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SMQLearn / Derivatives Pricing / Binomial Trees / The Binomial ModelSearch lessons⌘K

Lesson 1 · Intermediate · 18 min

The Binomial Model

✓Before this: Market Model and Risk-Neutral Valuation

One-step binomial model and replication

Over one period the stock moves $S_{0} \to S_{0} u$ or $S_{0} \to S_{0} d$ with $0 < d < e^{r Δ t} < u$ . Replicating the option with $Δ$ shares and cash forces

Δ = \frac{V ^{u} - V ^{d}}{S _{0} ( u - d )}, V_{0} = e^{- r Δ t} [q V^{u} + (1 - q) V^{d}], q = \frac{e ^{r Δ t} - d}{u - d} .

Here $V^{u}, V^{d}$ are the option’s known values in the two successor states, and $q$ is the risk-neutral probability: it is a genuine probability exactly when $d < e^{r Δ t} < u$ , which is forced by no-arbitrage. Real-world probabilities never enter.

Financial interpretation. The hedge ratio Δ is the discrete ancestor of $\partial V / \partial S$ . Pricing by replication and by discounted risk-neutral expectation give the same number because the price is the cost of the hedge.

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Programme · 35 lessonsDerivatives Pricing and Computational Finance
Programme · 62 lessonsFixed Income and Interest-Rate Models
Programme · 47 lessonsAdvanced Derivatives and Exotic Options
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Interview questions · 311 questionsOptions, Greeks and Black–Scholes

Your SelfMadeQuant programmeDerivatives Pricing and Model QuantFor: Front-office derivatives quant

01Close the gapsCalculus, probability and Python, only where the diagnostics say soLessons
02Price it four waysTrees, PDEs, Monte Carlo and Fourier on the same contractLessons
03Validate someone else’s modelCapstone IV, with a validation report and change-control logCapstone
04Defend it in the roomGreeks and no-arbitrage questions with full answersInterview

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Each pathway ends in capstones on real market data: a dashboard, a notebook that runs, tested code and a decision memo. This one replays 69 sessions of the Coinbase BTC-USD order book.

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SMQProjects / Applied Research Capstones / MM — Inventory-Aware Market-Making ResearchCapstone

MM inventory-aware market making dashboard

The evaluation at a glance, then the operator page for the latest replayed session.

Evaluation · 2023-01 to 2026-09▾

NoteDo not quote at the assumed zero maker fee: every policy lost on average under every fill bound; the fee tier decides the business.

Desk decision, evaluationPause in 44 of 45monthly rule on the previous 12 sessions

Always join the best bid/ask−261.1USD / sessionloss limit hit in 33 of 45

Least-losing policy (as_g0.1)−6.4USD / session341 fills a session

Adverse selection ÷ spread capture1.6–2.7×60-s mark-out against spread earned

Join with a 1 bp maker rebate+109.4USD / sessionfirst-order fee sensitivity

Intensity fit, λ(δ) = A e^(−kδ)R² 0.965median of 69 sessions

Policies compared, evaluation sessions (USD per session)

Policy	Mean P&L	sd	Spread	Adverse 60 s
Join the best bid/ask	−261	76	16	−271
Symmetric 1 bp	−201	106	118	−317
Symmetric 3 bp	−71	105	90	−152
Avellaneda–Stoikov γ 0.02	−73	85	74	−148
GLFT γ 0.02	−155	106	136	−293

SMQProjects / Applied Research Capstones / MM — Inventory-Aware Market-Making ResearchCapstone

H1 · The execution intensity decays exponentially with distance from the mid

Avellaneda–Stoikov assume λ(δ) = A e^(−kδ). We estimate the rate at which a quote δ bp from the mid is reached, per session, and fit A and k before any policy is scored.

In [14]:

1It = R("intensity")2F = R("intensity_fits", dates=["date"])3g = It.groupby("delta_bp").intensity_per_s.median()4ax[0].semilogy(g.index, g.values, marker="o", label="median across sessions")5d = np.linspace(g.index.min(), g.index.max(), 50)6ax[0].semilogy(d, F.A_own.median() * np.exp(-F.k_own.median() * d), ls="--")7save(fig, FIG / "mm_intensity.png")

Out[14]:

Supported. R² above 0.9 in 68 of 69 sessions (median 0.965); k = 0.66 per bp. The fit slightly understates how often deep quotes are hit in jumps.

SMQProjects / Applied Research Capstones / MM — Inventory-Aware Market-Making ResearchCapstone

engine.pytest_mm.py

70def as_quotes(mid: float, q_lots: float, pol: Policy, it: Intensity) -> tuple[float, float]:71    """Avellaneda-Stoikov (2008) quotes (distances from mid in bp) with a rolling horizon."""72    g, s2, T = pol.gamma, it.sigma_bp_s ** 2, pol.horizon_s73    r = -q_lots * g * s2 * T                          # reservation price offset (bp)74    half = 0.5 * (g * s2 * T + (2 / g) * math.log(1 + g / it.k))75    return half - r, half + r                          # bid distance, ask distance767778def glft_quotes(q_lots: float, pol: Policy, it: Intensity, qmax: int):79    """Gueant, Lehalle & Fernandez-Tapia (2013) asymptotic quotes with an inventory cap."""80    g, k, A, s2 = pol.gamma, it.k, it.A, it.sigma_bp_s ** 281    c1 = (1 / g) * math.log(1 + g / k)82    c2 = math.sqrt(s2 * g / (2 * k * A) * (1 + g / k) ** (1 + k / g))83    db = c1 + (2 * q_lots + 1) / 2 * c2 if q_lots < qmax else None84    da = c1 - (2 * q_lots - 1) / 2 * c2 if q_lots > -qmax else None85    return db, da

$ pytest tests/test_mm.py -v

test_as_and_glft_formulas PASSED

test_intensity_fit_recovers_exponential PASSED

test_inventory_cap_blocks_adding PASSED

test_loss_limit_cancels_flattens_and_stops PASSED

SMQProjects / Applied Research Capstones / MM — Inventory-Aware Market-Making ResearchCapstone

Decision memo

Should we make markets in BTC-USD on Coinbase?

To: Head of the proprietary desk and the risk manager
Basis: 69 full sessions rebuilt from every level-2 update; seven policies, three fill assumptions; fixed on 2021–22, scored once on 2023–26.

Recommendation

Do not approve the book at a zero or positive maker fee. Every policy lost money in both periods and under every fill assumption.
Revisit only with a confirmed maker rebate of about 1 bp or more. The whole business sits within 1 bp of fees.
If approved, quote inventory-aware and keep all five controls: stale-feed withdrawal, inventory cap, loss limit, end-of-day flattening, monthly review.

Where the P&L went: spread captured against adverse selection

Policy	Mean P&L	Loss limit	1 bp rebate
Join the best bid/ask	−261	33 of 45	+109
Symmetric 1 bp	−201	17	+31
Symmetric 3 bp	−71	4	−5
Avellaneda–Stoikov γ 0.02	−73	2	+9
GLFT γ 0.02	−155	10	+23

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Exercises · worked solutionsThe Binomial Model · Exercise 1

Every exercise, solved line by line.

Each lesson ends with an exercise and its full solution. Reveal one step at a time and check your working against ours.

Exercise 1. A stock trades at $S_{0} = 100$ . Over one month it moves up or down on a CRR tree with $σ = 20%$ and $r = 5%$ . Price an at-the-money call and find the hedge.

1
$u = e^{σ Δ t} = e^{0.2 1/12} = 1.0594, d = 1/ u = 0.9439$ CRR parameters: match the variance of GBM over one step.
2
$q = \frac{e ^{r Δ t} - d}{u - d} = \frac{1.00418 - 0.9439}{1.0594 - 0.9439} = 0.5217$ The risk-neutral probability. No view on the stock is needed.
3
$V^{u} = (105.94 - 100)^{+} = 5.943, V^{d} = (94.39 - 100)^{+} = 0$ Payoffs at expiry in the two states.
4
$Δ = \frac{V ^{u} - V ^{d}}{S _{0} ( u - d )} = \frac{5.943}{100 \times 0.1155} = 0.514$ Hold 0.514 shares per call to hedge.
5
$V_{0} = e^{- r Δ t} [q V^{u} + (1 - q) V^{d}] = 0.99584 \times 0.5217 \times 5.943 = 3.09 ■$ The price is the cost of the hedge.

Tools · options pricingFree, no account

Feel the model before you prove it.

Black–Scholes call prices across spot and volatility. Move the strike and maturity, then drag across the map.

Volatility σ

S = 60Spot140

Strike 100Maturity 0.50y

Spot: 100.0
σ: 25.0%
Call: 8.01
Δ: 0.580
Γ: 0.0221
Vega: 0.276

r = 4%, no dividends. One of seven free tools.

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Derivatives Pricing · Binomial Trees

The Binomial Model

Before this: Market Model and Risk-Neutral Valuation

Proposition 1.3 (node values and recombination). The node reached by $n$ up-moves out of $m$ steps has value

S_{n}^{m} = u^{n} d^{m - n} S_{0}, n = 0, 1, \dots, m .

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Desk decisionPause 44 / 45

Join the touch−261.1 USD/session

With 1 bp rebate+109.4

“Feel the market before you model it.”

7 free interactive tools: walk a limit order book, quote against an insider, shape a volatility surface, design option strategies and train your Bayesian intuition.

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Order bookSpread 0.05 · Mid 100.025

100.2555

100.2057

100.1552

100.1032

100.0521

Market buy 120 → fills 4 levels · avg 100.112 · slippage 8.7 bp

100.0022

99.9543

99.9048

99.8546

99.8062

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Probability · Conditional probabilityMedium

Odd coin flips

You flip $100$ fair coins simultaneously. What is the probability that you observe an odd number of heads?

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