Options Primer

2.7.1 Rights, not obligations: calls and puts

An option is a contract in which one side buys a right and the other side sells an obligation. Everything in this session — and most of Unit 10 — follows from that asymmetry.

A call option gives its buyer the right, but not the obligation, to buy an asset — the underlying — at a fixed price, the strike price $K$, at or before a fixed date, its expiry. A put option gives the right to sell at $K$. The buyer pays for the right up front; that price is the option premium. The seller, or writer, collects the premium and takes on the obligation: to deliver at $K$ if a call is exercised, or to buy at $K$ if a put is. Buying makes you long the option; selling makes you short.

At expiry the holder's decision is mechanical. A call is exercised only when the stock is above the strike, so it pays $\max(S_{\text{exp}} - K,\, 0)$; a put pays $\max(K - S_{\text{exp}},\, 0)$. Notation for this session: $S$ is the underlying's price today and $S_{\text{exp}}$ its price at expiry (as in Session 2.4), $C$ and $P$ are the call and put premiums (here $P$ is a put, not the price $P_t$ of Session 2.3), and $\tau$ is the time to expiry in years, because $T$ stays the number of periods. Drawn against $S_{\text{exp}}$, these payoffs are the famous hockey sticks — a payoff diagram:

The payoff ignores what you paid; the P&L at expiry subtracts the premium (like most payoff diagrams, we ignore the interest on it). The lab's running example is a $100 stock with 3-month options, a 4% risk-free rate and a 20% volatility — §2.7.6 explains where the premiums come from. The 100-strike call costs $4.49 and the 100-strike put $3.49, so the call buyer breaks even at 104.49 and the put buyer at 96.51:

Two things jump out. The short put column is the long put column with the sign flipped: before costs, options are a zero-sum game. And the call is leverage without a loan: if the stock rises 10% by expiry, the call returns +123%; if it does not move at all, the call returns −100%.

Reading a contract

Listed options are standardised and cleared by a clearing house — in the US, the Options Clearing Corporation (OCC). A standard US equity option covers 100 shares, and premiums are quoted per share, so a $4.49 call costs $449 per contract. There are two exercise styles. American options can be exercised on any business day up to expiry; European options only at expiry. At expiry, the OCC's exercise-by-exception procedure exercises any equity option that is at least $0.01 in the money unless the holder instructs otherwise; your broker may use a different threshold (Options Industry Council, as of 2026). Two ways to trade S&P 500 options — through an ETF or on the index itself — show both conventions for exercise and settlement (Cboe fact sheet, 2026; US examples):

Does the exercise style matter? For a call on a stock that pays no dividends, no: early exercise throws away time value and pays the strike sooner, so an American call is worth no more than a European one (Merton 1973). For puts, and for calls just before a dividend, early exercise can pay; Session 10.1 — No-Arbitrage & Binomial Trees prices that extra right on a binomial tree. Everything else in this session uses European options.

2.7.2 Moneyness, intrinsic value & time value

An option's price splits into what it would pay if exercised right now and what the remaining time is worth. Where the stock sits relative to the strike decides the split.

Moneyness describes where today's price sits relative to the strike. A call is in the money (ITM) when $S \gt K$ — exercising now would pay — at the money (ATM) when $S \approx K$, and out of the money (OTM) when $S \lt K$. For a put, the labels swap sides:

Intrinsic value is what the option would pay if exercised now: $\max(S - K, 0)$ for a call, $\max(K - S, 0)$ for a put. Time value is everything else in the price. With the stock at $100, three strikes from the lab:

Why does time value exist? Because the payoff is lopsided: below the strike, bad outcomes are floored at zero; above it, good outcomes count in full. Average that hockey stick over every price the stock might reach by expiry and the floor is worth money — more the wider the spread of possible prices, so more with higher volatility and more time. That is optionality under uncertainty. The lab checks it by switching the uncertainty off: at $\sigma \to 0$ the at-the-money call is worth only $0.995, which is $S - K e^{-r_f \tau}$ — the interest saved by paying the strike later — against $4.49 at 20% volatility. The other $3.49 is optionality. With less time left there is less room to move, so time value decays: with one month to go the ATM call is worth $2.47 (fig 2.7.3). The rate of that decay is theta, one of the Greeks of Session 10.4 — The Greeks & Hedging P&L.

2.7.3 Put-call parity: two portfolios, one payoff

You can build a call out of a put, a share and a bond. That single fact pins the price of every European call to the price of the put with the same strike and expiry — no model, no forecast, no volatility required.

Compare two portfolios. A holds one call and a zero-coupon bond that pays $K$ at expiry, which costs $K e^{-r_f \tau}$ today. B holds one put and one share. If the stock ends below the strike, A's call is worthless and the bond pays $K$, while B's put pays $K - S_{\text{exp}}$, which with the share makes $K$. If it ends above, both are worth $S_{\text{exp}}$. In every state both portfolios are worth $\max(S_{\text{exp}}, K)$:

Two things with identical payoffs must have identical prices; otherwise you buy the cheap one, sell the dear one, and keep the difference with nothing at risk — an arbitrage. Ruling that out is the law of one price, and here it gives put-call parity (Stoll 1969):

If the stock pays a continuous dividend yield $q$, start portfolio B with $e^{-q\tau}$ shares, so that the reinvested dividends grow it to one share by expiry. Then

where $F$ is the forward price of Session 2.4 with net carry $c = r_f - q$. For discrete, known dividends, replace $S e^{-q\tau}$ by $S$ minus the present value of the dividends. The forward form says it all: a long call plus a short put at the same strike is a forward contract to buy at $K$, and an at-the-money-forward call ($K = F$) costs exactly as much as the put.

Notice what the argument did not use: a model of how the stock moves, a volatility, anyone's forecast. Parity is model-free. The lab illustrates that claim by pricing each option on its own, by Monte Carlo: 400,000 simulated prices at expiry from a geometric Brownian motion drifting at the risk-free rate (why that drift is Session 10.1's story), each option priced as its average payoff discounted at $r_f$:

S_exp = terminal_gbm(N_PATHS, S0, TAU, R_F, SIGMA, q, rng_parity)
C, P, C_se, P_se, diff_se = mc_prices(S_exp, STRIKES, TAU, R_F)
rhs = S0 * np.exp(-q * TAU) - STRIKES * np.exp(-R_F * TAU)
# The payoff identity behind parity holds path by path, whatever the model:
assert np.allclose(call_payoff(S_exp, 100.0) - put_payoff(S_exp, 100.0), S_exp - 100.0)
# ... so the simulated C − P matches S·e^{−qτ} − K·e^{−rτ} within Monte Carlo error (4 standard errors),
assert (parity.z.abs() < PARITY_TOL_SE).all()

At the 100 strike the simulation prices the call at $4.489 (standard error 0.010) and the put at $3.496, so $C - P =$ 0.9933 against parity's 0.9950 — −0.11 standard errors away. With a 2% dividend yield it is 0.4764 against 0.4963 (−1.26 standard errors); the lab asserts agreement within 4 standard errors at all 7 strikes, with and without dividends (Session 3.5 covers the Monte Carlo standard error). The gap is identical at every strike, which is instructive: path by path, call payoff minus put payoff is $S_{\text{exp}} - K$, so all a simulation can get wrong is the average of $S_{\text{exp}}$ — the forward. Swap the lognormal world for one with crash risk (an 8% chance of a −22% gap over the three months, rescaled so the average $S_{\text{exp}}$ is still the forward) and individual prices move — out-of-the-money puts get dearer (§2.7.6) — but $C - P$ still matches parity, within 0.64 standard errors. The forward is the only property of the distribution that parity uses, and a model that got it wrong could be arbitraged with the stock and a bond alone.

Rearranging parity gives every position a synthetic twin — useful because traders use whichever is cheaper, and because it shows what a structure really is:

2.7.4 When parity breaks: the conversion

If quotes ever violate parity, the replication argument turns into a trade. Whether that trade is an arbitrage for you depends on your costs.

Parity is enforced by traders who check it constantly and trade any gap — the same logic as the cash-and-carry arbitrage of Session 2.4. The loop has five steps:

A worked example from the lab. The market quotes the 3-month 100 call at $5.09, $0.60 above the price parity allows, with the put fairly priced at $3.49. Then $C - P =$ 1.595 but $S - K e^{-r_f \tau} =$ 0.995: the call side is rich. The conversion sells the call, buys the put, buys the share for $100 and borrows $99.00, which leaves $0.60 per share — $60 per contract — in your pocket today. At expiry the short call, long put and share are worth exactly $K = 100$ wherever the stock ends, which repays the loan. Across 10,000 simulated endings the lab finds the same profit every time: $0.606 per share at expiry, the $0.60 plus three months' interest.

Now the costs. You sell at the bid and buy at the ask, so each leg costs half its bid-ask spread, plus commissions. With illustrative half-spreads of $0.05 per option and $0.01 per share, and $0.65 per option contract, the package costs $12.30 per contract and the net profit is $47.70. These costs assume you borrow the $99.00 a share at the risk-free rate, as a dealer can. Every extra percentage point of financing costs $24.78 per contract over three months — enough to erase the $0.25 row below — and two points ($49.63) erase the $0.60 row too. Smaller gaps do not survive even at dealer rates:

2.7.5 Why options exist: structures and views

People buy and sell options for three different reasons: insurance, leverage and volatility. Each has a classic structure, and every structure is a sum of pieces you already know.

Insurance. A shareholder who fears a fall buys a put: a protective put caps the loss at a known amount for a premium, like an insurance policy with a deductible; the put's seller is the insurer. Leverage. A call buys a large exposure for a small premium (§2.7.1). A vertical spread — buy one strike, sell another with the same expiry — cuts the cost by giving up the move beyond the second strike: the bull call spread below buys the 100 call and sells the 110. Volatility. A long straddle — a call and a put on one strike — profits from a big move either way: a bet on the size of the move, not its sign. On the other side sit the sellers; their classic structure is the covered call, shares plus a short call, which collects a premium in exchange for the upside above the strike.

All figures are per share, at expiry, from the lab. The table hides the most important dimension: buying an option pays off when the stock moves more than the option's price implied, selling one when it moves less. So every structure is a view on direction and a view on volatility:

The covered call, honestly

Covered calls are sold as "income", so look at the whole distribution. The lab simulates 200,000 three-month outcomes for the stock, now in the real world: it drifts at 8% a year with 20% volatility — the drift and volatility of Session 2.3's asset, with this session's 4% risk-free rate. You either hold it, or hold it and sell the 105 call at the model price.

The covered call wins more often and swings less, but it earns less on average: it gives away the right tail (the call is exercised in 37% of paths) and keeps nearly all of the left tail — its bad case is the stock's plus the $2.39 premium, exactly. The average shortfall, $0.41, matches the theoretical $0.42: when the stock is expected to beat the risk-free rate, a fairly priced call pays out more on average than its price, so selling it gives up expected return. Here the call is priced at the stock's true volatility, so it carries no volatility risk premium: selling it simply hands the buyer part of the stock's equity premium. Real index options differ, because implied volatility has tended to exceed the volatility that followed (§2.7.6); Israelov and Nielsen (2015) find that index covered calls have historically earned attractive risk-adjusted returns by collecting that volatility risk premium along with the equity premium — together with a risk they argue is not compensated. Session 10.7 — Trading Volatility takes that trade apart.

2.7.6 Implied volatility: the price, quoted in volatility

Option prices cannot be compared directly: options differ in strike, expiry and underlying. Professionals convert every price into one comparable number — its implied volatility — and quote and trade that.

So far the premiums came from a black box, the Black-Scholes-Merton formula, written in the lab as a single function and clearly flagged: Session 10.3 — Black-Scholes-Merton derives it and Session 10.4 — The Greeks & Hedging P&L takes it apart. It turns six inputs into a price: spot, strike, time to expiry, the risk-free rate, the dividend yield and the volatility $\sigma$ of the stock until expiry. You can look up the first five. The sixth cannot be observed in advance, and it matters most: higher volatility means a wider spread of outcomes and, through the hockey stick, more time value (§2.7.2). The price rises with $\sigma$ for every call and every put:

Because price is strictly increasing in $\sigma$, the map can be run backwards: for any market price there is exactly one volatility that makes the model reproduce it. That number is the option's implied volatility. It is not a forecast somebody typed in; it is a unit of quotation, the way a bond price is quoted as a yield. Traders say "the 3-month at-the-money calls are trading at 25 vol" and mean an implied volatility of 25%. Finding it is a one-dimensional root-find, and bisection is the most robust way to do it — the same halving you know from git bisect:

def implied_vol(price, S, K, tau, r_f, q=0.0, kind="call", lo=1e-4, hi=5.0, tol=1e-8):
    """Bisection: the σ at which bs_price(σ) = price. Returns (σ, iterations)."""
    f = lambda s: bs_price(S, K, tau, r_f, s, q, kind) - price   # noqa: E731 — increasing in s
    if not f(lo) <= 0 <= f(hi):
        raise ValueError("price outside the range the model can produce: check no-arbitrage bounds")
    n = 0
    while hi - lo > tol:
        mid = 0.5 * (lo + hi)
        if f(mid) > 0:
            hi = mid          # model price too high → the answer is below mid
        else:
            lo = mid
        n += 1
    return 0.5 * (lo + hi), n

A made-up market quote of $5.25 for the 3-month 100 call ($525 a contract) implies a volatility of 23.88%, found in 29 halvings to a tolerance of one millionth of a volatility point. For a check in your head, use the rule of thumb that follows from Brenner and Subrahmanyam (1988): an at-the-money straddle costs about $0.8\, S \sigma \sqrt{\tau}$ (0.8 ≈ $\sqrt{2/\pi}$. Brenner and Subrahmanyam's formula is for a single option at the money forward, $S = K e^{-r_f \tau}$; a straddle barely notices the difference, a single at-the-money-spot call does, and the rule loosens as $\sigma\sqrt{\tau}$ grows). Here that gives $8.00 against the lab's $7.98, and running it backwards recovers 19.9%. It also says what a straddle price means: roughly the size of move the market is pricing in — a risk-neutral average, not a forecast (see the pitfall below) — here about 8% of the stock price, either way.

Now the payoff of the quote convention. Which is dearer: a 1-month at-the-money call at $2.60, or a 1-year one at $9.90? By price the question makes no sense. By implied volatility it is easy: 21.1% against 19.9%, so the cheap-looking option is the expensive one.

One more experiment, as a teaser. Invert the crash-world prices of §2.7.3 strike by strike. In the lognormal world every strike implies 20%, within 0.03 percentage points. In the crash world no single volatility fits: the 85 put implies 25.3%, the at-the-money option 18.8% and the 115 call 16.6%. The formula assumes lognormal prices with one volatility; when the real distribution has a fatter left tail, the market has to use a different volatility at each strike to get the prices right. S&P 500 index options have shown this downward slope ever since the 1987 crash (Rubinstein 1994): the volatility skew; laid out across strikes and expiries it becomes the volatility surface of Session 10.5 — The Implied Volatility Surface.

2.7.7 Try it

Open the companion lab, predict before you run, and break things on purpose.

2.7.8 Glossary · Check yourself · Go deeper

Check yourself

📚 Go deeper

1. Hull, J. C. (2021). Options, Futures, and Other Derivatives, 11th ed., ch. 10 "Mechanics of options markets", ch. 11 "Properties of stock options" and ch. 12 "Trading strategies involving options". Pearson. ISBN 978-0-13-693997-9
2. Natenberg, S. (2015). Option Volatility and Pricing: Advanced Trading Strategies and Techniques, 2nd ed. McGraw-Hill. ISBN 978-0-07-181877-3
3. Stoll, H. R. (1969). "The Relationship Between Put and Call Option Prices." Journal of Finance 24(5), 801–824. doi:10.1111/j.1540-6261.1969.tb01694.x
4. Black, F. & Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81(3), 637–654. doi:10.1086/260062
5. Merton, R. C. (1973). "Theory of Rational Option Pricing." Bell Journal of Economics and Management Science 4(1), 141–183. doi:10.2307/3003143

Sources for the numbers

1. Ofek, E., Richardson, M. & Whitelaw, R. F. (2004). "Limited Arbitrage and Short Sales Restrictions: Evidence from the Options Markets." Journal of Financial Economics 74(2), 305–342. doi:10.1016/j.jfineco.2003.05.008
2. Israelov, R. & Nielsen, L. N. (2015). "Covered Calls Uncovered." Financial Analysts Journal 71(6), 44–57. doi:10.2469/faj.v71.n6.1
3. Brenner, M. & Subrahmanyam, M. G. (1988). "A Simple Formula to Compute the Implied Standard Deviation." Financial Analysts Journal 44(5), 80–83. doi:10.2469/faj.v44.n5.80
4. Carr, P. & Wu, L. (2009). "Variance Risk Premiums." Review of Financial Studies 22(3), 1311–1341. doi:10.1093/rfs/hhn038
5. Rubinstein, M. (1994). "Implied Binomial Trees." Journal of Finance 49(3), 771–818. doi:10.1111/j.1540-6261.1994.tb00079.x
6. Cboe (2026). "SPX Index Options" fact sheet and S&P 500 Index Options product specifications. https://cdn.cboe.com/resources/spx/spx-fact-sheet.pdf · https://www.cboe.com/tradable_products/sp_500/spx_options/specifications/
7. Options Clearing Corporation. Equity options product specifications (contract size), and Characteristics and Risks of Standardized Options (June 2024). https://www.theocc.com/clearance-and-settlement/clearing/equity-options-product-specifications · https://www.theocc.com/getcontentasset/a151a9ae-d784-4a15-bdeb-23a029f50b70/dfc3d011-8f63-43f6-9ed8-4b444333a1d0/riskstoc.pdf
8. Options Industry Council. "Options Exercise" FAQ (automatic exercise thresholds; American and European style). https://www.optionseducation.org/referencelibrary/faq/options-exercise