A very carefully written digestion of Chapter 3 of Options, Futures and Other Derivatives.

Fundamental question about why we hedge (why not just invest in Treasury bills)

A. The hedger could be uncertain about the makret, but can be more certain about the instruments being hedged outperforming the market B. Sometimes people also hedge to weather volatile periods for assets they want to keep for the long haul

Hedging The Whole Thing...

...With A Short Future Contract

The below is a simplified scenario where hedge (LOCK IN) your entire price by hedging everything.

The entire operation of a short future can be reduced to realising a profit through a price differential at two different times. If we go with the example of a short option

The long story is:

1. Borrow commodity (from somewhere else). Hence i have $+0 and j\ \text{moles} of commodity
2. Sell commodity at $x. Hence I have $+x and 0 commodity.
3. Buy commodity back at $y //closing the position, Hence I have $(x - y) and j\ \text{moles} of commodity
4. Return commodity. Hence i have $(x - y) and j\ \text{moles} of commodity

And the “short” (haha) story is: when you take the short position and close it your profit is (y - x)per barrel, then if you go short on an amount of barrels matching your spot agreement, you can use that to effectively get the original spot price you were trying to buy the barrel at

E.g. say you agreed on august spot. You hope its 79/barrel. It turns out to be 80/barrel. Tough luck.

But you’re awesome and stuff so you hedge by sending a short future at 79. Your profit (from forcing someone to buy at 79 when the price rose to 80) is then

and when the time comes to pay the spot price you use the +1 to cancel the -80 back to -79

Basis

Here's some stuff I wrote on a whiteboard. Context: Basis is S_T - F_T.

small errata

Two things to note to myself here:

1. during a future contract, we’re not literally borrowing a commodity and selling it. We’re borrowing the right to a delivery of the commodity at a future date.
2. i had the misconception that this story meant there were 3 seperate deals occuring: a. a spot transaction b. a short/long position futures contract c. a futures contract on the opposing side
   in actuality Claude says there’s just 2, i think. (a, b)

Risk - The reason why basis isnt always 0

In ideal scenario, basis converges to 0 because arbitrageurs exploit price differences between spot price and future price.

But basis is not necessarily 0 (basis is S_T -F_T), because though we wish it would be, those 2 terms are not necessarily the same. This uncertainty is known as basis risk and it happens because (according to Hull):

1. Cross Hedging
   The asset whose price is to be hedged may not be exactly the same as the asset
   underlying the futures contract
2. We Don’t Know When We’ll Close Our Position (in general)
   The hedger doesn't know in advance exactly when the real-world spot transaction will happen — since you close out the futures contract at the same time as the spot transaction, that uncertain date means you also don't know in advance which day's basis you'll end up locking in.
3. We Might Need To Close Our Position Too Soon
   The hedge may require the futures contract to be closed out before its delivery month.

It took me awhile to convince myself that the 3 risks even existed. These are the reasons I could prompt out for their existence

(1) Cross Hedge

Futures contracts are standardized, exchange-traded products — they only exist for a limited menu of specific, liquid underlyings (e.g. WTI crude, corn, gold, S&P 500 index, specific Treasury bonds). But the actual thing you're exposed to in real life is often something more specific or slightly different, and there's simply no futures contract written on that exact thing.

Example: an airline wants to hedge its exposure to jet fuel prices. There's no actively traded jet fuel futures contract with enough liquidity, so the airline instead hedges using heating oil futures (which move similarly to jet fuel because both are middle distillates).

(2) General Position Close Uncertainty

So the hedger sets up the futures position based on an estimate of when the transaction will happen. When the real-world transaction actually occurs, that's when they close the futures position whenever it turns out to be. Since the basis (gap between spot and futures price) changes day by day, and you don't know in advance which day you'll be closing on because which specific date in the month the spot goes through is out of your hands (vessels with the oil commodities arriving at different itmes etc.), you don't know in advance exactly what basis you'll be locking in.

(3) Specifc Position Closing Too Early Problem

In a perfect world where reality corresponded to the future contract:

On actual delivery month, its price and the spot price converge to same value (arbitrageurs causing basis → 0).

However, we don’t get that mathematically simple case very often:

A. Futures contracts only mature in a handful of fixed months (e.g., March, June, September, December for many contracts) — not "whatever day you need." If your transaction is expected in April, there may be no contract expiring in April at all. You have to pick the nearest available one, which usually means a contract maturing after your expected transaction date. B. Even when a well-aligned contract exists, Hull's own rule of thumb is to deliberately pick a contract that matures later than your expected transaction date — on purpose — and then close it out early. Why not pick one that matures right on time, or just before? Because of reason 2: you're not 100% sure when the real transaction happens or when you Close The Position.

If you picked a contract expiring right around your expected date and the real transaction slipped later than planned, the futures contract could expire while you're still exposed — leaving you with no hedge for the remainder, or forcing you to deal with actual physical delivery, which most financial/commercial hedgers want to avoid entirely.

Like. Ok like. Ugh.

Say today is January. You expect to sell your asset in May. You're choosing between:

• Contract A: matures in May (aligned with your expectation)
• Contract B: matures in September (deliberately later)

Now suppose the sale slips to July (later than planned — this is exactly the reason-2 uncertainty).

With Contract A: it matures in May, before your sale happens. You can't just "leave it open" past maturity — as a contract nears its delivery month, you're forced to either take/make physical delivery (which as a financial/commercial hedger, you almost certainly don't want) or close it out. So by May, you're forced to close out Contract A whether or not your real sale has happened yet. Now you have no futures position covering June and July, while you're still sitting on the actual price exposure.

Shakespare: Your hedge quietly stopped working two months before you needed it to.

Your only fix at that point is to open a brand new futures contract to cover June–July. That's called "rolling" the hedge — and it's not free or riskless: you're now transacting at whatever price/basis exists in May for that new contract, which is a fresh source of uncertainty layered on top of the original problem you were trying to solve.

With Contract B (matures September): your sale happens in July, contract is still very much alive, you just close it out early — a normal, unforced, everyday action. Nothing breaks. You had slack built in, so the slippage in reason 2 didn't cost you anything structural.

Cross Hedge P&L

The algebra is just a telescoping sum

The S_T^* terms cancel. So yes — identical to the non-cross-hedge formula. Hull writes it this way purely to split total basis risk into two independent sources:

• (S_T - S_T^*)— the mismatch between the asset you're actually exposed to and the asset the futures contract is written on
• (S_T^* - F_T) — the "ordinary" basis risk that exists even in a perfect hedge, i.e. the futures price not converging exactly to the spot price of its own underlying at time T

In a non-cross hedge, S_T=S_T^* by definition (you're exposed to the same asset the futures tracks), so that first term vanishes and you're left with just the ordinary basis risk term. Cross hedging adds the extra term on top. Decomposing it this way lets you reason about the two risks separately — e.g. the second term shrinks to zero as T\to delivery, while the first term depends on how correlated your asset is with the futures' underlying, and doesn't necessarily shrink at all

So around this part, I got so genuinely confused because this still means the P&L of the non-cross hedge instance hasnt changed (its a telescoping equation) and neither has the risk.

Metaphorically, it feels like pretending we introduce risk by drawing crayon lines here and there. but theres no change to the behaviour of the population because the new “risky” “spot on hedge asset” cancels.

It turns out the reason I was seeing it like this was because i thought S_T = S^*_T. It turns out actually F_T is the term substituted by S^*_T.

Where P&L used to be

and i thought it was being transformed into

the p&l equation is actually being transformed into

And the problem now is theres more uncertainty.

Back when Future Contract [F_T] close was tracking the Exposure Asset [S_T] F_T \rightarrow S_T was basically whittling the basis term down to 0

Now, F_T is tracking something different, the hedge asset, and F_T \rightarrow S^*_T can approach that asset, but since S^*_T is merely correlated to S_T, theres an additional discrepancy gap there, causing a bit more fluctuation in our P&L calculation

So, profit is the same, basis risk is a little different (dependent on S^*_T - F_T), and now theres a discrepancy term adding more risk (S_T - S^*_T)

linear regression again

h* is the hedge ratio you can choose to get the minimum variance, its the slope of the best-fit line from a linear regression of delta S against delta F

note: observations are made from time bins (non-overlapping, and as long as the period we’re hedging for)

so if h* is 2 then…

a. This means the delta S (S_T - S_0) vs. delta F (F_T - F_0) best-fit line has a slope of 2, so that fit shows delta S moving 2 units for every 1-unit move in delta F. b. Since spot moves twice as much as futures per the fit, you need twice as much in futures notional to offset your spot position — not because futures "react" to spot, but because both move together and you're scaling futures up to match spot's larger swings: N^* = h^* × (Q_A/Q_F).

(where Q_A is the number of items, Q_F is the number of items per contract, so Q_A / Q_F is number of contracts to hedge the whole position and h^* is the ratio of those total amount of those accounts)

Cross Hedge P&L final form

$$PT​=hF0​+ST​−hST∗​

Equities and Indexes

Siimilar concepts are extended by people who know what they're talking about from futures and underlying assets, to index portfolios and your personal portfolio

If you mirror an index portfolio, then the perfect amount to hedge is

If you do this, in laymans terms, it means you’re locking the amount you’re currently holding flat. Recall the start of the article where I mentioned to myself sometimes people will hedge when they’re unsure about the market but sure about their position relative to the market

If you have a personal portfolio, then

Beta is estimated by comparing a portfolio's excess return, R_p−R_f, to the market's excess return, R_m−R_f, where R_f is the risk-free rate (e.g., the T-bill rate) over the same period. Subtracting R_f from both removes the risk-free component of return, which reflects only the time value of money and is unrelated to risk-taking, so what remains is the compensation earned specifically for bearing market risk. Beta therefore measures how sensitive an investor's risk premium is to the market's risk premium, rather than how sensitive raw returns are to each other.

A hedge using index futures removes the risk arising from market moves and leaves the hedger exposed only to the performance of the portfolio relative to the market.

stuff i dont really understand yet/honestly i dont really want to write about:

1. tailing the hedge
2. stack and roll
3. changing beta (recalculate your new beta, if more/less than then take the same/opposite position on (b - b^*)(V_A / V_F) contracts to get to that ratio)