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06 / Carry, events, and liquidity provision

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Start with the idea

Carry, information surprises and liquidity provision can all produce recurring returns but expose a fund to different risks. Treating them as one generic technical score obscures who is paying the return and why losses may cluster.

Symbols, units & horizon

Π_h: currency profit over horizon h
carry_h: holding income in currency
Δprice_h: currency price P&L at the stated position size
funding_h,execution: currency expenses
uₜ: standardised event surprise
releaseₜ: announced value
consensusₜ⁻: forecast available strictly before announcement
σ̂_surprise: training SD in the announcement units
q*ₜ: model reservation price, not an order size
mₜ: quote midpoint
γ: absolute risk aversion in inverse currency
Iₜ: signed inventory units
σₜ: absolute price volatility per square-root time, not return volatility here
τ: remaining time in matching units

When and why to use this

Use separate return and risk accounting for each mechanism. Event surprise features need pre-release consensus; quoting rules need actual inventory and execution data.

Carry strategies seek return from holding a position if pricing conditions remain broadly unchanged. Event strategies seek a response to newly available information or predictable institutional flows. Liquidity provision earns a spread while bearing inventory and information risk. These are distinct mechanisms, even if a bot combines them.

𝔼 [Π_{h}] \approx {carry}_{h} + 𝔼 [Δ {price}_{h}] - {funding}_{h} - execution

Algebra and arithmetic

Separate holding income from price changes

Write holding-period P&L as cash income plus mark-to-market change minus funding and trading cash outflows. Take expectations term by term.
If your definition of carry already subtracts funding, combine those terms before computing net return so funding is not counted twice.

Work it by hand

Expected carry $20, price change −$5, funding $4, execution $3 yields expected net $8 over the defined horizon.

Define carry carefully so financing is not deducted twice. Examples include coupon and roll-down for bonds, forward discounts in currencies, and futures basis convergence. A high carry estimate may compensate for crash exposure, borrow scarcity, or a constraint that prevents the apparent arbitrage.

u_{t} = \frac{{release}_{t} - {consensus}_{t^{-}}}{{\hat{σ}}_{surprise}}

Algebra and arithmetic

Normalise an announcement surprise

Surprise is first-published release minus the consensus available immediately before it. Divide by the training-sample SD of comparable past surprises.
Rearrange to $release = consensus + u σ_{s u r p r i s e}$ . This standardisation compares magnitude, not whether a higher value is good or bad for an asset.

Work it by hand

Release 3.2%, consensus 3.0%, surprise SD .1 percentage point gives u=2. Consistent percentage-point units prevent a 100-fold error.

For an event surprise, use consensus known immediately before release and the first published value, not a later revision. The response may depend on valuation, policy expectations and positioning. The headline’s timestamp and the bot’s executable quote are different observations.

q_{t}^{*} \approx m_{t} - γ I_{t} σ_{t}^{2} τ

Stylised model; unit-consistent algebra

Interpret the inventory reservation-price adjustment

Write $q^{*} = m - γ I σ^{2} τ$ . Under this convention σ is absolute price volatility per square-root time, I is inventory units and γ is risk aversion per currency; the product is a price adjustment.
For positive γ and τ, increasing a long inventory lowers q*. A desired shift δ=m−q* implies $I = \frac{δ}{(γ σ^{2} τ)}$ .

Work it by hand

m=$100, γ=.01 per dollar, I=10 units, σ=$2/√day and τ=.25 day gives a $.10 adjustment and reservation price $99.90. Return volatility would require an additional price-scale conversion.

This stylised inventory adjustment lowers a liquidity provider’s reservation price when it is long inventory I. Risk aversion γ, variance and holding horizon τ must have compatible units. It is an illustration of inventory pressure, not a complete market-making algorithm; actual quotes also depend on fill intensity, tick size, fees, queue position and adverse selection.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def carry_pnl(holding_income, expected_price_pnl, funding, execution):
    return holding_income+expected_price_pnl-funding-execution

def event_surprise(release, prior_consensus, training_surprise_sd):
    if training_surprise_sd <= 0:
        raise ValueError("Positive surprise SD required")
    return (release-prior_consensus)/training_surprise_sd

def reservation_price(mid, risk_aversion, signed_inventory, absolute_price_vol, remaining_time):
    return mid-risk_aversion*signed_inventory*absolute_price_vol**2*remaining_time

print(event_surprise(3.2,3.0,.1), reservation_price(100,.01,10,2,.5))

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