Options From First Principles

IntroductionThe posting, and the method

Walleye Capital is hiring a Quantitative Trader for its Index Volatility team in New York, at a stated range of $180,000 to $200,000, with applications through the firm’s own site by 1 January 2027.

The posting asks for someone who can build alphas, and it asks applicants to read its questions carefully. This chapter takes that as an instruction about method: say exactly what was done, show how each number was produced, and mark everything that was not done. It follows the five steps of chapter 12: questions raised, analysis, recommendations, actions and follow-through.

The posting text is held by the author; this page paraphrases it and does not quote it.

Section 1 · The seatQuestions raised

Three questions an index-volatility desk would put to an applicant with this background.

1. Can the applicant price an option and its risks without a library? Section 2 answers this on one synthetic option, with every figure recomputed by script.
2. Can the applicant turn an idea into a testable signal and say plainly what it shows? The evidence is a network-game paper and three WorldQuant Challenge levels; Section 2 reports what each supports and what it does not.
3. What is missing? Market data, live risk, execution, and any record of trading real money. The Limits section lists them.

Section 2 · Mathematics and recordAnalysis

One synthetic option, one identity, one round trip, and a record read against its own weaknesses.

The option and its numbers

Invented inputs: spot 100, strike 100, 0.25 years to expiry, rate 4% a year, no dividend, volatility 20% a year. Black–Scholes gives:

Three checks that do not depend on the formula’s derivation

• Put–call parity. Call minus put is 0.9950, and spot minus discounted strike is 0.9950.
• Finite differences. Bumping spot and volatility by 0.001 and re-pricing reproduces delta (0.55962), gamma (0.03945) and vega (19.724) to about five significant figures.
• The Black–Scholes equation. Theta + ½σ²S²·gamma + r·S·delta − r·V comes to zero (residual below 1e-12). This is the identity behind a delta-hedged volatility position: the time decay paid is the gamma earned if realized variance equals the variance priced.

Implied volatility round trip

Given the price 4.4852 and the other inputs, bisection returns 0.2000, and the same from the put price. A small three-strike table with invented implied volatilities (90 at 24%, 100 at 20%, 110 at 18%) is fitted exactly by a parabola in log-moneyness; it gives 21.71% at strike 95 and 18.78% at strike 105. Three points and three coefficients mean the fit is exact by construction, so it tests nothing. It shows the mechanics, not a model of any real surface.

A parity flag, for example: if the put in this example were quoted at 3.60, it would sit 0.1098 above parity. That is a flag to investigate, not an arbitrage: borrow cost, dividends, bid–ask width and early exercise on American options each explain gaps like it.

The network-game paper, read against itself

The author’s paper on positional dominance in network games (a two-player model on an oil midstream network with a two-state volatility chain) derives a volatility threshold above which the player holding the intermediate node dominates. The existence argument uses the intermediate value theorem under two stated conditions; uniqueness is proved in a restricted case. These are results about a stylised model, and the model has not yet been tested on market data.

The author holds three certificates from the WorldQuant Challenge, each issued “in recognition for reaching” a level: Bronze, Silver and Gold. They certify reaching a level, not winning a contest. WorldQuant’s consultant page describes the levels as steps earned by accumulating points from submitted alphas. Levels are earned on the platform’s own simulator, not in live trading, and nothing here converts a level into a performance figure.

The three certificates carry no date. The author’s screenshots of the level notifications show May 15 (Bronze), May 31 (Silver) and June 22 (Gold); the screenshots show no year, and the author states that all three fall in 2026. From May 15 to June 22 is 38 days, a little over five weeks. The month and day were read from the screenshots by an assistant in another session and have not been checked against the emails themselves; the year is the author’s statement.

Section 3 · What would strengthen the fileRecommendations

A plan, not something done.

• Public record: read the firm’s public regulatory filings and state only what they show.
• Data test: fetch a public daily oil price series and test the model’s threshold under two written definitions of the volatility state, reporting the spread.
• Literature: place the work against the theory-of-storage and commodity-network literature.

Section 4 · Next stepsActions

In order.

1. Run this chapter’s script and confirm every check passes.
2. Run the oil-price test on public data and report the result under both definitions, whichever way it falls.
3. Extend the option section from one synthetic option to a small synthetic surface with a stated, falsifiable test.

Section 5 · MeasuresFollow-through

How the work could be shown to have held.

• Reproducibility: every figure on this page comes from ch14-verify.py, which uses only the standard library.
• Controls: the script runs each check on a wrong input and requires it to fail.
• Sensitivity: any result from real data is reported under at least two definitions.
• Review: a second reader re-derives the headline numbers before the page is sent anywhere.

LimitsWhat this does not show

• Any trading, market-making or risk-management experience. The author has none to claim.
• Any result on market data. Every quote is invented; the page proves the mathematics, not an edge.
• The firm’s strategy, systems or people. Nothing is known beyond the posting.
• The network-game model on real data. It has not been tested on market prices.
• American exercise, dividends, skew dynamics and index-specific features. Not covered.

UpdatesWhat changed after the first version