implying the cost of carry in options

the basis of arbitrage and vol surface modeling

Friends,

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.