Calls, puts, payoff diagrams, put–call parity and the strategy zoo — the contracts that turn an underlying $S$ into a derivative.
An option is a contract that gives its holder the right but not the obligation to trade an underlying asset $S$ at a fixed price $K$, on or before a fixed date $T$. The holder pays the writer a premium up front in exchange for that right.
"An option is the right, but not the obligation, to buy or sell an underlying at a prescribed price at a prescribed time in the future."
— Wilmott, Ch. 2That single asymmetry — right, not obligation — is what separates an option from a forward (where you must transact) and is the entire reason an option has a non-trivial value.
The right to buy $S$ at $K$.
Worth something when $S$ rises above $K$. Worthless if $S < K$ at expiry: don't exercise, walk away.
$$\text{payoff at }T = \max(S_T - K, 0).$$
The right to sell $S$ at $K$.
Worth something when $S$ falls below $K$. Worthless if $S > K$ at expiry.
$$\text{payoff at }T = \max(K - S_T, 0).$$
Every option strategy in this book is a sum of these four legs:
| Position | You pay/receive | Payoff at $T$ | View |
|---|---|---|---|
| long call | pay $C$ | $+(S_T-K)^+$ | bullish, limited downside |
| short call | receive $C$ | $-(S_T-K)^+$ | bearish/neutral, unlimited risk |
| long put | pay $P$ | $+(K-S_T)^+$ | bearish, limited downside |
| short put | receive $P$ | $-(K-S_T)^+$ | bullish/neutral, capped upside |
The buyer chooses whether to exercise; the writer must comply. The writer is paid the premium to take on that obligation.
The contract is fully specified by a handful of parameters; the rest of the book is about pricing them.
The book mostly prices Europeans; American early-exercise is treated by free-boundary problems (Deck 03 and onward).
| State | Call | Put |
|---|---|---|
| in the money (ITM) | $S > K$ | $S < K$ |
| at the money (ATM) | $S = K$ | $S = K$ |
| out of the money (OTM) | $S < K$ | $S > K$ |
The premium decomposes into
$$C = \underbrace{\max(S-K,0)}_{\text{intrinsic}} + \underbrace{C - \max(S-K,0)}_{\text{time value}}.$$
The intrinsic value is what the option would pay if exercised immediately. The time value is everything else — the option holder's chance that the underlying moves further in their favour before expiry. For a European option, time value $\ge 0$; it decays to zero as $t \to T$.
Even if $S < K$ today and the call is worthless if exercised now, there is still some probability the spot moves above $K$ before $T$. That residual probability — weighted by the discounted payoff — is the option's time value.
Payoff diagrams plot the value of the position at expiry as a function of $S_T$. They ignore the premium — that comes in when we plot profit.
$$V_T = \max(S_T - K, 0).$$
Flat at zero below $K$, slope $+1$ above. Unlimited upside; downside capped at the premium $C$.
$$V_T = -\max(S_T - K, 0).$$
Mirror image. Best case: keep the premium and walk away. Worst case: unlimited loss.
$$V_T = \max(K - S_T, 0).$$
Slope $-1$ below $K$, flat above. Best when the underlying collapses; bounded loss = premium.
$$V_T = -\max(K - S_T, 0).$$
Maximum gain = premium. Loss bounded only by $S_T \ge 0$; equivalent to being long the stock plus a fixed liability.
Buyer's payoff + Writer's payoff = $0$ at every $S_T$. The option is a zero-sum contract; the premium is the price at which both sides agree the bet is fair.
Buying an option is simple: pay the premium up front, the most you can lose is the premium. Writing an option is much more dangerous — you receive a small premium and take on a potentially large liability.
To stop writers walking away, the exchange demands margin — collateral posted with the clearing house. The margin is marked-to-market daily; if it falls below maintenance level the writer must top it up or the position is closed out.
Margin requirements for naked short calls are particularly punishing because the loss is unbounded.
One listed equity option in the US covers 100 shares. Quoted premiums are per share. FX options are quoted as a percentage of notional; index options are cash-settled because you can't deliver an index.
Expiry is almost always a Friday; American style is the norm for listed equity options, European for index options.
| Action | Cash today | What you owe |
|---|---|---|
| buy a call | $-C$ | nothing (you have a right) |
| sell a call | $+C$ | $\max(S_T-K,0)$, plus margin |
| buy a put | $-P$ | nothing |
| sell a put | $+P$ | $\max(K-S_T,0)$, plus margin |
"Writing options is not an activity to undertake casually." Long positions risk only the premium; short positions risk the entire payoff, which is why margin exists.
Calls and puts on the same underlying, with the same strike and expiry, are not independent — they are linked by a no-arbitrage identity. Consider two portfolios at $t = 0$:
Payoff at $T$:
$$\max(S_T-K,0) - \max(K-S_T,0) = S_T - K.$$
Payoff at $T$:
$$S_T - K.$$
Both portfolios deliver the same payoff at every $S_T$. By no-arbitrage they must cost the same today:
$$\boxed{\;C - P = S - K e^{-rT}.\;}$$
This is put–call parity. It is model-free — no stochastic process, no volatility, no Black–Scholes — it follows from no-arbitrage alone, exactly like the forward price in Deck 01.
Rearrange:
$$S = C - P + K e^{-rT}.$$
A long call plus a short put plus a deposit of $K e^{-rT}$ is a synthetic share. You can build any one leg from the others; that is why a market in calls fully determines the market in puts.
Traders watch the parity residual $C - P - S + K e^{-rT}$ in real time. A persistent non-zero value signals either a quoting error or a hidden cost (borrow fee, dividend assumption, hard-to-borrow stock).
A binary (or digital) option pays a fixed amount if the underlying ends in a specified region at expiry, and zero otherwise.
Pays $1$ if $S_T > K$, else $0$.
$$V_T = \mathbf{1}_{\{S_T > K\}}.$$
A pure bet on direction; no exposure to the magnitude of the move.
Pays $S_T$ if $S_T > K$, else $0$.
$$V_T = S_T\, \mathbf{1}_{\{S_T > K\}}.$$
The ordinary call is the difference: cash-or-nothing pays $K$ on the same event, so $C_{\text{vanilla}} = V_{\text{asset}} - K\, V_{\text{cash}}$.
Binaries are useful as building blocks. Any payoff $f(S_T)$ that is piecewise linear in $S_T$ can be written as a portfolio of digital and vanilla options, which is how exotic books are static-hedged in practice.
The payoff is a step function. As $t \to T$ and $S$ approaches $K$, the binary's delta and gamma blow up — tiny moves in the spot flip the payoff from $0$ to $1$. The retail-friendly "binary option" of online broker fame is the same instrument, and the same risk-management nightmare.
A vertical spread is two options of the same type and expiry, different strikes. The first family of strategies that bound both the upside and the downside.
Long a call at $K_1$, short a call at $K_2 > K_1$.
$$V_T = \max(S_T - K_1, 0) - \max(S_T - K_2, 0).$$
Payoff: $0$ below $K_1$, slope $+1$ between $K_1$ and $K_2$, capped at $K_2 - K_1$ above.
Cheaper than an outright call; you've sold the tail.
Long a put at $K_2$, short a put at $K_1 < K_2$.
$$V_T = \max(K_2 - S_T, 0) - \max(K_1 - S_T, 0).$$
Payoff: capped at $K_2 - K_1$ below $K_1$, slope $-1$ between $K_1$ and $K_2$, $0$ above.
Bearish view with bounded loss.
"Bull" = the spread profits if the market rises; "bear" = profits if the market falls. The names are loose — you can build a bull spread out of either calls or puts — what matters is the slope of $V_T(S_T)$.
These are volatility strategies: trades whose P&L depends on how much $S$ moves, not where it goes.
Long one call and one put, same strike $K$ (usually ATM).
$$V_T = |S_T - K|.$$
V-shape with minimum at $K$. Profitable if $S$ moves far enough in either direction to overcome the combined premium.
Long a call at $K_2$ and a put at $K_1 < K_2$.
Cheaper than a straddle (both legs OTM) but needs a bigger move to pay off. Flat between the strikes, V-shaped outside.
Long a call at $K_2$, short a put at $K_1 < K_2$ (or vice versa).
Used in FX. A synthetic forward with a "dead zone" between the strikes — cheap or even zero-cost depending on the skew.
| Strategy | You profit when… | Greek you're long |
|---|---|---|
| long straddle | realised vol > implied vol | vega, gamma |
| short straddle | $S$ stays near $K$ | theta |
| long strangle | large move in either direction | vega (cheaper) |
| risk reversal | directional view + skew view | delta + skew |
The point of slides 7-9 is that by combining options you can shape your P&L to bet on volatility, skew, or kurtosis — not just direction. Decks 6-8 explain how that exposure is priced.
One more turn of the screw: three options instead of two, four instead of three, different expiries instead of different strikes.
Long at $K_1$ and $K_3$, short two at $K_2 = (K_1+K_3)/2$.
Tent-shaped payoff peaking at $K_2$. Profits if $S_T$ lands near $K_2$. A bounded short-vol play.
Long at $K_1, K_4$; short at $K_2, K_3$ with $K_1 < K_2 < K_3 < K_4$.
Flat-topped trapezoid. Like a butterfly with a wider sweet spot.
Same strike, different expiries: short a near-dated option, long a far-dated one (or vice versa).
Trades the term structure of volatility and the differential theta decay.
Every payoff on this slide is a finite sum of vanilla calls and puts — the same four cardinal positions of slide 01. The whole catalogue of strategies is just linear algebra over those four basis vectors.
Add legs (long/short call/put), set each strike and premium, and watch the total payoff $V_T(S_T)$ (yellow) and the P&L (green) update. Breakevens and net premium appear in the metrics.
Try the presets: a straddle is V-shaped (long volatility); a bull spread is the truncated ramp of slide 07; a butterfly is the tent of slide 09. The P&L curve sits below the payoff by the net premium — that's all option pricing is about, ultimately: how much is "paying for the payoff" worth?