The math of trading

1. Percentages act on a changing base

Buy q units at price P₀, then sell at P₁. Your gross profit is q(P₁ − P₀). The asset’s simple return is the price change divided by its starting price:

From $100 to $120 is +20%. From $120 back to $100 is −16.67%, because the denominator is now $120. This is why adding percentages across successive periods does not track your money correctly.

For successive returns, multiply the growth factors: Wₙ = W₀ × (1 + r₁) × … × (1 + rₙ). A +20% day followed by a −20% day turns $100 into $96: 100 × 1.2 × 0.8 = 96. Reversing those two returns gives the same final wealth when there are no intervening deposits or withdrawals.

If you lose a fraction d, you keep 1 − d. To recover, you need d / (1 − d): a 20% loss needs a 25% gain; a 50% loss needs 100%. Log returns, ln(P₁/P₀), add across time because logarithms turn products into sums. They are useful accounting quantities, not a way to erase losses.

2. Leverage changes exposure, not predictive skill

Let E₀ be starting equity and N₀ be the position’s initial notional exposure. Entry leverage is L = N₀/E₀. For a fixed-size linear position, ignoring costs, profit is N₀r and return on starting equity is Lr.

A −2% move loses the same $100. Leverage changes the slope of the profit line. It does not improve the odds of choosing the right direction. As losses reduce equity, current leverage can increase even though you did not add contracts.

In a deliberately simplified model, E(r) = E₀ + N₀r. If maintenance is a fixed $200, then $1,000 of equity and $5,000 notional reaches that level at r = (200 − 1000)/5000 = −16%. This is not an exchange liquidation-price formula. Real systems may use mark prices, changing notional, maintenance tiers, fees, isolated or cross margin, and other collateral. A gap can skip the level at which you hoped to exit.

Also distinguish fixed contracts from a product that resets leverage daily. Daily resets compound a different sequence of exposures, so “five times the asset’s monthly return” is not generally its monthly result.

3. Futures turn price changes into contractual cash flows

A derivative’s value depends on an underlying asset, rate or index. Futures and options are two kinds of derivatives. A futures contract creates obligations under specified settlement rules; an option gives its buyer a right. Neither is automatically a purchase of the underlying asset.

For n long linear futures contracts with currency multiplier m, opened at futures price F₀ and closed at F₁:

One contract with a $10-per-point multiplier, moving from 100 to 106, produces $60 for the long and −$60 for the short before costs. The exchange’s actual contract specification determines the multiplier and settlement currency; this formula does not describe every inverse crypto contract.

Futures margin is collateral for performance, not a down payment that buys a commodity. Initial margin opens the position; maintenance margin is the minimum required balance. Falling below it can require additional funds or position reduction. Exchange and broker requirements can change. CME explains the distinction and margin process.

Daily settlement makes the path visible in cash. Starting with $200, a move from 100 to 104 credits $40; a later move from 104 to 101 debits $30. Balance: $210. With an invented $150 maintenance level, a fall to 94 leaves only $140. A later rebound may be irrelevant if the position has already been closed.

Hedging is a change in the shape of risk

Suppose you own one unit of an asset and short a matching linear future at 100. At a matched settlement price S, the idealized combined value is S + (100 − S) = 100. You traded price uncertainty for other issues: basis differences, collateral needs, counterparty arrangements and costs. This is a settlement illustration, not a claim that every hedge locks a guaranteed executable price.

Perpetual futures have no scheduled expiry and typically use periodic funding transfers to help align the contract with its reference market. Funding can be paid or received, can change sign, and enters net P&L separately. Dated futures, perpetuals, forwards and CFDs have different contractual mechanics; check the product you are modeling.

4. Options bend the payoff line

A call gives its holder the right to buy at strike K; a put gives the right to sell. Exercise style and settlement rules matter. European-style exercise occurs at expiry; American-style contracts can generally be exercised earlier. We first consider expiry values per underlying unit, before costs.

Pay $6 for a call with strike $100. At expiry price $104, it pays $4, so profit is −$2. You were right about the rise and still lost money. Expiry break-even is $106. Maximum loss on the purchased call itself is its $6 premium per unit, excluding costs and any exposure created by exercise. OIC’s long-call guide explains this payoff and risk.

A $6 put with strike $100 breaks even at $94 at expiry. At $80 it pays $20 and profits $14. Multiply per-unit figures by the actual contract multiplier and number of contracts; do not assume every market uses the same size.

The seller’s expiry profit is the negative of the buyer’s profit when entry terms match and costs are ignored. A short uncovered call has theoretically unlimited loss as the underlying rises. A short put can also suffer a large loss. Receiving a premium does not mean you have a bounded-risk position.

6. From financial derivatives to calculus derivatives

“Derivative” has two related but different meanings here. A financial derivative is a contract tied to something else. A calculus derivative measures how a function changes. We apply calculus to the contract’s value function V(S, t, σ, …).

For a small move, a local approximation is ΔV ≈ Δ·ΔS + ½Γ·(ΔS)² + Θ·Δt + Vega·Δσ, with other sensitivities omitted. Units must agree: theta quoted per day is not per year; vega quoted per one volatility percentage point is not per unit volatility. If using time remaining rather than elapsed calendar time, the theta sign convention changes.

A delta of 0.5 suggests roughly a $0.50 option-value move for a $1 underlying move, locally and with other inputs fixed. It is not a guaranteed change or automatically a real-world probability. Time passing or a drop in implied volatility can outweigh a favorable underlying move. OIC describes how these pricing inputs interact.

The model behind the curved line

The animation uses a European call under the Black–Scholes assumptions: no dividends, constant volatility, continuous trading, and a frictionless market. We set the risk-free rate to zero for the illustration. Let N be the standard normal cumulative distribution function and T time remaining in years:

At T = 0, use the expiry payoff directly rather than divide by zero. Before expiry this is a model value, not an executable market quote. Implied volatility is the volatility input that matches an observed option price; it is not a promise about realized volatility. Real markets add jumps, spreads, changing volatility and exercise details. OIC explains the model’s inputs and limits.

7. Costs and sizing close the loop

Net profit subtracts commissions, spread, slippage, borrowing charges and net funding paid from gross profit. If a $1,000 trade incurs $1 on entry and $1 on exit, it needs a 0.2% favorable move just to cover those two charges, before any other friction.

Expected value is probability-weighted profit, not the percentage of winning trades. A hypothetical 40% chance of winning $3 and 60% chance of losing $1 gives 0.4 × 3 − 0.6 × 1 = $0.60 before costs. That calculation is only as useful as its probabilities and loss model. A strategy with many small wins can still have negative expected value if rare losses are large.

For a spot example with $10,000 of equity, a chosen $100 risk budget and a $2 distance between entry and an intended stop, the arithmetic suggests 50 units. It does not guarantee a $100 maximum loss: a stop is a trigger and the fill can be worse. Position limits, liquidity and correlated exposures still matter.

The useful habit is to draw the payoff, identify its units, subtract costs, and ask what could force you out before your assumed horizon. Then test the arithmetic in a simulation before treating a forecast as a position.