In a random walk where trials are independent, variance scales linearly with time. Since standard deviation is the square root of variance, volatility scales with sqrt(T).

This sublinear power law scaling gets smuggled into option math that answers practical questions. For example, assuming implied vol is constant, a 12-month ATF straddle is twice the price of a 3-month ATF straddle because sqrt (12/3) = 2.

This scaling is commonly used to convert raw vega into weighted vega. Raw vega is an extremely low-resolution number. If you own 50k 12-month vega vs being short 40k 3-month vega then it appears like you are long vol. But 12-month IV doesn’t whip around as much as 3-month IV, so this position will not act like it’s long vol on a large move higher in vol as the term structure will not “parallel shift” higher. The 3-month will increase faster as the term structure steepens into a downward sloping shape. A shape referred to as “inverted” or “backwardated”.

A simple way to modify raw vega is to scale all your monthly vegas by 1/sqrt(T) by normalizing them to a fixed DTE, for example 3 months. In that case, using the same math we did above, a 12-month vega is cut in half relative to the 3-month.

So your re-weighted vega is now short 15k vega instead of being long 10k vega!

12-month vega x scaling factor relative to 3m vega = +50k * 1/sqrt(12/3) = +25k

3-month vega x scaling factor relative to 3m vega = -40k * 1/sqrt(3/3) = -40k

Net: -15k

That volatility changes should move in proportion to 1/sqrt(T) is not a commandment brought down from Moses. It’s a convenient scaling factor that corresponds better, even if imperfectly, to empirical vol surface behavior. It also has a handy interpretation. If IV’s change in proportion to 1/sqrt(T) then ATM time spreads are unchanged (net of theta). In other words, the 3m/12month straddle spread is unchanged in such a regime.

Again, this scaling doesn’t need to hold. Sometimes we have parallel shifts in term structure and sometimes term structures steepen faster or slower than sqrt(T) scaling would predict. But the scaling is still a better prediction than the raw vega measure, which would have you believe IVs from all months are directly comparable without adjusting for how slow long-dated IVs change or how fast a weekly IV can move.

Random walks and the derivative pricing theory built upon them assume returns are independent. In hindsight, random walks still exhibit stretches that can be labeled “trend” (like a run of heads) or “mean reversion” (period of frequent alternating). But it’s one thing to label these stretches and hindsight vs predict them.

It should be self-evident that being able to predict trends or reversion would be marvelously profitable for a directional trader. But, direction aside, it would be a gift to volatility traders as well. It would influence not only how they priced vertical spreads and time spreads but the deltas in their models and their delta-hedging strategies. In other words, it would change everything if you had an edge on the probability of the next move being up or down, even if you did not have an edge on the fair value of the stock (this would occur if you had an edge on probability but not on the magnitude of up move vs down move). Option structures allow fine-grained bets that can isolate probability from magnitude.

If an asset trends over weeks or months, you will underestimate its volatility by scaling its daily volatility by sqrt(T). That makes sense. If it trended, that’s similar to saying the moves were auto-correlated and therefore dependent. Again, this is descriptive, not predictive, but relating measures of volatility to this interdependence lets us see how sensitive option pricing is to the random walk assumption. A few articles I’ve written in this vein:

• how a high implied vol can be cheap | 9 min read
• The Option Market’s Point Spread (Part 2) | 11 min read
• Thinking In N not T | 6 min read

These articles have a unifying concern. If prices are random, then sure, the power function that specifies how volatility scales is the familiar:

But if prices trend or mean-revert, the exponent is no longer 1/2.

Over any historical sample, H can be observed to be something other than 1/2. For it to be 1/2 would mean that annualized volatility over 2 different sampling windows was identical. In hindsight, that will rarely occur. But it’s also true for any exponent you pick. It’s hard to make the persistent case for a value other than 1/2, especially when it carries the financial totem of randomness.

In Retail Options Trading, Euan Sinclair says markets aren’t random, but they’re close to random. The question of whether there’s enough life growing in the gap between “random” and “almost random” for a skilled hunter to eat is existential professional investors’ careers.

We need to examine randomness.

Returning to the context of volatility scaling and its relationship to randomness, Euan reaches for a popular quant tool. The Hurst exponent. That’s why I picked H for the exponent in the general version of the volatility power law.

Euan’s definitions:

• H = 0.5 is a random walk. No memory.
• H < 0.5 is mean-reverting. Up tends to be followed by down.
• H > 0.5 is trending, or “persistent.” Up tends to be followed by more up.

It’s time to do some learning moontower-style and start with the basics.

What The Hurst Exponent Actually Measures

Our Favorite Starting Point: Coin Flips

Flip a fair coin 100 times. Score +1 for heads, −1 for tails, and keep a running sum.

After 100 flips, how far from zero is that running sum?

Three stylized regimes to compare:

• Perfectly correlated flips (every flip copies the last one): the running sum after 100 flips is ±100. It grows linearly with N.
• Perfectly anti-correlated flips (+1, −1, +1, −1, …): the running sum never escapes ±1. It doesn’t grow with N at all.
• Independent flips: the running sum lands around ±√N or in this case ±10.

Think of these as regimes that correspond to three scaling exponents:

• Correlated (trending) N^1
• Anti-correlated (mean-reverting): N^0
• Independent (random walk) N^0.5

The exponent is the answer to “what power of N does the cumulative range scale with?”

Strip out the step size to isolate the regime

The ±1 coin gave a running sum with range around √N. If the coin paid ±10 instead, the range would be 10·√N. Bigger steps, bigger range. We want to strip out that distortion. If we measured price range on raw market data, a jumpy stock would always look more “trending” than a calm one, just because its steps are bigger. We’d be measuring volatility tangled up with regime, when we want regime alone.

The fix is to divide the range by the standard deviation of the steps: R/S

For the ±1 coin, R ≈ √N and S = 1, so R/S ≈ √N.

For the ±10 coin, R ≈ 10·√N and S = 10, so R/S ≈ √N. Same answer. The step size cancels out.

That’s the rescaled range. R/S only cares about the regime of the series, not its scale.

From coins to assets

Now we can adapt this to asset returns.

So we have two measurements over a window of T days of log returns:

• S = the standard deviation of the returns (the step size in the coin example)
• R = the range (max − min) of the cumulative sum of the de-meaned returns. How far the running total wandered between its high and its low.

We de-mean before computing R, so we strip out drift. We don’t care that the thing went up over the window, we care how it wandered around that trend. We divide by S to strip out the volatility scale.

The √T Benchmark

If returns are independent, R/S also grows like √T for the same underlying reason:

The variances of independent things add, so the spread grows by √T.

Now generalize it. Instead of forcing the exponent to be 0.5, let the data tell you:

R/S ~ T^H

• H = 0.5: matches √T. Independent.
• H > 0.5: R/S grows faster than √T. Trending. Moves reinforce each other.
• H < 0.5: R/S grows slower than √T. Mean-reverting. Moves fight each other.

Reading H Off A Plot

The scaled range takes the functional form of a power law. If we take logs of both sides, the power law becomes a straight line, and the exponent H becomes the slope of the line.

log₂(R/S) = H · log₂(T)

Compute R/S at a few different T’s, plot them log-log, and the slope is H. It doesn’t matter which type of log we use. We could choose log₁₀ or ln, but using log₂ gives a clean way to narrate it: every time you double T, R/S multiplies by 2^H.

• H = 0.5: each doubling multiplies R/S by √2 ≈ 1.41
• H = 1.0: each doubling doubles R/S
• H = 0.0: each doubling leaves R/S untouched

The Implementation Recipe

1. Pick several T’s (say 5, 10, 20, 40).
2. At each T, chop the sample into non-overlapping chunks. (see appendix)
3. For each chunk: de-mean, cumulative sum, R = max − min, S = std dev, then R/S.
4. Average R/S across the chunks at that T.
5. Fit a line through the (log₂T, log₂(R/S)) points. The slope is H.

Worked Examples

Computing one R/S by hand

Take a single 5-day chunk of returns, in %: +1, +3, −2, +4, −1.

1. Mean: (1 + 3 − 2 + 4 − 1) / 5 = +1%
2. De-mean (subtract the mean from each): 0, +2, −3, +3, −2
3. Cumulative sum (running total of the de-meaned series): 0, +2, −1, +2, 0
4. R is the range of that running total: max − min = (+2) − (−1) = 3
5. S is the standard deviation of the original five returns ≈ 2.28 (population stdev, STDEV.P)
6. R/S = 3 / 2.28 ≈ 1.32

That 1.32 is one chunk’s R/S.

Notice that since √5 ≈ 2.24, this little stretch wandered less than a random walk would, so it reads mean-reverting

We just repeat this for several windows.

Say you’ve got 80 days of returns.

Compute R/S at T = 5, 10, 20, 40:

The Hurst exponent, H ≈ 0.43, is extracted as the slope from the log-log plot, which is is linear transformation of a power function.

H<.50 corresponds to mean-reversion. Every doubling of T multiplies R/S by 2^0.43 ≈ 1.35, a hair under the 1.41 you’d get from a pure random walk. The wandering is growing slower than random diffusion would predict.

Applications of H

If H isn’t 0.5, then √T annualization is wrong for that asset. H > 0.5 means your long-horizon vol is higher than √252 × daily vol claims. H < 0.5 means it’s lower.

The articles I linked to in the intro wrestle with this same idea but in a simpler point-to-point manner in the form of a trend ratio (ie vol sampled weekly ÷ vol sampled daily).

If you assume the asset is “self-similar,” then the exponent H governs the scaling at every horizon then besides looking for trend or mean reversion strategies you can now research a world of option relationships that are potentially mispriced if the assumption of independence is strongly embedded in volatility scaling models.

To be reductionist, my trend ratio calcs were a two-point estimate of H. Autocorrelation patches function as a lagged estimate of the same thing. Hurst is the version that uses the whole curve instead of two points or one lag.

The assumption that markets are self-similar is wrong. The more wrong it is, the less you have to gain from Hurst vs point-to-point extrapolations, but all of this is dominated by the biggest elephant in the room. Can past data help you predict trend or mean-reversion at all? Which just circles back to Euan. If you are going to bother trading, you must believe, at worst, they are merely “almost random”.

A Sense Of Proportion

H looks like a number between 0 and 1, so a move from 0.50 to 0.55 feels insignificant. The vol-annualization lens is the cleanest way to debunk that.

Consider a stock with 1% daily vol.

• At H = 0.50: 1% × 252^0.5 = 15.9% annual
• At H = 0.55: 1% × 252^0.55 = 19.4% annual

A 0.05 bump in H means a 22% increase in annualized vol. This obviously affects your opinion of option prices but it’s also meaningful for position sizing and risk or VaR.

Most equity-index Hurst estimates sit in a narrow-looking 0.45 to 0.55 band, but that “small” band obscures significant differences.

The Catch: The Naive Number Lies

Now go back to Sinclair’s warning, because this is where it earns its keep.

Classic R/S — the recipe above, the one in his book, the one everybody reaches for first — is biased. Run it on a series you know is a memoryless random walk, at a 252-day window, and it does not hand you back 0.5. It hands you back something noticeably higher. The estimator manufactures a little fake memory all on its own, before the data even gets a vote.

So when SPY’s rolling H sits below 0.5, you have to ask how much of that is the market and how much is the ruler. This isn’t a fringe complaint. Lo built a modified R/S statistic back in 1991 precisely because the classic version confuses genuine long memory with garden-variety short-range stuff like volatility clustering, and equity returns are drowning in volatility clustering.

The fix is not exotic. Simulate a big pile of random walks the same length as your estimation window, run the exact same R/S recipe on them, and see what H the estimator coughs up on data you built to have none. Whatever offset it shows is the lie. Subtract it. Now a true random walk reads 0.5, and a reading that survives the correction is one you can actually look at.

This is the same humility you already preach about your own VRP work. A single rolling-window H is one draw. Treating it as gospel is exactly the “sample size of 1” trap. Calibrate it or don’t believe it.

Sandbox

I’ve heard of many traders, including option traders using Hurst in their research. It feels like it’s accelerated in the past 5 years. I didn’t take a harder look at it until Euan gave a brief intro to it in Retail Options Trading and LLM’s made it easier to tutor yourself on a quant method. It’s a technique that’s well-known, but anecdotally I’ve heard a wide range of mileage from it (I’m guessing every pro option trader in a seat today has at least heard of it in trading contexts).

If autocorrelation adnrealized vol ratios at different frequencies are worth looking at then Hurst is worth at least “spaghetti on the wall”. I built a Jupyter notebook to tinker using yfinance data. You can use it, fork it, whatever:

https://colab.research.google.com/github/Kris-SF/data-pipelines/blob/main/quant-analysis/hurst_analysis.ipynb

If I were to bring this “in the lab” to see how it can become a metric or even signal I’d start with tinkering to see how it its output jives with my intuition of how a certain asset behaved over a particular period.

Once I had a feel for it, I’d throw the metric up on a scatterplot against other metrics to develop a sense of what is normal. Are there any correlations between H and IV skews or IV term structures? How do changes in Hurst coincide with changes in realized vol (rv is an input to R/S therefore and ultimately H so maybe we are hunting for a residual variable to track?)

If you have organized data, in the world of LLMs all of this work is more fun and faster. For now, I hope this primer on Hurst was a digestible first step for explaining the theory behind it and why it can be relevant.

You can find additional notes below.

Appendix: What “chop into non-overlapping chunks” really means

T is a window length, just how many days of wandering you measure at once. You pick several because H isn’t a property of any single window. It’s the rate at which R/S grows as the window lengthens. A handful of T’s gives you points to fit a slope through.

You have 251 daily returns. You want one number, H. That’s the entire goal.

Pick a few window sizes: 5, 10, 20, 40.

For each window size you do the exact same thing:

• T = 5: chop the 251 days into back-to-back groups of 5. You get 50 groups. Compute R/S for each group, then average all 50. That’s your R/S at 5.
• T = 10: chop into groups of 10. You get 25 groups. R/S for each, average them. R/S at 10.
• T = 20: groups of 20, so 12 groups. Average. R/S at 20.
• T = 40: groups of 40, so 6 groups. Average. R/S at 40.

Now you have four points: (5, R/S@5), (10, R/S@10), (20, R/S@20), (40, R/S@40). Plot them log-log, draw the best-fit line, and the slope is H.

You want enough windows to fit a line, but longer windows are comprised of fewer blocks (like the T=40 window) so they’re shakier sample from which you are computing an average R/S.

Appendix: Bias

The body said classic R/S reads high on a random walk.

The finite-sample problem

Even on a true coin-flip walk, R/S over a short window doesn’t average to exactly √T. It sits a little above. Hurst, Anis, and Lloyd worked out the expected R/S of a random walk in closed form back in the 70s, so one fix is to divide your measured R/S by that expected value at each T before you fit. It’s conceptually similar to the familiar Bessel n−1 adjustment done to sample variance since we don’t know the true population variance.

Claude suggested 2 ways to apply a correction:

• Use the closed-form expected R/S directly
• Simulate a pile of random walks and measure what your exact regression spits out.

They differ because the log of an average isn’t the average of a log (Jensen’s inequality). The closed-form route leaves a residual bias of a few hundredths. The simulation route, because it runs the identical regression you use in practice, lands a true random walk back at 0.5.

After much back-and-forth, I took Claude’s rec and had the notebook use the simulation route.

The nice thing about LLMs is they know a lot of the academic history of a measure. Like I said this is a starting point for your own exploration.

Better estimators exist.

Classic R/S is the cleanest to teach and the weakest to trade. Lo’s modified R/S (1991) is built to ignore short-range dependence like volatility clustering, which plain R/S happily mislabels as memory. Detrended Fluctuation Analysis (Peng et al., 1994) is the workhorse in the econophysics literature. If you ever size a position off an H, cross-check it with one of those rather than lean on R/S alone.

US bond yields are rising as inflation re-enters the conversation. The 10-year yield is up to 4.65% and 30-year bonds have just crossed 5%, a nearly 20-year high.

This isn’t surprising. 6 weeks ago, in Trading As A Sudoku Puzzle With Prices As The Given Numbers, I talked about how 1-year gasoline futures were trading at a 1/3 discount to prompt pricing, but if gasoline prices remain high, this will roll up. If spot prices stay high for a year, those back-month futures will converge to current prices. Even though energy is only about 5% of CPI, the size of such a sustained move would easily transmit 1.5% to inflation indices and that is just due to direct energy effects and ignoring indirect effects on food, construction, and transport.

We’ll switch the conversation to crude oil just because it’s more widely tracked and the specifics of the contracts aren’t critical to where we’re going. Prompt oil is roughly in the same place vs 7 weeks ago, but the contract that was 12-months out and is now 11-months out has rolled up >6%. Meanwhile, another month of sustained high oil prices has pushed the 10-year yield up 30 bps from 4.3% to 4.6% with IEF price returning about -1.6% a bit better than what’s expected by its duration.*

*There’s some leeway since I’m using an index for the yield which may have a different set of weighted maturities than IEF holds. Also, IEF total return is closer to -1.1% because you earn interest for 7 weeks.

So far, so good. The reaction function in bonds makes sense. But my Sudoku post claimed that an inflation-induced yield bump would transmit to real equity risk premiums. In other words, I would expect equities to sell off with bonds, or heck, at least not have such a sharp rally.

This is not quite the puzzle it appears to be. The equity exuberance is actually quite limited if you look under the hood of the index.

From Shannon’s substack:

The internals are doing something the people who watch this for a living have never seen. The S&P is up 4.2% month-to-date with 209 stocks up and 295 down. The NASDAQ is up 8% month-to-date on a near-even split (51 up, 50 down). The index is 9% above its 50-day moving average while only roughly half the components are above their own 50-day; at that distance you’d normally expect 80% breadth. Four days running, more S&P stocks hit new 52-week lows than 52-week highs, with the index at all-time highs and up 30% year over year. Yesterday 9% of the index hit new lows. None of this happens together in a healthy tape.

I’ve noticed many market people interpret these “internals” as bearish. I’m not sure this is bearish for the index. It just is. A few companies are eating everything else. We get it. At this point, the low cross-correlation of the components is common knowledge (isn’t this what managers call a “stock pickers market”?).

Rather than use the term “bearish” which has a predictive slant I can’t justify, we can just accept that the sustained oil price, inflation jitters, and rise in yields are being reflected in prices broadly. SMH (semis ETF) is near 1-year highs while XHB (homebuilders) are near one-year lows.

The AI story is in a parallel vacuum, indifferent to relics like discount rates or identities such as spending = income, but stocks overall are not being indiscriminantly bid. SPY has returned nearly 2x RSP, the equal-weighted SP500 index, over the past year. So the loving arms of our cap-weighted benchmarks hold us tight, shielding our eyes from the turmoil within. Trepidation over supply-side inflation is confirmed by bond and non-AI stocks alike.

Concerned with inflation, I dust off some old posts, like What I Learned About TIPs which I wrote when I bought when breakevens shrunk to about 2.2% (green box).

(When breakevens are skinny, TIPs are relatively cheap compared to nominal bonds, and when they are fat, they are relatively expensive. The way to think of that is if you buy TIPs at say 2% breakevens, then you are better off with the TIPs if CPI realizes more than 2% and vice versa.)

Breakevens are currently matching 3-year highs so TIPs don’t look attractive on a relative basis, but that’s only one lens. The real driver of my decision to buy TIPs in Oct 2023 was the absolute real rate which was ~2.45% which still stands as the peak for most investors under age 40’s working life.

Remember that’s 245 bps of return above inflation for no risk and if you hold them in an IRA, no tax drag. Historically speaking, equity real returns have been in the range of 3-6%, but recent years have been quite a run of heads. Whether the coin is biased now is a question for someone smarter than I. But I digress.

The point is I’ve started once again to consider inflation-aware trades. 10-year TIPs don’t stand out as a bargain relative to nominal bonds. I’m wary on gold and silver because of how well they’ve performed recently, but also historically, they have not been great to own when real rates increase and we can see from the absolute TIPs rate that, despite breakevens not breaking out, real rates are crawling higher, approaching an 18-month high.

So I dust off yet another post, this one from 2 years ago: Inflation Replicator. I show how a portfolio of oil futures plus nominal bonds mimics the behavior of inflation-indexed bonds like TIPs. I constructed it in Composer using USL, which holds a strip of oil futures maturing within the next 12-months, plus TLH, a bond ETF holding bonds with 10-20 year maturities. The portfolio is inverse-vol weighted, rebalanced quarterly.

This is the out-of-sample performance since I published the post (green line).

That portfolio is a set-and-forget inflation hedge if you don’t like TIPs.

[Speaking of “tips”, here’s a general one. If you have a portfolio that rebalances, it is often selling winners to re-invest in losers. This keeps you diversified and avoids the volatility tax that comes from concentration, but it’s not tax-friendly unless you do it in a sheltered account. To do it in a regular account, you can consider a tax-loss overlay where instead of buying more of the losing position, you actually sell the losing position and another ETF that has a highly correlated exposure. So, for example, if TLH is the losing side and you need to add more on the rebalance, you actually tax-loss harvest the TLH and replace it with TLT length. It’s a similar exposure, but you can now use the TLH capital loss to offset the gain on the USL win you trimmed.]

The specific inflation replicator I composed was TLH + USL. But if we abstract it to “bonds + oil”, it invites us to think about risk premia that exist in both asset classes in the current market.

In the remainder of this post, I’ll narrate layering a couple of edges onto a core portfolio idea. By following along, you’ll get concrete ideas for measuring and managing risk and open your mind to the different Legos available to build the portfolio and ultimately express the trade while targeting the carry embedded in the asset’s pricing complex.

Inflation Replicator with positive carry

Let’s talk about our baseline exposures: oil + bonds

Instead of building the inflation replicator portfolio with the USL ETF, we want to isolate a carry-rich version of “oil”.

The oil leg

As I write on 5/20/26, the prompt WTI future, CLM6 (expires in May), is $98.

CLZ6, expiring in November, is $81.75.

If the spot oil market is unchanged over the next 6 months, CLZ6 will “roll up” nearly 17% (~34% annualized).

The bond leg

Long TLT shares. You collect the ~5% annual yield as carry. That’s the simplest expression and what we’ll size against.

Reiterating the core idea of the inflation-protected bond we are creating

We are pairing oil and bonds together because high oil prices are a major driver of inflation and the accompanying weakness in bonds. In other words, the bonds and the oil hedge each other if we own both.

They are coupled antagonistically. Look at the correlation of TLT (longer-dated bond ETF) and USO, which holds prompt WTI futures.

Before the war, the rolling 21-day correlation of returns between TLT and USO ranged from about zero to -.50, spending the bulk of the time between 0 and -.25.

Since the war, the correlation range abruptly shifted lower, recovered a bit and has now collapsed to -.75.

💡Does it matter that we are comparing TLT with prompt WTI via USO when we want to express crude length with the deferred Z26 contract? The vol of the two contracts is very different, which matters for sizing reasons and would show up in the beta, which is vol ratio * correlation. But correlation alone is still tight across the futures strip with M1 to M6 easily above 0.90. It’s safe enough to infer the correlation of TLT to Z6 futures from its relationship with USO.

You will see how the inflation replicator portfolio benefits from the negative correlation when we get to sizing. Understanding the correlation range will also be key, as it’s a critical input to risk management.

Sizing the core portfolio

We begin with a risk target expressed as a fraction of a portfolio. We’ll choose $100k of annualized volatility allocated to this trade. Feel free to pick your own number, the method is what matters.

A $100k annual vol target is easier to reason about if I convert it to a daily number, because daily P&L swings are what I actually watch on the screen. Annual vol scales with the square root of time, so:

$6,300 of daily swings is for the portfolio of oil futures + TLT. We need to size the individual legs of the trade.

Step 1: convert each leg’s vol to a daily number

Again, we are converting annual vols to daily by dividing by √252

CLZ6 has 43% implied vol → daily vol ≈ 2.71%

TLT has 11.5% implied vol → daily vol ≈ 0.72%

Oil is about 3.7x as volatile as TLT on a same-dollar basis.

Step 2: inverse-vol weight the two legs

Inverse-vol weighting means I want each leg to contribute the same daily dollar volatility to the portfolio. Not the same notional, the same risk. The high-vol leg (oil) gets less notional, the low-vol leg (bonds) gets more, until they pull equal weight in risk terms.

Mechanically:

The daily dollar vol due to either asset should be equal. We’ll set that dollar vol equal to S.

Step 3: solve for the portfolio vol as a function of S and correlation

This is the two-asset portfolio variance formula:

 

The w’s are dollar weights, the vols are in daily percent, ρ is correlation.

Inverse-vol weighting forces w₁σ₁ = w₂σ₂ = S, therefore every term becomes a multiple of S²:

 

Note how correlation has such a large impact on the portfolio vol. At today’s ρ = −0.75, the multiplier √(2(1−0.75)) = √0.5 ≈ 0.71. The portfolio is less volatile than a single leg.

Step 4: invert to find the leg size

I want σ_p = $6,300. Solving the formula above for S gives S = $6,300 ÷ √0.5 ≈ $8,900. So each leg should carry about $8,900 of daily dollar vol.

Convert that back to notional: oil notional = S ÷ daily oil vol = $8,900 ÷ 2.71% ≈ $328,000.

• CLZ6 is $81.75 and each contract is 1,000 barrels, so one contract is ~$81,750 of notional. $328,000 ÷ $81,750 ≈ 4 contracts.
• TLT is .72% daily vol, so we need $8,858/.72% or ~ $1.22mm of notional or about 14,500 shares because TLT is $84

[$8,858 instead of the $8,900 we solved for comes the fact that we need 4 contracts that are not divisible any further. Note how the bond notional is ~3.7x the oil notional, exactly the inverse of the vol ratio.]

And the resulting portfolio daily vol at ρ = −0.75 is σ_p = $8,858 × √0.5 ≈ $6,263. Right on our $6,300 daily risk target, which corresponds to $100k of annual vol.

The beauty and danger of correlation

Let’s appreciate what’s happening here by considering monthly risk and reward.

Let’s start with risk.

Scale daily risk to monthly:

$6,300 *√(252/12) = $28,870

Now for the expected reward.

Oil: 2.5% roll up * $327,000 notional = $8,175

TLT: 5% yield * $1.22mm / 12 months = $5,083

Total expected return = $13,258

Monthly sharpe ratio = $13,258/$28,870 = .46

Annualize the SR:

.46 * √12 = 1.59

This is possible because we get to be long quite a bit of assets in notional terms, but the volatility of the portfolio is small.

The reason it’s so small is that the correlation is very negative.

But ρ = −0.75 because the war pushing oil up is adding a risk premium to bonds (ie pushing their price lower).

To understand the risk, we must stress-test correlation. We fix S and vary ρ.

[Risk should really be treated like a matrix since changes in the correlation will coincide with the vol of the legs moving, thus changing the vol ratio between them. For example, if the war relaxes and the correlation heads back towards 0, oil prices likely fall, bonds likely rally. That’s ambiguous for the p/l, but since oil’s vol is the one that’s more stretched from “normal” you are underweight the falling asset which is good. However, the increased correlation means total portfolio risk is more than you intended]

Your portfolio risk doubles if corr goes back to 0.

Juicing the bond leg with options

So far the bond leg is plain-vanilla long TLT shares earning the ~5% yield. But given the sell-off and inflation fears, the bond option market is also offering risk premia as vols have increased and put skew has steepened.

Let’s talk about the vol first.

VRP

As I write on 5/20/26, the ATM 1-month put is around $1.20, corresponding to 11.5% IV. TLT’s realized vol has been running below its implied. 1m realized vol is ~8%, 1-week rv is closer to 9% and median 1-month rv for the past year is about 10.5%.

Call it a 15% vol risk premia:

• Put premium: $1.20/share
• Fair value: $1.20 ÷ 1.15 ≈ $1.043/share
• VRP edge: $1.20 − $1.043 ≈ $0.157/share, or 15.7 cents per share

The practitioner’s way to carry that number in your head is per contract. Each contract is 100 shares, so the VRP per contract is $0.1565 × 100 = $15.65 per contract per month.

The bond leg’s delta target was the equivalent of +14,556 shares of TLT. An ATM put has a delta of about −0.50, so selling one put gives you +0.50 deltas per share, or +50 deltas per contract (100 shares × 0.50). To replicate the share position’s delta: contracts = 14,556 deltas ÷ 50 deltas/contract ≈ 291 contracts.

291 × $15.65 ≈ $4,555 per month, or ≈ $54,700 annualized.

If you sell ATM puts to express the same long-delta exposure. You collect the put premium, and the portion of that premium above fair value is vol risk premia stacked on top of the yield carry.

💡It’s never that simple when it comes to options. The yield carry is the yield * notional but as TLT moves around, you are short gamma so as the stock falls you are longer TLT and as it goes up, you become less long TLT, so the yield due to bond income is a moving target.

Be careful. 291 contracts on an $84 stock is $2.44mm of gross notional, even if it’s still $1.22mm share-equivalent notional. The local delta exposure is identical, but by swapping the expression to pick up VRP we added non-linear risk to the position.

Skew

TLT’s 1-month risk reversal is at the 94th percentile of the trailing year. The put skew is rich, call skew is depressed.

You can express the delta by selling OTM puts, which will make the risk non-linearities even more concave. You can also sell put/buy call on risk reversals to take advantage of the stretched skew in both directions. All of this is changing the shape of the p/l and risks dramatically. The best way to get your arms around it is to construct a matrix of scenarios.

Oil options

The bond leg harvests rich skew by selling puts but the oil leg can do the same thing in the opposite direction.

Oil call skew has a war premium. That makes a call spread an attractive way to express the long oil leg: buy a closer-to-the-money call and sell a further-OTM call at a stretched IV against it, financing part of your long with the fat skew you’re selling.

For example, instead of buying 4 Z26 futures, you could buy the Z26 88/98 call spread. With the underlying at $81.75 this OTM structure costs ~ $2.35. It has a .12 delta, so to get 4 contracts worth, you’d need to buy ~33 call spreads (4/.12).

You’re long the rollup-and-supply-scare upside, but you’ve capped your gain to $7.65 (about 3.2-1 odds on your premium), but your downside is limited to the debit if Hormuz de-escalates and oil pukes.

Trade management

It’s well understood that when it comes to options your risk is changing as assets move around, as time passes, and as implied vol fluctuates.

A more subtle risk is how your exposure changes on the oil leg even without options. The oil future becomes more volatile it approaches maturity ages. The 6-month oil future currently has a 43% implied vol but the near-dated future can be twice the vol in times of stress. So even if nothing moves, the oil leg’s daily dollar vol creeps up over the life of the trade. This might be partially mitigated by the roll-up amount becoming steeper as you approach the front of the curve.

The 1-month rollup from M2 to M1 is twice as steep as the 1-month rollup from M6 to M5.

 

The good news: the same risk framework that sized the trade also manages it. Re-run σ_p = S·√(2(1+ρ)) with fresh inputs whenever the market moves:

• Oil vol rose? Each leg’s S is no longer balanced. Trim oil contracts (or tighten the option overlay) and add bonds to re-equalize the legs and pull portfolio vol back to the $6,300 daily target.
• Correlation drifting toward zero? The shock table prescribes how to proportionally hold both legs to maintain the vol target.
• If you use options and your total risk or relative leg risks get out of tolerance bands, you can reassess to see if you should roll, add, or even close.

Because the trade is a living position, you may want to treat the target risk as an upper bound, and initiate the trade at smaller sizing giving you wiggle room to rebalance less often.

A summary of stacked edges

The bond leg is long carry because the position has a net long delta (yield) and short rich puts (VRP).

The oil leg is long carry (rollup) and short rich calls (skew).

You’ve taken a simple “buy oil, buy bonds” inflation replicator and layered distinct edges onto it, each one sourced from a risk premium in the pricing complex:

1. Oil rollup carry (term structure)
2. Bond yield carry
3. Bond VRP + put skew
4. Oil call skew

On the carry side, you are monitoring VRPs, term structure, and coupons, while on the risk side you are monitoring the volatility of the legs as well as the correlation which has a major impact on the portfolio risk.

How big a portfolio does this need?

I sized everything to $100k of annual vol, but I never said what size account sits behind this trade. Vol targets don’t specify a portfolio on their own — they specify a portfolio once you decide what fraction of your risk budget the trade gets. If you want this to be a 10% vol sleeve, you’re implicitly running it against a $1mm book. A 5% sleeve implies $2mm.

You’ll immediately notice a problem if you consider the 10% / $1mm case. The bond leg alone is $1.22mm of TLT shares, which exceeds the entire account. You can’t fund it with cash, you need leverage. Futures are inherently levered as you only need to post initial and possibly variation margin. For the equity portion, portfolio margin can allow you to post even less than a 50% haircut.

But leverage introduces path risk. Your position is changing with market conditions, especially if you use options. But this portfolio sizing is resting on a large position in a low-vol asset as well as a negative correlation. The simplest way to appreciate the risk is to notice that a mere $100k of annual vol rests on ~$1.5mm gross exposure. If you run this portfolio at $100k annual vol with only a $1mm account, you are managing both risk and margin closely.

Recall the portfolio expected Sharpe was 1.59. So for $100k annual vol, we expect $159k in profits or 15.9% on a $1mm account. The expected return halves to ~8% if you run it in a $2mm account.

The best but most complicated choice is to run a strategy like this in a diversified account where the other moving parts interact with the portfolio margining computations such that the required haircut is small and therefore efficient.

Let’s leave it there for today.