Options — the right, not the obligation

Every contract in M0.7 obliges both sides. The farmer must deliver, the miller must pay, and the future is just that arrangement industrialised. An option breaks that symmetry, and it’s the only member of the family that does. Get the asymmetry and the premium, the jargon and the odd hockey-stick payoff all follow from it.

The two words that define it

An option gives its buyer the right, but not the obligation, to trade the underlying at a fixed price (the strike) by a fixed date (the expiry). The seller — the writer — has no such luxury: if the buyer chooses to exercise, the writer must deliver.

• A call is the right to buy at the strike.
• A put is the right to sell at the strike.

If the memory hook helps: you call the asset away from the writer, or you put it to them.

Why the premium exists

A future costs nothing to enter. That’s not a promotional offer — it’s because the deal is symmetric: at the agreed price, both sides face the same shape of risk, so neither owes the other anything to get started.

An option is not symmetric at all. The buyer keeps every bit of the upside and simply walks away from the downside. The seller has the mirror image: a capped gain and a large, sometimes unlimited, loss. Nobody takes that side for free. So the buyer pays a premium on day one.

That’s the whole reason an option costs money up front, and it’s worth stating plainly because it’s often mistaken for a transaction fee. It isn’t a fee. The premium is the price of the asymmetry.

What it looks like with numbers

ACME trades at £100. You buy a call: strike £100, three months to expiry, premium £5 per share.

   ACME at expiry    exercise?    payoff    premium     net P&L
   ─────────────────────────────────────────────────────────────
      £ 80              no         £  0       −£5        −£5
      £ 95              no         £  0       −£5        −£5
      £100              no *       £  0       −£5        −£5
      £105             yes         £  5       −£5         £0    ◀ breakeven
      £120             yes         £ 20       −£5       +£15
      £200             yes         £100       −£5       +£95

   * at exactly the strike, exercising gains you nothing

Three things fall straight out of that table:

• Your loss is capped at the premium. ACME can go to zero; you lose £5 and no more, because you are never forced to buy.
• Your gain has no ceiling. That open-ended upside is precisely what the £5 bought.
• Breakeven is the strike plus the premium — £105, not £100. This catches almost everybody once: being right that ACME would rise, and watching it reach £103, still loses money.

Drawn out, it’s the shape every options textbook opens with:

   LONG CALL — strike £100, premium £5

      +15┤                            ╱  upside open
         │                         ╱
         │                      ╱
        0┼───────────────────╱─────────────  breakeven £105
       −5┤━━━━━━━━━━━━━━━━╱
         └────┬─────┬─────┬─────┬─────┬───▶  ACME at expiry
             80    90    100   110   120

   flat line on the left = loss floored at the premium, forever

The put is the mirror. Buy a put at strike £100 for £5 and you break even at £95, your loss is again capped at the £5, and you gain as ACME falls — up to a maximum of £95 if the company goes to zero. Not unlimited, because a share price can’t go below nothing.

Four positions, not two

Every option has two sides, so the basic vocabulary is four positions. This table repays memorising:

                          you want         max gain          max loss
   ────────────────────────────────────────────────────────────────────
   long  call  (buy)    price up, a lot    unlimited          premium
   long  put   (buy)    price down, a lot  strike − premium   premium
   short call  (sell)   price flat/down    premium            UNLIMITED
   short put   (sell)   price flat/up      premium            strike − premium

The rows pair off exactly: the buyer’s loss is the seller’s gain, penny for penny. Between the two parties an option is strictly zero-sum.

Look hard at the short rows, because they explain why selling options is so seductive and so dangerous. Max gain = the premium. You collect a small sum very often, and pay out a large one rarely. Sold with discipline that’s a real business — it’s the same payoff shape as market making in M0.4, where steady winnings from uninformed flow cover occasional beatings from the informed. Sold without discipline, it looks like free money right up until the day it isn’t.

The vocabulary, briefly

• In the money (ITM) — exercising right now would pay: a call with spot above the strike, a put with spot below it.
• Out of the money (OTM) — it wouldn’t. If nothing changes, the option expires worthless.
• At the money (ATM) — spot is roughly at the strike.
• European vs American — exercisable only at expiry, or any time before it. Nothing to do with geography.
• Intrinsic value — what the option would pay if exercised this instant (never less than zero). Time value — everything else in the premium: the chance things improve before expiry.

Premium = intrinsic + time value, and time value grinds down to exactly zero at expiry. An option is a wasting asset; the buyer is fighting the clock and the seller is being paid by it.

What moves the premium

You don’t need the pricing formula to reason about direction. Five things move the price of a call:

   more time to expiry     ▲    more chances to end up in the money
   more volatility         ▲    bigger swings — see below
   spot rises vs strike    ▲    more likely, and more deeply, ITM
   higher interest rates   ▲    mild: paying the strike later is worth more
   dividends coming        ▼    the option holder doesn’t receive them

The one genuinely worth internalising is volatility raises the price of calls and puts alike, which sounds wrong the first time you hear it. It follows from the asymmetry: the downside is already floored at the premium, so widening the range of possible outcomes only fattens the tail you actually get paid on. A more volatile stock is a more valuable thing to hold an option on, whichever way the option points.

Which means an option is, in a real sense, a position on volatility itself rather than only on direction — you can be right about where ACME ends up and still lose, or wrong about direction and win, depending on what happened to volatility in between.

That is the doorway to the rest of the course, and it’s where M0 deliberately stops. Turning that intuition into an actual number is Black–Scholes; running it backwards — taking the market price and asking what volatility would justify it — gives implied volatility; and measuring how the price responds to each input gives the greeks. All three need machinery M0 hasn’t built, so they wait for M2 and M3.

Source: John Hull, Options, Futures, and Other Derivatives, ch.10–12 (mechanics and properties of options). Read the payoff diagrams closely and skip the mathematics on a first pass — the asymmetry does more work than the algebra. Pricing is deliberately deferred to M2–M3.