Desk reference
print it · pin it next to the tape
The formula sheet
Everything quantitative from the course on one card: payoffs, parity, Black–Scholes,
the Greeks with their exact conventions, the rules of thumb with their derivations, the P&L identity,
the strategy grid, surface vocabulary, risk one-liners, cross-asset conventions and the desk
dialect with its morning screen. Every number here
reproduces what the course engine (OL.math.bs) computes.
House conventions — read once, they apply everywhere
- Defaults: ACME at S = $100, r = 0, no dividends unless stated. T is in years; σ is annualized (σ = 0.25 = “25 vol”). Prices are per share; one listed contract = 100 shares, so multiply P&L by 100.
- Theta is per calendar day (annual rate ÷ 365 — rent accrues all 365 nights). Vega is per vol point (÷ 100: IV 25 → 26). Rho per rate point. Vanna per vol point; volga per vol point²; charm per day.
- Day counts: realized vol, the rule of 16 and breakeven moves use 252 trading days; theta accrual uses 365. Rent per trading day = |θ| × 365/252.
1 · Payoffs
2 · Parity
3 · Black–Scholes & Greeks
4 · Rules of thumb
5 · P&L identity
6 · Strategy cards
7 · Surface vocabulary
8 · Risk
9 · Cross-asset
10 · Desk dialect
1Payoff, P&L and the four positions
call payoff = max(ST − K, 0)
put payoff = max(K − ST, 0)
ST = stock at expiry · K = strike. Payoff is gross;
P&L is net of premium: long P&L = payoff − premium, short P&L = premium − payoff.
Per share; × 100 per contract.
| Position | P&L at expiry | Breakeven | Max profit | Max loss | You want |
|---|---|---|---|---|---|
| Long call | max(ST − K, 0) − prem | K + prem | unlimited | premium | a rally past breakeven |
| Short call | prem − max(ST − K, 0) | K + prem | premium | unlimited | ST ≤ K (expires worthless) |
| Long put | max(K − ST, 0) − prem | K − prem | K − prem (stock → 0) | premium | a fall past breakeven |
| Short put | prem − max(K − ST, 0) | K − prem | premium | K − prem (stock → 0) | ST ≥ K |
Short positions are the long's mirror across the zero line — same breakeven, profit and loss regions swapped. Long options: loss capped at premium. Short options: gain capped at premium.
2Put–call parity & synthetics
C − P = S − K·e−rT
European call and put, same strike K, same expiry T. Model-free — no
volatility, no formula, just no-arbitrage. With dividend/carry yield q: C − P = S·e−qT − K·e−rT.
House default r = 0: C − P = S − K.
| You want | Build it from | Reading |
|---|---|---|
| Synthetic long stock (forward) | +C −P (same K, T) | committed to buy at K either way — a straight line |
| Synthetic short stock | −C +P | the mirror |
| Synthetic call | +P + stock | protective put ≡ call + K bond — insurance is a call |
| Synthetic put | +C − stock | call − stock ≡ put − K bond: a put's shape, financed by borrowing K |
| Covered call (stock − call) | ≡ short put + K bond | “income” = a short put's premium |
| Protective put (stock + put) | ≡ long call + K bond | same hockey stick, same price to bear |
Desk view
If C − P ≠ S − K·e−rT, buy the cheap side, sell the rich side, lock the difference (conversions / reversals). Market makers harvest these in microseconds — retail never sees the free money.
3Black–Scholes & the Greeks
d₁ = ln(S/K) + (r + ½σ²)·Tσ·√T
d₂ = d₁ − σ·√T
S = spot · K = strike · T = years to expiry · σ = annualized vol · r = rate ·
N(·) = standard normal CDF · φ(·) = its density
C = S·N(d₁) − K·e−rT·N(d₂)
P = K·e−rT·N(−d₂) − S·N(−d₁)
Read right to left: (what you might pay, probability-weighted) subtracted from
(what you might receive). N(d₂) = risk-neutral P(ST > K) — the probability of exercise.
N(d₁) = the call's delta — P(ITM) tilted up, because you receive the stock precisely in the states
where it finished high. They are not the same number (interview classic).
| Greek | Measures | Call | Put | Course/engine units | Where it lives |
|---|---|---|---|---|---|
| Delta Δ | ∂V/∂S | N(d₁) | N(d₁) − 1 | per $1 of spot | S-curve in spot; ATM ≈ 0.5; sharpens near expiry |
| Gamma Γ | ∂Δ/∂S | φ(d₁) / (S·σ·√T) | per $1 of spot | bell at ATM; explodes short-dated ATM | |
| Theta θ | ∂V/∂t | −S·φ(d₁)·σ2√T − r·K·e−rT·N(d₂) | −S·φ(d₁)·σ2√T + r·K·e−rT·N(−d₂) | ÷ 365 → per calendar day | gamma's negative image; ATM, accelerating into expiry |
| Vega | ∂V/∂σ | S·φ(d₁)·√T | ÷ 100 → per vol point | ATM; grows ~√T — long-dated options are vol instruments | |
| Rho ρ | ∂V/∂r | K·T·e−rT·N(d₂) | −K·T·e−rT·N(−d₂) | ÷ 100 → per rate point | long-dated, ITM |
| Vanna | ∂Δ/∂σ = ∂vega/∂S | −φ(d₁)·d₂ / σ | ÷ 100 → per vol point | ≈ 0 ATM, peaks in the wings — OTM put deltas inflate as vol rises | |
| Volga | ∂vega/∂σ | S·φ(d₁)·√T·d₁·d₂ / σ (= vega × d₁d₂/σ) | ÷ 100² → per vol point² | ≈ 0 ATM; wings love vol-of-vol | |
| Charm | ∂Δ/∂t | φ(d₁)·d₂ / (2T) (r = 0 form; call = put) | ÷ 365 → per day | ≈ 0 ATM; wings — passing time melts deltas toward 0 (OTM) or ±1 (ITM) | |
Sign & unit traps
- Long any option: +Γ, +vega, −θ. Short: the mirror. There is no structure with +Γ and +θ at the same spot and time — convexity is rented, never free.
- A theta chip of −$0.048 means the option sheds 4.8¢ per calendar day — weekends included. Desks think in trading days: rent per trading day = |θ| × 365/252.
- Vega $0.114 means +$0.114 if IV goes 25 → 26 (one point), not 25% → 50%.
The house card — verify your calculator against it
| ACME $100 · 30d · 25 vol · r = 0 | Price | Δ | Γ | θ / day | Vega / pt | Notes |
|---|---|---|---|---|---|---|
| 100 call (ATM) | $2.86 | 0.51 | 0.056 | −$0.048 | $0.114 | d₁ = 0.036, d₂ = −0.036, N(d₂) = P(ITM) = 0.49 |
| 100 put (ATM) | $2.86 | −0.49 | 0.056 | −$0.048 | $0.114 | same price as the call at r = 0 — parity |
| 90 put (wing) | $0.21 | −0.066 | 0.018 | −$0.015 | $0.037 | vanna −0.0074/pt (delta inflates as vol rises) · charm +0.0031/day (delta melts toward 0) · P(ITM) 7.6% |
4Rules of thumb — with their one-line derivations
| Rule | Statement | One-line derivation |
|---|---|---|
| Rule of 16 | typical daily move ≈ IV / 16. IV 32 ≈ 2%/day · VIX 16 ≈ 1%/day · VIX 80 ≈ 5%/day |
variances add over independent days, so σdaily = σannual/√252, and √252 ≈ 15.9 ≈ 16. |
| ATM option price | ≈ 0.4 · S · σ · √T. $100, 25 vol, 30d → 0.4 × 100 × 0.25 × √(30/365) ≈ $2.87 (exact BS: $2.86) |
exact ATM (r = 0): C = S[2N(½σ√T) − 1] ≈ S·φ(0)·σ√T, and φ(0) = 1/√(2π) = 0.399 ≈ 0.4. |
| ATM straddle | ≈ 0.8 · S · σ · √T — the market's expected absolute move. house example: ≈ $5.73 (BS: $5.72) |
call + put = 2 × 0.4·S·σ√T; equivalently E|move| of a normal = √(2/π)·σ√T·S ≈ 0.80·σ√T·S. |
| ATM vega | ≈ 0.4 · S · √T per 100 vol points — ÷ 100 for per-point. $100, 30d → $0.115/pt (BS: $0.114) |
differentiate price ≈ 0.4·S·σ·√T with respect to σ — the σ drops out, the rest stays. |
| Breakeven daily move | ≈ S · σimp / 16 per trading day. 25 IV on $100 → ±$1.56 pays the day's rent |
set gamma gain = theta rent: ½Γ(ΔS)² = ½ΓS²σ²Δt → ΔS = S·σ·√Δt = S·σ/√252 ≈ S·σ/16. |
| Implied move | ≈ ATM straddle price / S (to that expiry, or for an event). | invert the straddle rule: the straddle is the price of the expected move, so straddle/S ≈ 0.8·σ√T reads it back out. |
These approximations sit within ~1% of the exact Black–Scholes values for at-the-money options at everyday vols and tenors — they degrade far ITM/OTM and at extreme σ√T.
5The daily P&L identity
daily P&L ≈ ½ Γ S² [ (ΔS/S)² − σimp²·Δt ]
a delta-hedged option book, one day at a time — the single line the whole vol
business runs on. Long gamma: you win the days the bracket is positive. Short gamma: the reverse.
| Term | Name | Meaning |
|---|---|---|
| ½ Γ S² | dollar gamma | your stake on today's variance coin — the house ATM call's is ½ × 0.0556 × 100² ≈ $278 |
| (ΔS/S)² | realized variance | the day's move that actually happened, squared — a 2% day contributes 0.0004 |
| σimp²·Δt | implied variance budget | the day's rent, set by the vol you traded at; Δt = 1/252 in trading time (25 vol → 0.000248) |
| Σ over days | the vol trade | a strip of daily variance bets, each weighted by that day's dollar gamma — which is why the same realized vol can pay differently on different paths |
Worked day (house ATM call, 25 IV): a 2% day earns 278 × (0.0004 − 0.000248) ≈ +$0.04/share ≈ +$4 per contract; a flat day bleeds the full rent, 278 × 0.000248 ≈ −$0.07/share — exactly |θ| × 365/252. The move that balances the two is S·σ/16 = $1.56: rule 5 above, rediscovered.
full P&L ≈ Δ·ΔS + ½Γ·(ΔS)² + θ·Δt + vega·Δσ + residual
the un-hedged version — the desk's morning attribution report. Small residual =
healthy book; large residual = you don't know your risk.
6Strategy quick cards
Per share (× 100 per contract). D = net debit paid, Cr = net credit received, w = distance between adjacent strikes (for the condor: the wing width, K₂ − K₁ = K₄ − K₃), S₀ = stock price at entry. Greek signs are at inception, near the money; they migrate as spot and time move.
| Structure | Legs | View | Greeks | Max P&L | Breakeven(s) |
|---|---|---|---|---|---|
| Covered call | +stock −C(K), K > S₀ | mildly bullish; rich IV; income with upside sold away | +Δ −Γ +θ −vega | +: (K − S₀) + Cr −: S₀ − Cr (stock → 0) | S₀ − Cr ≡ short put (parity) |
| Cash-secured put | −P(K) + cash for K | neutral-bullish; happy to own stock lower | +Δ −Γ +θ −vega | +: Cr −: K − Cr (stock → 0) | K − Cr |
| Bull call spread | +C(K₁) −C(K₂), K₁ < K₂ | moderately bullish, target near K₂ | +Δ; Γ, θ ≈ flat — long near K₁, short near K₂ | +: w − D −: D | K₁ + D credit twin: −P(K₂) +P(K₁), BE K₂ − Cr |
| Bear put spread | +P(K₂) −P(K₁), K₁ < K₂ | moderately bearish, target near K₁ | −Δ; Γ, θ ≈ flat — long near K₂, short near K₁ | +: w − D −: D | K₂ − D credit twin: −C(K₁) +C(K₂), BE K₁ + Cr |
| Long straddle | +C(K) +P(K), K ≈ ATM | big move, direction unknown; long vol into events | ≈0Δ +Γ +vega −θ (double rent) | +: unlimited (K − D down) −: D | K ± D price ÷ S = the implied move |
| Long strangle | +C(Kc) +P(Kp), Kp < S₀ < Kc | same, cheaper — needs a bigger move | ≈0Δ +Γ +vega −θ | +: unlimited −: D (between strikes) | Kc + D / Kp − D |
| Long butterfly | +C(K−w) −2C(K) +C(K+w) | pinned at K; sell the distribution's center, cheap | near K into expiry: −Γ +θ; ≈ −vega | +: w − D (at ST = K) −: D | K − w + D / K + w − D tent inflates only near expiry |
| Iron condor | +P(K₁) −P(K₂) −C(K₃) +C(K₄) | range-bound between K₂ and K₃; collect rent, wear a helmet | ≈0Δ −Γ +θ −vega | +: Cr −: w − Cr | K₂ − Cr / K₃ + Cr |
| Calendar | −Cfront(K) +Cback(K) | pinned near K into front expiry; front IV rich vs back | ≈0Δ −Γ +θ +vega (back-month) | +: greatest with S at K at front expiry −: D | no closed form — depends on back-month value |
| Collar | +stock +P(Kp) −C(Kc) | keep the stock, insure the downside, sell the upside to pay for it | +Δ (reduced); tails clipped both sides | +: (Kc − S₀) − D −: (S₀ − Kp) + D | S₀ + D “zero-cost”: strikes set so D ≈ 0 |
| Risk reversal | +C(Kc) −P(Kp) | bullish; finances calls by selling rich put-skew | ++Δ; long upside tail, short downside tail; +vanna/skew | +: unlimited above Kc −: Kp + D (or − Cr) at stock → 0 | zero-cost: flat between strikes, wins above Kc, owns the downside below Kp |
Structure-picking order (m07s06): (1) direction? (2) my vol view vs implied? (3) horizon / event inside? (4) defined risk budget? — then choose the shape. Premiums in the course are always Black–Scholes prices at a stated IV, never invented.
7Vol-surface vocabulary
| Term | What it is |
|---|---|
| Smile | IV plotted across strikes at one expiry. Its existence is the market correcting Black–Scholes' thin tails — one σ per strike, not per stock. |
| Skew / smirk | the smile's tilt. Equity indexes: OTM puts over OTM calls (crash-insurance demand + leverage effect). Desk-speak: “the 90s trade 7 over” = 90-strike IV is 7 vol points above ATM. |
| 25Δ risk reversal | RR = IV(25Δ call) − IV(25Δ put) — the skew ruler, quoted at the standard ±25-delta wing addresses. Equity RR is usually negative (puts over). Moneyness-normalized, so it compares across assets. |
| 25Δ butterfly | BF = ½[IV(25Δ call) + IV(25Δ put)] − IVATM — wing convexity: how much the smile curls above its tilt; the price of both tails. |
| Term structure | ATM IV across expiries. Contango when calm; inverts in stress (front end explodes, long end barely moves). Compare expiries in variance, never raw IV. |
| Forward vol | σ²fwd = (σ₂²T₂ − σ₁²T₁) / (T₂ − T₁) — the vol the curve implies between two future dates. 1m at 20, 2m at 25 → the second month is priced at ≈ 29. Variances add; vols don't. |
| Event vol | event variance ≈ (short-expiry total variance) − (baseline variance for the other days); implied event move ≈ that expiry's ATM straddle ÷ S. |
| Sticky strike | each fixed strike keeps its IV as spot moves — pinned/range regimes; your BS delta is roughly honest. |
| Sticky delta | the smile rides with moneyness — ATM vol travels with spot; trending regimes. Which regime holds is empirical and messy. |
| Smile-adjusted delta | Δtrue = ΔBS + vega · (∂σ/∂S). With equity skew (∂σ/∂S < 0), true call deltas sit below BS deltas — “what's your delta assuming the surface?” is the desk interview classic. |
| VRP | variance risk premium = implied − subsequently realized. Persistently positive on average (the insurance margin hedgers willingly pay); inverts violently in crises. It is the “income” in every income strategy. |
| VIX | 30-day SPX variance priced from a strip of OTM options (1/K² weights), reported as vol. VIX/16 ≈ implied daily move. Under skew the variance strike sits above ATM IV (the wings count). |
| Implied correlation | ρimp ≈ (σindex / σ̄single)². Dispersion = short index vol vs long single-name vol = short correlation. In crashes ρ → 1 — index-put buyers get paid twice. |
8Risk quick-list
| Tool | One-liner |
|---|---|
| VaR (95%, 1d) | the loss exceeded on ~1 day in 20. A fence, not a worst case — it says nothing about the size of what lies beyond it. |
| Expected shortfall | the average loss across the worst 5% of days — the tail's mean; answers exactly what VaR refuses to. |
| Scenario grid | P&L across spot × vol shocks. The killer cell is spot down, vol up. Desks cap every Greek and the worst cell. |
| Stress tests | named nightmares — 1987 repeat, rates +200bp, corr → 1 — imagination as a risk tool, because VaR is calibrated to yesterday's world. |
| Sizing | the master control. Kelly-lite: optimal fraction ∝ edge/variance; pros run fractional Kelly because edges are estimated and tails are fat. Ruin risk outranks expected value. |
| Path | path kills before destination: margin and liquidity are paid along the way; being right at expiry is no defense in between. |
The blowup genome
Every canonical disaster shares three genes:
- Leverage — the path is amplified until it is lethal;
- Short liquidity — in stress, you are the one paying for exits;
- Size — too big to hold through the path.
One sentence: “short liquidity, in size they couldn't hold through the path.”
Same genome: 1987 portfolio insurance · LTCM 1998 · Volmageddon/XIV 2018 · OptionSellers 2018 · Allianz Structured Alpha 2020. Audit any “income” plan against all three genes before sizing it.
9Cross-asset conventions
| Asset | Underlying | Quoting | Smile shape | Watch out |
|---|---|---|---|---|
| Equity / index | shares, indexes, ETFs | strikes in $ (desks: % of spot or forward, “95%F”); IV in % | left smirk — puts over calls | index skew steeper than single names; dividends & borrow shift the forward |
| FX | a currency pair — every put is a call on the other side (EUR put = USD call) | delta-quoted smile: ATM (DNS) + 25Δ RR + 25Δ BF; strikes recovered from deltas | smile-ish; RR sign and size flip by pair and regime | two rates (rdom − rfor) drive the forward; premium currency conventions differ |
| Rates | forward swap rates (swaptions); caps/floors = strips of caplets | normal vol in bps/yr — survived zero and negative rates; cube = expiry × tenor × strike | quoted vs strike offsets (±100bp); shape varies by cycle | rule of 16 still works: 100bp/yr normal vol ≈ 6bp typical daily move |
| Commodities | options on futures, month by month | IV %, strikes on the future | often reversed — upside/call skew in energy (spike fear) | seasonality; Samuelson effect (front vol > back); storage & squeeze quirks |
| Crypto | coin-settled “inverse” contracts (premium & P&L in BTC/ETH) or stable-quoted | strikes in $; 24/7/365 calendar — theta never takes a weekend | fat both tails; RR flips sign with the regime | funding/perp basis leaks into forwards; venue margin rules differ wildly |
The invariants
Across every asset class the same physics holds: no-arbitrage and replication set prices; vol is the native unit; the smile points at the feared tail; the Greeks mean the same thing everywhere. Only the dialect changes.
10Desk dialect & practical screens
How strategists actually write a trade (m11s01) — so a line like “buy the 110–125% call spread vs 80% put, indic zero premium, 34d” reads natively. Examples use the house card: ACME $100 · 30d · 25 vol · r = 0.
| Convention | How it reads | House example |
|---|---|---|
| Strikes as % of spot / forward | “110%” = strike at 110% of spot — $110 on ACME, $55 on a $50 stock, never a dollar figure. “95%F” = 95% of the forward; ATMF = at-the-money-forward. Comparable across names; forward ≈ spot at r ≈ 0. | “the 90% put” = the 90-strike wing put |
| Vols as “v” | IV quoted in vol points: “25v”. Spreads and richness in points too: “+2v” = 2 vol points over the comparator (another expiry, realized, or history). | ATM 25v; “1m +2v to 3m” → 1m 25, 3m 23 (inverted) |
| Premium in bps of spot | bps = 10000 × premium / spot — 100bps = 1% of the underlying, instantly sizeable against a portfolio. | ATM call $2.86 on $100 → 286bps |
| Payout multiple | max payout ÷ net premium — “2.8x net”. Max-case arithmetic, not probability-weighted. | 100/105 call spread: $5 wide for $1.78 → 2.8x |
| Package delta | net delta of the whole structure, quoted in delta points: “25d” = 0.25 per share of the package. | that call spread: 0.51 − 0.26 → 25d |
| Structure shorthand | CS call spread · PS put spread · CSvP call spread vs put (“vs” = funded by selling) · collar · RR risk reversal · fly butterfly. “Zero-cost” = net premium ≈ 0 — the short leg is the cost. | “110–125% CSvP 80%” = three legs, one line |
| Implied move | ≈ ATM straddle ÷ spot (rule 6, §4) — the market's expected absolute move to that expiry, in %. | $5.72 ÷ $100 → 5.7% over 30d |
The morning screen — six lines before the open
- IV %ile vs 3y — where does today's ATM IV sit in its own 3-year range: rich, cheap or mid?
- IV − RV spread — implied minus realized, the carry (§4, §5): who is paying whom to hold gamma?
- Skew by wing %ile — 25Δ RR and wing vols vs their own history (§7): which tail is bid today?
- Term structure vs events — contango or inverted (§7)? does every known event date carry its vol bump?
- OI walls — the big open-interest strikes near spot: pin candidates and hedging-flow magnets.
- Cross-index vol spreads — index vs index, single-name vs index (implied correlation, §7): what is rich against what?