Core answer: Black-Scholes prices a European option from five inputs — spot, strike, time to expiry, risk-free rate and volatility — with volatility the only unobservable one; among the Greeks, Delta≈0.5 marks at-the-money and Theta is the buyer’s daily rent. For grid trading, the iron rule is spacing > round-trip fees: 0.1% fees demand at least 0.3–0.5% spacing.
Two tools at a glance
| Tool | Question answered | Key inputs |
|---|---|---|
| Black-Scholes | what is this option worth? | spot, strike, T, r, σ |
| Grid trading | how do I automate range trading? | range, grid count, spacing, per-grid capital |
Black-Scholes inputs and Greeks
Call price C = S·N(d₁) − K·e^(−rT)·N(d₂); put via put-call parity.
| Greek | Measures | ATM reading |
|---|---|---|
| Delta | price change per $1 of spot | call ≈ +0.5, put ≈ −0.5 |
| Gamma | Delta’s own sensitivity | peaks near ATM |
| Vega | price change per 1 vol point | peaks near ATM, grows with T |
| Theta | daily time decay | worst near ATM in final weeks |
| Rho | rate sensitivity | minor at short maturities |
Volatility sensitivity table
ATM call, spot 100, strike 100, T=0.25y, r=2%:
| Implied vol | Call price | Vega per vol point |
|---|---|---|
| 15% | ~3.15 | ~0.19 |
| 25% | ~5.05 | ~0.20 |
| 40% | ~7.95 | ~0.20 |
Price scales almost linearly with volatility — which is why “buying options before earnings” is mostly a bet on IV not collapsing after the announcement.
Grid trading parameters
- Range: historical support/resistance; leave the range and the grid idles (or holds a loss)
- Grid count: 20–50 typical; more grids = smaller per-trade profit
- Per-grid profit: spacing − round-trip fees. 1% spacing at 0.1% fees nets 0.8%
- Capital split: base position (usually 50%+) + grid ammunition
Arithmetic vs geometric grids
| Type | Spacing | Suits |
|---|---|---|
| Arithmetic | fixed price step (every ¥0.10) | narrow ranges, low-priced assets |
| Geometric | fixed ratio step (every 1%) | wide ranges, trending assets |
At ¥10 with 1% spacing: arithmetic steps ¥0.10 everywhere; geometric steps 0.10 at 10 but 0.12 at 12 — keeping percentage returns constant.
Example: pricing a call option
Stock at ¥100, strike ¥105 call, 3 months to expiry, r=2%, σ=25%:
The B-S calculator returns C ≈ ¥3.0 with Delta ≈ 0.40 — the option moves ~¥0.40 per ¥1 of stock, so a 5-lot position behaves like 200 shares of directional exposure. If IV jumps to 35%, C ≈ ¥4.1 — +37% with the stock unchanged: pure volatility exposure.
Example: ETF grid setup
An index ETF ranging ¥0.90–1.10 for a year, ¥50,000 capital:
- Geometric grid, 2% spacing, ~10 levels from 0.90 to 1.10
- Round-trip fees 0.1% → per-grid net ≈ 1.9%
- Base position ¥25,000 at ¥1.00; each grid trades ¥2,500
- Monthly turnover ~6 round trips → grid harvest ≈ ¥285/month on the ammunition portion; the base position tracks the index
Discipline: pause and reassess if price exits the range — never widen the grid to chase it.
Common mistakes
- “An option’s price is intrinsic value”: deep-OTM options are pure time value, and Theta bleeds them fastest near expiry — buyers fight the clock daily.
- Grid spacing below the fee line: 0.2% spacing against 0.1% round-trip fees donates most profits to the broker. Keep spacing ≥3× fees.
- Running grids on trending assets: a strong one-way trend forces continuous buys (holding a falling asset) or early full exits — grids fit mean-reverting, range-bound instruments.
- Ignoring volatility regimes: the same option priced at 15% IV versus 40% IV differs 2.5× — check the IV percentile before judging “cheap”.