrasies in a calendar.

But let’s put that aside and assume a calendar with no known events.

We want to build some intuition of expensiveness or cheapness in a time spread. We can use a toy calculation to do this in the context of a common scenario:

“I am buying a time spread to harvest the front-month VRP”

If you sell the front month and buy the back, it’s a hedged way to capture the premium of implied to realized.

Note this premium, despite being empirical, is neither constant in magnitude nor size, but simply “on average”. Still, we can use this type of trade to explore the tradeoffs in month selection.

When we buy say, an ATM time spread, we are going to be long vega, long time premium, short gamma, and generally collecting theta.

Wait.

Why am I bolding “generally”?

It’s a clue to the tradeoffs involved.

Let’s just jump right into the toy model to see what we can learn about time spreads. This is stuff that excites, transfixes, and haunts every vol trader.

“What needs to be true for this time spread to be cheap or expensive?”

This will be the first in a few posts as I try to eventually turn this into a webapp.

 

We need to build a simulation.

We are angling towards an appreciation for how a time spread behaves by decomposing how a time spreads perform until the first month’s expiration as a function of:

1. implied vs realized
2. change in M2 vol if it “rolls” towards M1 vol by the time it becomes M1

We understand that buying a time spread should win if the realized vol underperforms the M1 vol you are short, but you are often paying a premium vol for M2 (contango term structure) and we want to see for various combinations of steepness between the 2 months and VRPs how attractive it looks to buy/sell time spreads.

The toy simulation

Scenario:

Consider a $100 stock that moves at 16% vol.

You buy a 40 DTE /20 DTE 100-strike calendar spread delta-neutral. 19.2% in the back month and sell 19.2% vol in the front month. Hedge daily at the close.

It’s a flat vol time spread.

You are trying to capture a VRP by being short 19.2% vol when the moves will be sampled from a 16% vol return distribution. Since you are short the nearer-dated option you should win if implied vol > realized vol since the moves are not large enough to cover the option theta.

How can we model this?

We can look at the daily delta-hedged p/l of a time spread assuming M2 vol converges to M1 vol by the time M1 expires. In our current example, M2 vol = M1 vol so there’s no “roll down”.

Today, we will simply zoom in on what a single simulation looks like.

Step 1: Stock process

Assuming we pull a random return from a Brownian motion diffusion process with RFR = 0% and a daily vol of 16%/sqrt(251) or 1.01%

This is simple to implement. The details are here:

🔀Stock diffusion model explained step-by-step

Step 2: Daily strike reset

While the simulation has randomness based on the draw of daily moves, I’d like to minimize the dependency on path. If you trade the $100 strike and the stock trends away from your strike, your greeks “go away”. To get rid of this noise, we will reset the stock to $100 each day so that the options are once again at-the-money.

We will also assume the stock only goes up. We get the random return and assume the stock goes up by that amount. We are resetting the strike each day to whether the move is up or down makes no difference.

Bringing step 1 and step 2 together is an example of a single draw:

💡Quiz question

I computed a realized vol of 16.1% from that return stream. Do you get that same value? If not, what do you think I did and why? The answer is one of those things that makes you realize just how subtle choices in measures can have surprisingly large impacts.

 

Step 3: Compute daily p/l

 

We assume a constant implied vol for M1.

If M2 differs from M1 we assume a linear glide path for M2 to approach M1 by the time M1 expires. For example, if M1 has 20 dte and is 20% while M2 is 22%, then M2 vol will fall by .10 per day (2 vol points / 20 days).

Each day we start with a delta-hedged position on the $100 strike. We are short M1 and we assume the stock goes up by the randomly drawn amount from the diffusion process. So we will lose on the short call and win on the long stock. For M2, the opposite will happen — we win on the long call and lose on the short stock.

(I broke them out separately because it will be handy for decomposition but in reality, the long and short stock legs would be close to canceling out since it’s an ATM call spread.)

We can understand how this works for both long and short legs by just zooming in on M1 as an example.

Being short M1 at 19.2% IV while it realized 16.1% (20% vrp) yielded a profit of about 11 cents. You can see the messy p/l path and this is with the strike resetting to ATM each day which sterilized the stock path!

Doing the same procedure on M2, we can see the breakdown:

The long leg broke even while the short won. Although the win was driven by very small moves towards expiration.

I can’t overemphasize how noisy this is. The stock can sit unchanged for 19 days then on day 20 your short ATM call, which starts the day worth $.57, can blow out on a $4 rip, leaving you net loser on the whole trade. You were short vol at 19.2% and that hypothetical return stream is about 14% vol. Bad beat on the river.

(Yes, $4 is extreme but so is being unchanged for 19 days.)

The stock returns of those last few days before expiry get averaged into the realized vol calcs the same as any other day, betraying lived experience — your dollar gamma is so high near expiry that those returns have a disproportionate impact.

Which is exactly why nobody get outs alive from the fact being short where lands and vice versa will dominate everything else…hence my IBIT calls that were directionally right AND priced cheaper than the vol realized and experienced a total loss.

Here’s another draw, this one snatches defeat from the jaws of victory…

Healthy VRP. Sitting pretty with a cumulative profit 1 day before expiry and boom a 2.3% pop in the stock. A bit over 2 st devs. On that day you make 1/2 the move in share p/l because you do have the .50 delta hedge but the call you are short goes from $.57 to $2.30. Your M2 options have far less firepower so they only mitigate about 1/4 of the day’s loss.

 

So far we’ve stuck to the case where the time spread has flat vol (ie M1 IV = M2 IV). We’ll change that next week, when we get into “shadow theta” as we push ahead to understand calendar spreads.