Free lesson · Crypto derivatives

Solve a simplified liquidation boundary

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Start with the idea

Solving the equity-maintenance equality helps explain why higher leverage leaves a smaller price buffer.

Symbols, units & horizon

P_liq: simplified long-position liquidation mark USD/base
q: positive base units long
P_0: entry USD/base
E_0: initial isolated cash USD
m: constant maintenance fraction between 0 and 1
funding/fees zero in this derivation

When and why to use this

Explain leverage sensitivity and compare simplified stress buffers before adding a venue-specific margin engine.

Solving the equity-maintenance equality helps explain why higher leverage leaves a smaller price buffer.

For a long linear isolated position with no funding or fees and constant maintenance fraction, equate cash plus marked P&L to required maintenance. This produces a threshold useful for sensitivity analysis.

The equation gives a checkpoint under ideal continuous marking, not the final executed liquidation price. Gaps, liquidation charges and inability to replenish collateral can worsen outcomes.

P_{liq} = \frac{q P_{0} - E_{0}}{q (1 - m)}

Algebraic boundary under an isolated linear margin model

Solve a simplified liquidation boundary

Set E_0+q(P−P_0)=mqP.
Collect qP(1−m)=qP_0−E_0.
Divide by q(1−m); if numerator is nonpositive, no positive-price boundary exists under this toy model.

Work it by hand

q=.2 BTC, entry $50,000, cash $1,000, m=.02: threshold=(10000−1000)/(.2×.98)≈$45,918.37/BTC.

Apply it in a strategy

Explain leverage sensitivity and compare simplified stress buffers before adding a venue-specific margin engine.
Record the input timestamp, executable quantity, currency and horizon. Reconcile the result with a cash-flow or state table.
Stress this failure condition: Cross margin, fees, tier changes and collateral haircuts require a different boundary.

Research deliverable

Build and explain a solve a simplified liquidation boundary worksheet. Explain leverage sensitivity and compare simplified stress buffers before adding a venue-specific margin engine.

Evidence boundary: Synthetic arithmetic and scenarios illustrate mechanics. They are not historical returns, a paper replication, or evidence of an executable edge. Research sources and their access limitations are recorded at the end of this module.

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.

# Python 3.10+; standard library unless NumPy is imported below.
# Inputs and outputs use the units defined in this lesson. Synthetic teaching example.
def long_liquidation(q,entry,cash,maintenance):
    if q<=0 or entry<=0 or cash<0 or not 0<=maintenance<1: raise ValueError("Invalid isolated-long inputs")
    return max(0,(q*entry-cash)/(q*(1-maintenance)))

print(long_liquidation(.2,50000,1000,.02))

Continue learning

Crypto Derivatives: Carry, Funding and Liquidation — all lessons

Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations