5. Options

Algo-wide warnings.

1. Greeks are first-order approximations — valid for small moves; they change as spot/vol/time move (that's what gamma/vanna/charm measure). Re-price for large scenarios.
2. IV is the market's price, not a forecast. Don't feed realized vol into a pricer expecting market prices.
3. American vs European exercise changes pricing/early-exercise (dividends). Black–Scholes is European.
4. Quotes: use mid-price, check open interest & spread for liquidity; stale/wide quotes wreck options backtests.

Notation: S = spot, K = strike, T = time to expiry (years), r = rate, q = dividend yield, σ = implied vol.

Core Concepts

Call / Put

Def. Call = right to buy at K; Put = right to sell at K (by expiry). Formula. Payoff at expiry: call $\max (S-K,0)$, put $\max (K-S,0)$, minus premium paid. Signal. Long call = bullish/long vol; long put = bearish/hedge. Algo. Buyer's loss capped at premium; writer's risk can be unbounded (naked call) — model margin & assignment.

Moneyness

Def. Strike position relative to spot. Formula. ITM / ATM / OTM. Quant moneyness: $\ln (S/K)$ or $K/S$, "delta-moneyness" uses Δ. Signal. ATM has max gamma/theta; OTM is cheap, low-probability, high-leverage. Algo. Build vol surfaces in delta- or log-moneyness space (stable across spot levels), not raw strikes.

Intrinsic vs Time Value

Def. Intrinsic = in-the-money amount; time value = premium beyond intrinsic. Formula. $\text{Premium}=\text{Intrinsic}+\text{Time Value}$, ${\text{Intrinsic}}_{\text{call}}=\max (S-K,0)$. Signal. Time value decays to 0 at expiry (theta); all premium on OTM options is time value. Algo. Time value is what theta erodes; track it to reason about decay P&L.

Put–Call Parity

Def. No-arbitrage link between call, put, spot, and a bond. Formula. $C-P=S-K{e}^{-rT}$ (European, no divs; subtract PV of divs if any). Signal. Lets you synthesize positions (synthetic long stock = long call + short put). Algo. Violations = arb (rare, fleeting); useful sanity check on a quoted chain / to back out implied dividends or borrow cost.

The Greeks

Delta (Δ)

Def. ∂Price/∂Spot — sensitivity to underlying move; also ~ risk-neutral prob of finishing ITM. Formula. Call Δ $=e^{-qT}N(d_1)$ ∈ (0,1); put Δ = call Δ − e^{−qT} ∈ (−1,0). Signal. Δ 0.50 ≈ ATM. Position delta = portfolio's stock-equivalent exposure. Algo. Delta-hedge by holding $-\Delta$ shares per option. Δ itself moves (gamma) → rehedge.

Gamma (Γ)

Def. ∂Δ/∂Spot — how fast delta changes; curvature. Formula. $\Gamma =e^{-qT}\phi (d_1)/(S\sigma \sqrt{T})$ (same for calls/puts). Signal. Peaks ATM and near expiry. High gamma → delta unstable → frequent rehedging. Algo. Long gamma profits from realized moves (you buy low/sell high rehedging) but pays theta; short gamma is the opposite (the classic "picking up pennies" risk).

Theta (Θ)

Def. ∂Price/∂Time — time decay per day (usually negative for longs). Formula. Closed-form BS expression; commonly quoted per calendar day = annual θ / 365. Signal. Decay accelerates as expiry nears (esp. ATM). Option sellers harvest theta. Algo. Long options bleed theta daily; the long-gamma/short-theta tradeoff is the core of vol trading. Note calendar vs trading-day convention.

Vega (ν)

Def. ∂Price/∂IV — sensitivity to a 1-vol-point change in implied vol. Formula. $\nu =S{e}^{-qT}\phi (d_1)\sqrt{T}$ (per 1.00 of σ; divide by 100 for "per 1%"). Signal. Highest for ATM, longer-dated options. Long options = long vega (profit if IV rises). Algo. Not a true Greek (vol isn't a model parameter in BS) but essential. Vega risk dominates long-dated positions.

Rho (ρ)

Def. ∂Price/∂Rate. Formula. Call $\rho =KT{e}^{-rT}N(d_2)$. Signal. Matters mainly for long-dated options and in high/volatile-rate regimes. Algo. Usually negligible for short-dated; don't ignore on LEAPS.

Second-order (Vanna, Charm, Vomma)

Def. Vanna = ∂Δ/∂vol; Charm = ∂Δ/∂time; Vomma = ∂Vega/∂vol. Signal. Drive hedging flows; "charm/vanna flows" cited around large-OI expiries (OPEX). Algo. Relevant for dynamic hedging and dealer-positioning models; ignore for simple strategies.

Volatility

Implied Volatility (IV)

Def. The σ that makes the BS price equal the market price. Formula. Solve $\text{BS}(\sigma )=\text{market price}$ numerically (Newton/bisection). Signal. Market's expected vol; rises into events/fear (VIX is index-level IV). Algo. No closed form — invert per option. Watch for no-solution cases (price outside arb bounds).

IV Rank / IV Percentile

Def. Where current IV sits in its own trailing range. Formula. $\text{IV Rank}=(\text{IV}-{\text{IV}}_{\min })/({\text{IV}}_{\max }-{\text{IV}}_{\min })$ over ~1yr; IV %ile = fraction of days below current IV. Signal. High rank → options "expensive" → favor selling premium; low → favor buying. Algo. Normalizes IV across assets/regimes; percentile is more robust to outliers than rank.

Volatility Skew / Smile

Def. IV varies by strike; equities show higher IV for low strikes (crash fear). Formula. Plot IV vs strike/delta; "25-delta skew" = IV(25Δ put) − IV(25Δ call). Signal. Steep put skew → demand for downside protection / bearish positioning. Algo. Never assume flat vol across strikes — interpolate a surface (in delta space) for accurate Greeks.

Term Structure

Def. IV across expiries. Formula. Plot ATM IV vs T. Usually upward (contango); inverts before events. Signal. Backwardation (front > back) → near-term stress/event premium. Algo. Calendar spreads trade term structure; VIX futures curve is the index analog.

VIX

Def. 30-day implied vol of S&P 500 options ("fear gauge"). Formula. Model-free variance swap calculation over the SPX option strip. Signal. Spikes in risk-off; mean-reverting. VIX >30 = stress, <15 = calm. Algo. You can't trade VIX directly — only futures/options/ETPs, which carry roll cost (VIX futures usually in contango → VXX bleeds).

Pricing

Black–Scholes–Merton

Def. Closed-form European option price under lognormal spot, constant vol/rate. Formula. $C=S{e}^{-qT}N(d_1)-K{e}^{-rT}N(d_2)$, $d_1=\frac{\ln (S/K)+(r-q+{\sigma }^2/2)T}{\sigma \sqrt{T}}$, $d_2=d_1-\sigma \sqrt{T}$. Signal. Benchmark fair value & Greeks source. Algo. Assumes constant vol (false — hence skew), no early exercise, lognormal returns (fat tails ignored). Good baseline, not reality.

Binomial / Trinomial Trees

Def. Discrete lattice pricing; handles American exercise & dividends. Formula. Build up/down spot tree, backward-induct payoffs with risk-neutral probs. Signal. Use when early exercise matters (American options, dividend stocks). Algo. More nodes → more accurate but slower; converges to BS for European as steps→∞.

Monte Carlo

Def. Simulate many price paths, average discounted payoff. Formula. Simulate $S_{t+dt}=S_t\exp ((r-q-{\sigma }^2/2)dt+\sigma \sqrt{dt}Z)$, price = $e^{-rT}\text{mean}(\text{payoff})$. Signal. Needed for path-dependent/exotic (Asian, barrier) options. Algo. Slow & noisy (error ~1/√N); use variance reduction (antithetic, control variates). Overkill for vanillas.

Common Strategies (payoff shapes)

Algo for spreads: model both legs together (net Greeks, net margin), include commissions per leg and bid-ask on each leg — multi-leg slippage compounds fast.