Free lesson · Markets & returns

Cash securities, derivatives, and financing

Open interactive lesson Practice calculations Explore labs

Start with the idea

Separate the price quoted on the screen, the amount of economic exposure and the cash required to carry it. Contract multipliers translate points into money; settlement determines when the money moves. A short position reverses price exposure but introduces borrow and distribution obligations.

Symbols, units & horizon

N: signed futures contract count
M: currency multiplier per quote point per contract
Fₜ: futures quote in points
ΔF: current minus previous quote
Q: positive shares sold short
P₀, P_T: initial and covering prices per share
P&L or Π: currency profit or loss
|N|: absolute contract count
K: desired currency sensitivity per point in the rearrangement
Costs: borrow, dividend compensation and trading expenses in currency

When and why to use this

Use the contract P&L identity when sizing futures, checking execution reports and reconciling cash. Use the short-sale ledger before testing relative-value or long–short strategies.

Equity is a residual claim on a business. A bond promises contractual cash flows subject to default and other terms. A future is a contract whose price changes create gains and losses, usually settled daily. An option gives an asymmetric payoff in exchange for a premium. Each instrument needs its own P&L, settlement, and exposure conventions.

Δ Π_{futures} = N M (F_{t} - F_{t - 1}), notional = | N | M F_{t}

Algebra and arithmetic

Convert a price move into futures dollars

One contract gains $M Δ F$ dollars for a one-period quote change. N signed contracts gain $N M Δ F$ ; replace $Δ F$ with $F_{t} - F_{t - 1}$ .
The magnitude of quoted exposure is contracts × dollars per point × quoted points: $| N | M F_{t}$ . For a desired P&L sensitivity K dollars per point, solve $N = \frac{K}{M}$ .

Work it by hand

With N=10, M=$50/point, and ΔF=−20 points, P&L = −$10,000. At F=4,000, notional is $2,000,000.

$N$ is signed contract count, $M$ is the currency value of one quoted point, and $F$ is futures price. Ten contracts with a $50 multiplier lose $10,000 when price falls 20 points. Initial margin is collateral, not the maximum loss or the instrument's notional.

Π_{short} = Q (P_{0} - P_{T}) - borrow - dividends - e x e c u t i o n c o s t s

Algebra and arithmetic

Balance the short-sale cash flows

Sell Q borrowed shares for $Q P_{0}$ ; buy them back for $Q P_{T}$ . Subtract the repurchase from sale proceeds: $Q (P_{0} - P_{T})$ .
Deduct borrow fees, manufactured dividend payments and execution costs. Setting net profit to zero and solving gives $P_{T} = P_{0} - \frac{costs}{Q}$ , when total costs can be treated as fixed.

Work it by hand

Short 100 shares at $50, cover at $47, pay $40 total costs: 100×3−40=$260. Break-even cover price is $49.60.

A short sale requires available borrow and may be recalled; a correct thesis can still be expensive to maintain.
Daily settlement creates a cash requirement even when a hedge is expected to work at maturity.
Options require delta, gamma, and vega exposures alongside notional. Equal notionals need not mean equal risks.

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 futures_pnl(contracts, multiplier, previous_quote, current_quote):
    pnl = contracts*multiplier*(current_quote-previous_quote)
    notional = abs(contracts)*multiplier*current_quote
    return pnl, notional

def short_pnl(shares, sell_price, cover_price, total_cost):
    if shares <= 0:
        raise ValueError("Short share magnitude must be positive")
    return shares*(sell_price-cover_price)-total_cost, sell_price-total_cost/shares

print(futures_pnl(10, 50, 4020, 4000))
print(short_pnl(100, 50, 47, 40))

Continue learning

Markets, Returns & Capital — all lessons

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