implying the cost of carry in options

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.