Futuresintermediate

Continuous Compounding Futures Pricing

Under continuous compounding with risk-free rate $r$ and continuous income yield $q$ , the no-arbitrage forward or futures price is $F_0 = S_0 e^{(r - q) T}$ . The exponential factor $e^{(r-q)T}$ is the limit of $(1 + (r-q)/m)^{mT}$ as the compounding frequency $m \to \infty$ , and it is the convention used throughout Hull (2022) because it linearises log returns and simplifies the Black-Scholes framework that builds on the same financing logic.

Why it matters

Continuous compounding treats interest as accruing in infinitesimal slices, so the growth factor over time $T$ is $e^{rT}$ rather than $(1+r)^T$ . Picture a savings balance whose interest is added to the principal every instant. For derivatives pricing, this convention is preferred because log returns are additive and the financing curve fits naturally into expectations under the risk-neutral measure. The fair futures price is then the spot price scaled by $e^{(r-q)T}$ , where the income yield $q$ is the dividend rate for equity indices, the foreign rate for FX, or the convenience yield net of storage for commodities.

Formulas

Non-dividend asset

F_0 = S_0 e^{rT}

Pure financing case from Hull (2022) §5.4.

Asset with continuous income yield

F_0 = S_0 e^{(r - q) T}

Equity index with dividend yield

q

, or FX forward with

q = r_f

Continuous vs periodic rate conversion

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

Convert a rate

r_m

compounded

m

times per year into continuously compounded

r_c

. Hull (2022) §4.2.

Worked examples

Scenario

A non-dividend stock trades at $S_0 = 50$ . The continuously compounded three-month rate is $r = 4\%$ .

Solution

Fair three-month futures price $F_0 = 50 \times e^{0.04 \times 0.25} = 50 \times e^{0.01} = 50 \times 1.01005 \approx 50.50$ . Any market price meaningfully above this allows a cash-and-carry arbitrage.

Scenario

An equity index trades at $S_0 = 4{,}500$ with continuous dividend yield $q = 1.8\%$ and continuously compounded risk-free rate $r = 5.0\%$ . Time to maturity is six months.

Solution

Net carry $r - q = 0.032$ . Fair futures price $F_0 = 4{,}500 \times e^{0.032 \times 0.5} = 4{,}500 \times e^{0.016} \approx 4{,}500 \times 1.01613 \approx 4{,}572.6$ index points. The premium of about 72 points reflects net financing minus dividends over the half-year holding period.

Common mistakes

✗Continuous compounding produces materially different prices from discrete. For short maturities the gap is tiny. At $r = 5\%$ and $T = 0.25$ years, $e^{rT} = 1.01258$ versus $(1+r)^{T} = 1.01227$ , a difference under 0.03 per cent on the futures price.
✗Continuous compounding is a real-world cash-flow process. It is a mathematical convention that makes growth factors multiplicative in log space. Bank deposits still pay interest in discrete periods, so applied work must convert quoted rates with $r_c = m \ln(1 + r_m/m)$ .
✗The formula always assumes zero dividends. The general form is $F_0 = S_0 e^{(r-q)T}$ . Setting $q = 0$ is the no-income special case used for non-dividend stocks and pure commodities without convenience yield.

Revision bullets

• $F_0 = S_0 e^{rT}$ for a non-dividend asset
• $F_0 = S_0 e^{(r-q)T}$ with continuous income yield
•Compounding factor $e^{rT}$ is the limit of $(1 + r/m)^{mT}$
•Convert quoted $r_m$ to $r_c$ via $r_c = m \ln(1 + r_m/m)$
•Standard convention in Black-Scholes and academic derivatives work

Quick check

Continuous compounding, $S_0 =$ US$100, $r = 6\%$ , $T = 1$ year, no dividends. The fair futures price is:

An equity index has $S_0 = 4{,}000$ , continuous dividend yield $q = 2\%$ , continuously compounded rate $r = 5\%$ , and $T = 0.5$ years. $F_0$ is closest to:

Connected topics

Cost of Carry Discrete Pricing BSM Intro

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.
Section 4.2 introduces continuous compounding and the conversion to discrete rates. Section 5.4 derives the cost-of-carry futures formula.
Black & Scholes (1973)
Black, Fischer, and Myron Scholes. The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 1973, pp. 637 to 654.
Original derivation that established continuous compounding as the convention for derivatives pricing through risk-neutral arguments.
CME Group, Calculating Fair Value
CME Group. Equity Index Fair Value Calculation. CME Group Education, 2024.
Working examples of $F_0 = S_0 e^{(r-q)T}$ applied to S&P 500 futures fair value.

How to cite this page

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

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