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:
- Start with a puzzle
- Show how the solution informs arbitrage theory
Next week, we go from theory to practice:
- 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.
- 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
You notice the following market prices:
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”.