Lottery tickets and put-call parity

Two ideas finish the pricing picture. The first explains why cheap options are usually cheap for a reason. The second is a relationship that ties calls, puts and the stock together, and it will save you from several bad trades once it is in your head.

Why far-OTM options are lottery tickets

The $56 call from the last lesson costs $0.25 with XYZ at $50 and 30 days left. That is $25 per contract, and if XYZ finishes at $60 the call is worth $4.00, a sixteen-fold return. Cheap and huge upside: that is the pitch, and it is exactly the structure of a lottery ticket.

Put some rough numbers on it. Suppose XYZ has about a 6% chance of finishing above $56 in 30 days, and when it does, the average finish is around $58, so the call is worth about $2.00 on average in that case.

• Expected value at expiration: 0.06 × $2.00 + 0.94 × $0.00 = $0.12.
• You paid: $0.25.

Even before the bid-ask spread, you are paying roughly twice the expected payoff. Why would anyone sell it for $0.25 then? Because the seller also bears the small chance of a much larger loss than $2.00, and because market makers price a demand premium into far-OTM options; buyers like them, so they trade rich. That is the option version of the long-shot bias at the racetrack: the worst-odds bets are the most overpriced.

The figures are illustrative, not from a model, but the pattern is what you see in real markets: the further out of the money, the more you pay per unit of expected payoff. None of this means far-OTM options never work. It means that buying them as a habit is a strategy with negative expected value, and that occasional big wins are what keeps people doing it.

There is a legitimate use: when you believe the market is badly underestimating the chance of a specific large move, and you can afford to lose the premium many times over while you wait to be right. That is a rare situation, not a Tuesday afternoon.

Put-call parity in plain words

For European options on the same stock, same strike and same expiration, this always holds (ignoring dividends and interest for a moment):

call price − put price = stock price − strike

Or: being long a call and short a put at the same strike is the same thing as owning the stock at that strike. That combination is called a synthetic long stock.

Check it against the chain from the first lesson in this module. XYZ at $50, strike $53, call $0.75, put $3.70:

• Call − put = 0.75 − 3.70 = −2.95.
• Stock − strike = 50 − 53 = −3.00.

The five-cent difference is interest on the strike (you would earn a little on the $53 you did not have to spend) less any dividend expected before expiration. On a longer-dated option or a high-yield stock the gap is bigger, but it is always explained by those two things.

Why parity matters to you

Calls and puts are the same thing. A put is a call plus a short stock position; a call is a put plus a long stock position. So a put at a given strike has the same extrinsic value as the call at that strike, adjusted for interest and dividends. If someone tells you "puts are expensive right now but calls are cheap" at the same strike and expiry, one of you is reading the chain wrong.

Covered calls are short puts. Long stock plus a short call equals a short put at the same strike. Anyone who will not sell a naked put but happily writes covered calls is taking the same risk with a different name. Module 7 uses this heavily.

Synthetic positions. Every basic position can be built two ways. Long call = long stock + long put. Long put = long call + short stock. Sometimes one route is cheaper or has better liquidity than the other, and knowing that gives you options in the plain sense of the word.

Spotting errors. If the parity relationship is broken by more than interest and dividends explain, either you have the wrong strike, the wrong expiration, a stale quote, or an adjusted contract. It is a fast sanity check on any chain.

Key idea: Far-OTM options are priced above their expected payoff because people like lottery tickets. Put-call parity says a call and a put at the same strike are the same option wearing different clothes; every position has a synthetic twin.

The parity formula with interest and dividends

For completeness, the exact version is:

call − put = stock − strike × discount factor − present value of dividends

where the discount factor is slightly below 1 and reflects the interest earned on the strike until expiration. With rates near zero and no dividend, it collapses to the simple form. With rates at, say, 5% and a year to expiration, the strike is discounted by about 5%, which on a $100 strike is $5, a noticeable difference between the call and put prices.

Try it: On any liquid chain, pick a strike close to the money. Compute call − put and compare to stock − strike. Then pick the same strike a year out and do it again. Note how the gap grows, and work out whether interest or dividends account for it.

Recap

• Far-OTM options are usually priced well above their expected payoff; buying them habitually has negative expectancy.
• The rare justification is a strong, specific view that the market is under-pricing a large move, sized so that many losses are affordable.
• Put-call parity: call − put = stock − strike, adjusted for interest and dividends.
• Every position has a synthetic twin: a covered call is a short put, a long call is long stock plus a long put.
• If parity looks broken by more than interest and dividends, check the quote, the strike, the expiry and whether the contract is adjusted.