Futuresintermediate

Discrete Compounding Futures Pricing

Under discrete compounding, the no-arbitrage futures price for an asset with no income is $F_0 = S_0 (1 + r)^T$ , where $r$ is the annual rate compounded once a year. For compounding $m$ times per year the formula is $F_0 = S_0 (1 + r/m)^{mT}$ . Discrete compounding is the convention used by Australian money market quotes such as BBSW and the 90-day bank accepted bill futures on ASX 24, and matches how bank deposits actually accrue interest.

Why it matters

Discrete compounding pays interest at fixed intervals: once a year, every quarter, every day. After each interval the new interest earns interest itself, so wealth grows by the factor $(1 + r/m)^{mT}$ over time $T$ . For derivatives pricing, the no-arbitrage condition is unchanged in spirit. Money used to buy spot today must grow to $F_0$ by delivery, otherwise borrowing and short-selling close the gap. The choice between $e^{rT}$ and $(1+r/m)^{mT}$ is purely conventional, and the two formulas converge as $m \to \infty$ .

Formulas

Annual compounding

F_0 = S_0 (1 + r)^T

Compounding $m$ times per year

F_0 = S_0 \left(1 + \frac{r}{m}\right)^{mT}

Set

m = 2

for semi-annual,

m = 4

for quarterly. Hull (2022) §4.2.

Continuous-equivalent rate

r_c = m \ln \left(1 + \frac{r_m}{m}\right)

Convert a discretely compounded

r_m

to continuous

r_c

for use in

e^{r_c T}

formulas.

Worked examples

Scenario

$S_0 = 50$ , annual rate $r = 4\%$ compounded annually, $T = 0.25$ years.

Solution

$F_0 = 50 \times (1.04)^{0.25} = 50 \times 1.00985 \approx 50.49$ . The continuous formula gives $50 \times e^{0.04 \times 0.25} = 50 \times 1.01005 \approx 50.50$, a difference of about one cent.

Scenario

An Australian three-month bank bill yields $r = 4.35\%$ per annum quoted with $365$-day simple interest. The futures contract on ASX 24 is priced from this convention.

Solution

For a 90-day bank bill with face value A$1,000,000, price $P = \dfrac{1{,}000{,}000}{1 + 0.0435 \times 90/365} = \dfrac{1{,}000{,}000}{1.01073} \approx$ A$989,386. The futures price is quoted as $100 - $ yield, so a yield of $4.35\%$ implies a futures quote of $95.65$. This is a simple-interest pricing rule, the discrete cousin of $F_0 = S_0 (1 + r)^T$ , adapted to Australian money market conventions.

Common mistakes

✗Discrete and continuous compounding always give the same answer. They are close but not identical. For one-year horizons at $r = 5\%$ , the gap is about 0.13 per cent in growth factor, which matters for large notional fixed income trades.
✗Compounding convention is irrelevant. It changes the quoted rate even when the underlying instrument is the same. A 4.00 per cent continuously compounded rate equals 4.08 per cent semi-annually compounded, and ignoring this distorts forward and futures prices in textbook exercises.
✗Discrete pricing only applies to short maturities. It applies wherever conventions specify it. Australian bank bills, US Treasury bills, and most money market futures use simple or discrete compounding regardless of the horizon.

Revision bullets

• $F_0 = S_0 (1 + r)^T$ for annual compounding
•Generalises to $F_0 = S_0 (1 + r/m)^{mT}$ for $m$ periods per year
•Converges to continuous $e^{rT}$ as $m \to \infty$
•Standard convention in bank bill and US T-bill futures
•Always check which convention the question or quote uses

Quick check

Discrete annual compounding, $S_0 =$ US$200, $r = 8\%$ , $T = 0.5$ years. $F_0 = ?$

An asset has $S_0 = 100$ , $r = 6\%$ compounded semi-annually, $T = 1$ year. The futures price is:

Connected topics

Cost of Carry BB Futures

In learning paths

Futures Path

Sources

Hull (2022), §4.2 and §5.4
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Treatment of discrete versus continuous compounding conventions and their equivalence in cost-of-carry pricing.
ASX 90 Day Bank Bill Futures
Australian Securities Exchange. ASX 90 Day Bank Accepted Bill Futures and Options Factsheet. ASX, 2024.
Specifies the simple-interest pricing convention used for the contract priced from BBSW, the canonical Australian discrete-compounding example.
ASX Interest Rate Derivatives Pricing Guide
Australian Securities Exchange. Interest Rate Derivatives Price and Valuation Guide. ASX, 2025.
Formal pricing manual covering all ASX 24 interest rate futures including 90-day bank bills and three- and ten-year Treasury bond contracts.

How to cite this page

Dr. Phil's Quant Lab. (2026). Discrete Compounding Futures Pricing. Derivatives Atlas. https://phucnguyenvan.com/concept/discrete-futures-pricing

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