Recently, I’ve been writing a lot about option funding — implied rates, reversal/conversions, and financing stock positions, and even a touch on early exercise. The posts from earliest to most recent:

• the easiest win in options is for stock traders
• implying the cost of carry in options
• what hides in the option chain
• the art of paranoia

You can’t really overstate how important this is, especially to pros.

On the same theme, Stanford lecturer and HF manager Kevin Mak published an outstanding and detailed post:

Holding a high cost to borrow stock? Here’s how to collect the borrow fees using the options market (link)

I want to quote a few key parts:

I cannot stress enough that if you want to be competitive in capital markets, you cannot afford to make these types of mistakes and forgo this return.

Kevin is quite blunt:

If you think this is too complicated to follow, to be blunt, you likely do not have the mental fortitude to have alpha in markets (you may still make money via luck). If you CAN figure this out, but couldn’t be bothered to spend time on it, that’s probably fine, but I suggest staying away from holding a stock with a high cost to borrow. If you insist on holding high cost to borrow stocks, and “not worry” about collecting the free borrow/lending fees, you really should be playing triple-zero roulette, or splitting every pair of 10’s at Caesar’s Palace instead because that would have a lower edge, and be way more fun.

I suggest that by using options to refinance your position, you “inherit the market maker’s funding rates” which are almost certainly better than yours, whether you are long or short.

This is Kevin’s way of saying this which might land better for some (emphasis mine):

If this is arbitrage, it shouldn’t exist right? This unique situation is you “arbitraging” your own holdings. You’re basically holding an inefficient asset since you can’t lend it (or collect the lending fees) so this lets you own it in a more efficient manner. Nobody is going to compete away this arbitrage because nobody can access your holdings except you. In fact, it’s the arbitrage happening in the open markets which is pushing the value of synthetic long to be equal to the long stock (and collect borrow) position.

Kevin wisely tucks all the brain damage into the endnotes to not ruin the flow and central message of the post. I love his intro to them:

This is a beginner treatise on a synthetic long position and covers ~98% of what you need to know about it. The last 2% I could write 25,000 words about and not be finished.

In the spirit of the endnotes, I want to share a recent real-world example of a scenario that resides in that annoying “2%”

Alex, a mutual of mine on X, asked the following:

I bought some deep itm $AIV calls, I think Jan exp, a couple weeks ago because I thought they were cheap vs my financing cost. Not much extrinsic value… Underlying had silly spike after hours, up then down.

Was the right way to trade this, to short the underlying after hours? I thought about exercising to sell, but didn’t want to burn a few months of time value for nothing, even if I didn’t really pay for it.

I’m going to step through the conversation, injecting meta-commentary to bring it down a level.

Me: Your trade expression was because of financing but you are asking about what you should have done with your delta. I’m not being pushy, I’m just trying to help you answer your own question.

There are 2 things I can see so far:

• Alex is conflating his strategic financing decision with tactical delta management. I do the Socrates thing as a habit when learners entangle multiple issues into a single question. If I just jump ahead and answer each part of the implied questions, I rob Alex of a valuable opportunity — to debug his own thinking by breaking the big question into its components. I notice many people getting option or trading concepts rolled into a ball of yarn, which tells me they don’t understand each string as well as they should. When you’re forced to decompose your own question, the deficiency is self-apparent.
• He’s showing good instincts about not wanting to exercise the calls because he’d be “burning a few months of time value.”

💡Option Pricing Clarifications

1. The first [but not final] check on whether you should exercise a call is if the value of the OTM option on the strike is worth more or less than the cost of carry.
   1. For an ITM call, is the dividend you are exercising for worth more than the put on the same strike?
   2. For an ITM put, is the interest on the stock short if you exercise greater than the call on the same strike?
2. Alex said he thought about “exercising to sell,” but to be clear, you can’t exercise the calls after-hours and have it settle the next day. There is a cut-off for exercise notices. He would have needed to short the stock after hours, then exercise the next day, meaning he would have to carry a short for 1 day. This tactic would have flattened his delta, but he would have to incur any financing ramifications for the 1-day short.

This simple scenario highlights a risk to using heavily-discounted synthetic futures to express a long:

If you plan on liquidating the length before expiration, you are exposing yourself to the change in funding that you otherwise locked in.

In this case, Alex is exposed to the borrow he may have to pay on short shares if he chooses that route to cut length. One day’s borrow, even if steep, is unlikely to make or break the whole trade, but the bigger issue is the “locate”. There might not be any shares to borrow from your broker. If Alex was long the stock instead of the synthetic stock, he could just sell it in the after-market.

If Alex chooses to cut length by selling the ITM call, it may be for the same discount that attracted him to buying them in the first place. It’s like buying a home for a discount because it has lots of street noise. It’s a problem you can’t outrun because you will be on the losing end of the discount when you sell the house.*

Stay groovy

☮️

 

*Tangential, but I see people misunderstand this in the private school vs public school debate. If you pay up to live in a good school district, you are exposed to 2 risks:

1. The carry cost on the premium. If you pay 20% more for a good school district, the funding cost on the extra ~20% of mortgage is the cost. If schools are the deciding factor between an $800k or $1mm house and mortgage rates are 5% then your pre-tax cost of the better schools is ~$10k per year. This is probably much less than private school tuition, especially if you have multiple kids.
2. The perception of school quality. The “school premium” can expand or contract over time. You can win or lose to this when you sell.

In general, the true cost of the premium you pay for good school districts is very small relative to private school tuition, making your VORP hurdle for private school choice high (although worth it for many people).

Here’s an FYI that reinforces a lot of moontower 2025 writing on option synthetic futures.

This is from my IBKR screens from 10/22:

Spot VIX was 18.4

I highlighted the March VIX future. It had a mid-market of 21.725

The ATM strike for options on VIX expiring in March is 22.

The combo or price of the 22 synthetic =

call price – put price = 3.50 – 3.78 = -.28

Synthetic future = Strike + Combo = 22 -.28 = 21.72

No arbitrage available folks (as expected).

The synthetic future on VIX and the actual VIX futures trade in line.

💡The VIX options and futures typically expire on the Wednesday morning preceding monthly option expiry cycles. The options cannot be traded between the prior night's close and the Wednesday settlment but the futures can.

We’ve been talking about option funding stuff recently in the paid Thursday issues. Recently, I had a trader ask for some help making sense of an option expiry in a single name that trades by appointment but where some chunky size goes through.

It’s a name with lots of hair on it with respect to events and distribution.

[The current mark of a big option trade that went through a few weeks ago is still rattling in my head. I’m looking forward to where the roulette wheel is gonna land on this thing!]

I’m obviously not going to give away the name, but I can recycle some of what I explained to the client using a fake stock.

It’s rooted in funding and why understanding it is frankly critical for making sense of names that have wide markets. You’ll see :

• the first thing that caught my eye when I looked at the option chain
• put-call parity’s relationship to a vol curve
• how to avoid making really dumb trades (or if you’re a broker how to look like a hero to your client)

We can do this with screenshots and commentary to make this tour brisk but rich.

We begin with an invented option chain for our fake stock. I chose these values to be in keeping with the quality of the real stock’s markets without giving anything away.

For any junior traders or trainees this is good diagnostic practice — to eyeball an option chain and take notice of what’s interesting.

Relevant background info:

✅European-style expiry (it’s complicated enough without early exercise)
✅No dividends
✅RFR: 4%
✅DTE: 43
✅Stock price: $108.50

What do you notice:

Don’t start all nerd mastermind. Instead observe. These markets are wide!

Well, before you start thinking “The 125/130/135 call fly is negative, yay free money”, you should recognize that the market widths are obscuring this vol surface. I mean, if you think you can trade at mid-market, there’s free money all over this board. All kinds of bells should be going off but just as a surgeon has a checklist, there is definitely a priority thing to look for.

Think a bit before I offer a hint.

 

Ok, here are 2 columns that should help:

IVM = “IV Mid”

Categorically, the call IVs are greater than the put IVs on the same strike.

 

It’s safe to assume the stock is $108.50 as I indicated in the setup.

So what’s the likely culprit?

The rate.

We used RFR = 4% but with that rate put/call parity is not holding.

This is messy since we are using the mid of wide markets, but I didn’t contrive this situation from scratch— it is based on a real snapshot the client showed me on a screenshare, so it’s an opportunity to address real-world complications.

If call IVs is categorically higher than put IVs then the IV is being computed from a rate that is too low.

Instead of imposing a rate, let’s try something else. We will require that put/call parity hold at each strike.

💡Review the method: implying the cost of carry in options

This table is a handy way to start:

I highlighted the 110-strike because it’s closest to the $108.50 spot price.

The right-most column shows the implied yield of each strike. By computing an implied yield the call and put IVs are forced to be the same but I left the stale ones in the table for the sake of this chart:

It demonstrates that a difference in call and put IVs is another way of saying the implied yield or cost of carry on each strike is different.

If you impose put/call parity, forcing the IVs on the strikes to be the same (I didn’t recompute the IVs on each strike here with the implied rates from their strike), then instead of seeing a chart with call and put IVs not lining up you’ll get some implied rate curve across strikes like you see here.

Let’s look at this like a checklist:

✔️If the call and put IVs differ across the strikes when imposing a cost-of-carry parameter (as we did with 4%) then the market is telling you your cost-of-carry parameter is wrong.

✔️Instead, impose a market-based yield by starting with the no-arbitrage assumption of put/call parity to get call and put IVs to line up.

✔️But if this leads to large disparities in implied rates across strikes, well, we still have a puzzle.

Looking at our rate curve…we still have a puzzle.

Experienced traders know why, but just to bring it along gradually, here’s another table that will look very familiar to anyone with an ETF, index, or options arbitrage background:

Computing the implied market by calculating the implied synthetic stock futures bid and offer.

Remember, the synthetic is just the combo price (c-p) plus the strike price. In the prior table we based our synthetic prices on the midmarket values of the options.

Here we want more detail. For each strike, we calculate:

synthetic bid = call bid – put offer

Take the 110-strike as an example to consider what this means…if you hit the screen bid on the calls AND simultaneously lifted the screen offer on the puts, effectively crossing 2 $3 wide markets (before you got fired), you have sold the synthetic future at $107.

We compute the implied yield bid/ask using the implied combo bid and offer with the same logic. [Again, to turn option combo prices into implied yields see this post.]

The puzzle as to why we are getting a ridiculous range of implied yields is not too mysterious — the markets are just too wide. Garbage.

We will do our best with what we have because there’s still plenty to see.

The art of computing the vol surface

The preferred way to set a vol curve is to imply the rate, then impose that on the surface to generate strike vols. Since the implied rate on each strike based on mid-market won’t be perfectly uniform (although likely much better than this stock) you will still get different call and put IVs on the same strike but they are not likely to “cross”. In other words, you won’t be able to lift a call option on a strike for a lower IV than you can sell on the put (or vice versa). The error in IV should be within the market widths.

To give you a flavor of how you impose an implied rate on the strikes across the same expiry we can consider a few methods.

The tightest market

We cobble together the best bid and offer from any of the strikes to imply a yield. I don’t love this method for actually estimating the rate, but it’s the fastest way to spot an arb! Look at all the implied rates…the 75 strike really sticks out like sore thumb. If you hit the call bid and lift the put offer you have synthetically sold the stock at $109.80. If you buy the shares for $108.50, borrowing to finance them until expiry in 43 days, you will have legged a “conversion” trade for a fat profit.

Trader math — I borrow $108.50 for 2 months at 5% (notice conservative assumptions on both days and rate) that’s 1/6 * 5% or 80bps on $108…call it 90 cents. So I buy stock for $108.50, sell it at $109.80 and pay $.90 in interest…$.40 pure profit. Manage to get filled on 50 combos? That’s $2k in 2 seconds. Annualize that.

I’m getting carried away. This kinda thing is never just sitting there because it’s easy to program a bot to “eye” for it and then if it does find it, it’s because you ingested stale data. Still, I hope I conveyed the benefit of the “tightest market” method even if the benefit accrues to speed demons.

Weightings

How else can we find an implied rate to impose across all strikes? We could average the implied yields we find at each strike but give more weight to strikes with tighter markets (in this case, every market is $3 wide so the strikes would get equal weight and given the widths — still garbage).

We can just choose to look at a range of strikes near-the-money. We can weight their implied rates by inverse distance to the stock price. We can exponentially weight tight markets. We can draw hard cutoffs on strikes that exceed specified widths. You can use a solver across strikes but even then you probably filter the strikes according to criteria that come from experience.

[And when it comes to American options, may god have have mercy on our souls. The value of rev/cons can vary widely across strikes as the probability of early exercise differs. You are very much triangulating across several unknowns because the probability of an option being exercised also depends on its vol so you end up falling back on some ordinal relationships that bound early exercise relativity between strikes. In English, it’s easier to say relative things about early exercise adjustments to rev/cons than it is to absolutely value a rev/con. If there’s one area that even experienced traders trip up on its American options. Through the grapevine, from multiple sources, I’ve heard that one of the largest market makers in the biz lost a meaningful proportion of their annual profits because their mispricing of early exercise was exposed during the rate hikes in 2022.]

Anyway, there are enough choices involved that no 2 firms compute vol surfaces starting with implied rates in exactly the same way. There’s no benefit to bogging down on a specific method for this post so I’m just going to impose the 11.65% rate from the 110-strike and re-compute IVs.

With the single rate, the call and put IVs come closer together especially for the near-the-money options. The deep OTM options are going to stay a problem because a $3 wide market on a .15 delta option is just a lot of vol points of noise. [The vega on those options is much smaller than ATM, so $3 represents more vol width.]

Towards a single vol curve

Usually when you look at a surface, it’s just a single vol curve through all the strikes for a given expiry. A common way to do this is to simply use the IV from the OTM option. That option tends to have a tighter bid/ask width since it has less delta risk for the quote streamer.

If you don’t want to dismiss all the information from the ITM option on the strike, you could weight the IV inversely to the market widths or even to the options’ contribution to the straddle price on the strike. Here’s the vol curves using the OTM method and the Inverse Contribution To Straddle method:

Looks like a “W”.

If you have been paying attention to all the stuff I’ve written about vertical spreads and butterflies, you can guess the implied distribution:

bimodal

And yes, the original stock I helped the client with does indeed look bimodal. This is also a distribution you often see on stock earnings.

[My group used to call it the “teepee” and I heard from some transplants that it was another famous market maker who made a lot of money “teaching” the market that this was the right shape for a vol surface in particular situations.]

While this fake stock is the spawn of a real bimodal stock, this is not the most interesting thing about the surface.

By far the most important thing to see is that the implied rates are totally jacked!!! The calls are leaned incredibly high relative to the puts. You have to be able to see this right away (or infer it from the call IVs being high relative to the put IVs if you use a platform that doesn’t impose put/call parity on ATM mid vols).

Calls should never be too high relative to puts because conversions are easy arbitrages especially in non-dividend-paying stocks.

[In conversion trades, you sell call, buy put, buy stock. You must fund the stock purchase so you are exposed to rising interest rates. But that is the only material risk.

Reversals which entail buying call, shorting puts and shorting stock are exposed to declining rates but also any suprise dividends or the stock becoming harder-to-borrow. That’s why you almost never see implied rates trade much higher than SOFR — it’s easy to arbitrage via conversions. But implied rates often trade far lower than SOFR because borrow is uncertain. If you do a reversal trade because you want to exploit the implied rate, your most likely outcome is to find out the rebate you anticipated on your short stock was wishful thinking].

What can you do if you notice the calls are too high relative to the puts?

The eager beaver is going to say “do a conversion arbitrage”. I appreciate the optimism. But those markets are wide. Those mid prices are fake. You can’t get filled anywhere near mid if you try to sell calls or buy puts.

But that’s the clue.

What do you do with the knowledge that the implied rate is too high if you can’t do conversion arbs?

You simply don’t buy calls or sell puts anywhere near mid-market. You are walking into a trap. If you are a broker or advising someone, you explain this to them as well. You’ll save them a bunch of money and they’ll appreciate that you know your stuff.

[This also helps manage expectations. If you are a broker and given an order to sell calls into this market, you should point out that the implied rates are high, meaning the calls are leaned up, and the customer shouldn’t expect to get filled near mid. Similarly, they wouldn’t be able to buy puts near mid either as those are leaned down.]

 

In closing, when you look at an option surface there are so many invisible decisions about how to compute the IVs. When you see call and put IVs that vary greatly, your instinct should be to imply the rate. This post has been in the recent tradition of “options are ALWAYS about vol EXCEPT when they are about funding” but I hope today’s effort has actually shown that we can’t actually compute the vols without understanding funding.

One of the funny things about options is that while variance is an abstract concept to trade (the square of standard deviation??) it’s a straightforward bet — the outcome of the trade is tied to the intent. The payoff reflects the expression. If the realized volatility will be low, sell this.

Meanwhile, listed options, literally called “vanilla”, these American-style shape shifters tradeable from your phone are a pile of path-dependent, hard-to-solve “halting” problems, being discussed by weekend-house-flipper salespeople because there are so many ways to win or lose that are unmoored to your original intent that the randomness of the experience makes them perfectly marketable despite their basics being inscrutable to their average user.

I hope this post made them a little less inscrutable to you.

This is the follow-up to last week’s the easiest win in options is for stock traders.

In that post, we started with a puzzle that leads to a critical insight:

The collective pursuit of option arbitrage means that we can use put-call parity in reverse — to imply the cost of carry instead of assuming one, THEN trying to impose put-call parity.

In the example of the $100 stock and 4% SOFR rate, we computed the cost of carry or what we formally call the “reversal/conversion” or R/C was $3.92.

synthetic future = C – P = intrinsic Value + R/C

where:

C = call value on the 100-strike

P = put value on the 100-strike

The fair value of the synthetic future in our example is therefore:

→ synthetic future = intrinsic Value + R/C

→synthetic future = (S – K) + R/C

→ synthetic future = (100-100) + 3.92 = $3.92

I expect the call to be trading for $3.92 MORE than the put on the 100-strike if the stock is $100.

If it’s trading for a larger premium than $3.92 then there should be an arb:

• Sell call, buy put [short the synthetic future]
• Buy the stock

This is a “conversion trade and since the cost to finance the long shares is the 4% we used to compute fair value, I should have a profit left over.

If the call is trading at a discount to $3.92 vs the put then I should be able to do a “reversal” arbitrage where I:

• Buy call, sell put [long the synthetic future]
• Short the stock

The interest I collect on the proceeds of the short sale should exceed the premium I paid for the synthetic.

That’s the theory.

Of course, if you’re fair value differs from market pricing, guess who’s probably wrong.

Instead of using some assumption about the cost-of-carry, we invert:

“What does the cost-of-carry need to be for put-call parity to hold?”

It’s hard to overstate how powerful this inversion is. It has profitable applications to retail option traders, directional stock traders, both long and short, quants modeling option surfaces, and even fundamental investors concerned with dividends.

Conveniently, the lowest-hanging fruit affects the largest groups — directional stock and option traders. We will cover this in detail while keeping explanations shorter for the more professional applications.

We start with a question:

Have you ever noticed that the call IV and put IV for the same strike on an option chain are NOT equal?

This is all going to make sense soon. With some basic mechanics and simple algebra we are going to discover a whole new order book for stocks.

Solving for r: volatility is not the only thing we imply

We are going to take this journey in small steps.

We start with our identities to build our “if-then” muscles:

where:

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

If r increases, R/C increases as the gap between the strike and strike discounted to PV widens.

Let’s re-arrange the synthetic future identity which includes the R/C to be in terms of the call and put respectively:

→ Synthetic future = Intrinsic + R/C

→ C - P = (S-K) + R/C

If r increases, R/C increases, therefore, calls go up in value while puts go down in value.

The heuristic:

When interest rates are higher the opportunity cost of buying shares increases or the cost of leverage increases if you buy on margin. Arbitrage ensures these costs are passed into the value of calls just as they are passed into the basis of futures over cash in any forward market.

Volatility

The inputs to the Black Scholes pricing formula are:

• stock price
• strike price
• DTE (as fraction of a year)
• RFR
• volatility

For a given volatility, you can compute the call value, then, without using an option model, use put/call parity identities to compute the put from the call.

Note these call and put values are generated by the same volatility. We used the vol to get the call and then computed the put.

But this workflow isn’t typical. Instead, we are usually looking at option prices from a chain with implied volatility. In other words, the workflow is inverted. Instead of inputs generating option values, we see option values and imply inputs.

Notably, implied volatility.

Implied volatility is computed by fixing the option price and letting the volatility be the unknown.

[The solution is usually computed with a simple search algo like the Newton method which starts with a guess, then iterates until you are “close enough”.]

The RFR will be fixed to compute the implied vol, but when you observe the option prices you may find that the call and put have different implied vols. Another way to interpret this:

Put/call parity is not working.

But here’s the thing — put/call parity must work. If it doesn’t “work” there’s an arbitrage.

• If the call IV is lower than the put IV, you can do that reversal trade: buy call, sell put, short stock
• If the call IV is greater than the put IV, you can do the conversion: sell call, buy put, buy stock

What do you think is going to happen?

You will discover that a key assumption in the formula for generating those implied vols is wrong. The strike and DTE are in the contract specs. The stock price and option prices are observable from the marketplace.

The only variable remaining is the interest rate.

You can certainly call your broker to verify the interest rate, but they won’t be able to tell you tomorrow’s rate or any day after that.

What does this mean?

If you impose the rate and the call IV > put IV, then the market’s implied rate is lower than your assumption [and vice versa].

By assuming put/call parity must hold we are saying that the IV on the call and put of the same strike must be equal. But the only release valve for this constraint is we must accept that the market-implied rate can be different from what we think it is.

This is exactly what we should do.

By measuring the market rates by assuming no-arbitrage, we can then decide if a trade is attractive given our own funding rates. If the market implied interest rate is lower than what our broker offers (ie calls look cheap and puts look expensive or said otherwise the synthetic future looks discounted), then instead of buying the stock, we can buy the synthetic.

In fact, this is what professional option desks are doing all the time — they compare their funding costs from their brokers to the market-implied funding costs. If they can “refinance” their position in the options market, they effectively “go around” their broker. The implied funding market in options, including box rate markets, is often tighter than the spread of your broker’s long vs short rates. For a large enough desk it is not uncommon to have a trader whose entire job is to “manage funding” by trading rev/cons across the portfolio to reduce gross notional balances (ie if they are long lots of stock they will look to reverse or swap into futures if the cost of carry is cheaper than what the broker charges to borrow).

Solving for implied rate

Back to something we can easily see in the market — the price of the synthetic future (also known to older traders like myself as a “combo”):

Synthetic future = Intrinsic + R/C

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

I’ll use the examples from the webinar.

On 7/18/25, USO was trading $76.06

I pulled up the closest ATM strike in each month — the 76 line — and computed the synthetic future as the call – put.

I then subtract the intrinsic value of $.06 from each combo. The remainder is the R/C or cost of carry.

Remember:

We just rearrange this to solve for r, which gives us the implied rate.

Notice that the implied rates are below the Fed Funds curve at the time.

If you started with “I’m certain that the Fed Funds curve reflects my funding rate” then the combos would all look too cheap. When you “reversed” to do the arbitrage by buying the synthetic future and shorting the stock you’d discover why your Fed Funds assumption was faulty.

You will find that you are earning less than Fed Funds on your short stock proceeds.

But this gets better.

This is a perfect demonstration of why understanding this concept is immediately profitable. On 7/18, Interactive Brokers was charging 5.93% annualized to borrow USO. But you could short the stock via options to collect the rev/con instead of paying fees!

Consider the October expiry:

You could sell the synthetic futures at $.98 or $.91 more than intrinsic value, effectively collecting 3.3% annualized to be short USO instead of paying 5.93%. This is more than a 9% swing in carry costs (which is about 1/3 of the stock’s annual vol to put it in context).

Even though the funding rate from your broker stinks, you can “inherit” the market-makers rates by trading the options. The market-makers battling for arbitrage is a giant peace dividend to the rest of us who cannot access the same rates and borrow that institutions can. But even if you are a professional, the implied rates are often out of sync with the rates you can access, so there’s ample opportunity to refinance your positions in the synthetics market. The implied rate curve in the term structure is effectively an order book for a stock through time.

💡Refresher on how shorting works
→ If a stock is easy to borrow, you might earn a positive rebate (e.g., SOFR – 25 bps) on collateral of short proceeds
→ If the stock is hard to borrow (high demand, low supply), the rebate can be negative. This means you pay to borrow the stock (sometimes called the borrow cost)

Discussion

It should be a revelation to realize that the physical shares market is only one price for a stock, but the derivatives markets offer many others. The USO example showed how you can short USO at a higher synthetic price than if you borrowed the shares directly. Similarly, if a synthetic future trades far below the stock price, reflecting a high borrow cost, anyone who cares to buy the stock will get a massive discount in the options market.

The “real” market

When BYND went public, the peanut gallery (ie twitter) was all screaming how they wanted to short this fake meat company, but this was a consensus view — the shares were impossible to get your hands on to short. I’m going off memory, but the options market was pricing the synthetics at about ~40% discount to the ordinary shares. So the question for the peanut gallery isn’t “Do you still want to short the shares at the market-clearing price where the stock can be both bought and sold?”

Because that price is 40% lower. And if you were a long-term bull, what are you doing buying the ordinary shares? Just take the 40% discount and buy the synthetic.

[BYND has lost most of its value since it went public 6 years ago, but I wonder if a trader who but synthetic futures and rolled the position at each expiry would have actually won. I really hope so since that would be one of my favorite case-studies on the nature of trading.]

Backtesting

Directional traders who test both long and short strategies should be using the option synthetics market to reflect tradeable prices because those prices “lock in” a funding rate. Otherwise, backtests not only require borrow rate data sets but also need to deal with the fact that borrow rates change daily. A wicked backtesting concern.

Funding all the way down

In etf fair value, I mentioned an old habit from my arb days: computing the premium/discount on an ETF before trading options on it. I’ll leave this for you to ponder:

If an ETF trades 1% above its NAV, where should the synthetics on the ETF trade?

Vol modeling

When computing option surfaces, it’s common practice to imply the rate, THEN use that rate in the implied volatility formula to compute the IVs across the skew. This ensures that each strike has a single IV, and when charting the skew, we use the implied vol from the OTM option — its mid-market willbe more reliable because of narrower bid/ask. Having a single IV per strike also ensures the absolute delta of the call and put sum to 1.

In the examples I gave above, we implied the rate from a single strike that was closest to ATM. But a more robust method would average (or weighted average) the implied rate from more than 1 strike in case the bid/ask on any single strike was shaded too much in one direction. Multiple strikes would minimize the impact of those artifacts.

Dividends

I only addressed dividends in the appendix of the prior post to keep all of this a bit easier. For our current purpose, just recall that dividends decrease the cost of carry or R/C since the call owner misses out on the dividend but still experiences the stock falling by the amount of the dividend. Meanwhile, the put owner forgoes owing the dividend on the counterfactual short shares position and benefits from the stock falling by the amount of the dividend when it “goes ex”.

→The rev/con falls pushing puts up relative to calls

Easy enough.

However, the idea of an implied rate gets more complicated when we solve for r in the presence of dividends. Although it’s not hard to understand conceptually.

Consider a situation where Fed Funds (I’ve been using FF and SOFR interchangeably), is 4%, we expect the stock pays a 1% dividend, but the rev/con is 2.5% instead of something closer to 3% that we would predict from the general shortcut of cost of carry is interest – dividends.

Is this because the options market is saying your short rebate is 50 bps less than Fed Funds OR the stock’s dividend is expected to be 50 bps more than it has been in the past OR some blend of a dividend increase and rebate difference?

Do you see how incorrect assumptions here change the implied rate, which in turn affects all the implied vols?

Not only is there an implied rate, but an implied dividend. Whenever we get multiple unknowns, we need multiple lenses to triangulate. This is the realm of quantitative vol surface modeling, a task that many professional option traders outsource to specialty firms especially if their trading must discern the value of a penny in the premium.

This concludes the 2-part series on options as funding markets. As I said in part 1, this topic represents the largest gap between what people know and what they should know about options. It affects pricing, it’s highly actionable, and by allowing anyone to “refinance” a position at professional rates, it stands as one of the easiest win-wins in trading.

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.