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Common Formulas

A consolidated exam reference of the most-used formulas in a first derivatives course. The set spans futures pricing, option payoffs, put-call parity, Black-Scholes-Merton, binomial trees, hedge ratios, and money-market bill pricing. Each formula is listed in its standard textbook form. Memorise the formula, but learn what each symbol means and when the formula applies. Most exam errors come from misapplying the right formula in the wrong setting, not from forgetting an equation.

Why it matters

A formula sheet is only as useful as your judgement about which formula to reach for. Before the exam, group these into families. Pricing formulas link spot, futures, and option prices through no-arbitrage. Payoff formulas compute terminal cash flow given $S_T$ . Hedging formulas size positions to neutralise risk. Probability formulas (BSM, binomial) come from risk-neutral valuation. If you can name the family of every problem you face, the correct formula falls out immediately.

Formulas

Forward price, non-dividend stock

F_0 = S_0 e^{rT}

Continuous compounding. With known income

I

, use

F_0 = (S_0 - I) e^{rT}

. With dividend yield

q

, use

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

. See Hull (2022), §5.4 and §5.10.

Call payoff at expiry

\text{payoff} = \max(S_T - K, 0)

Profit equals payoff minus initial premium

C

paid. The break-even is

S_T = K + C

Put payoff at expiry

\text{payoff} = \max(K - S_T, 0)

Profit equals payoff minus initial premium

P

paid. The break-even is

S_T = K - P

Put-call parity (European)

C + K e^{-rT} = P + S_0

Holds for European options on non-dividend stocks. With dividend

D

, becomes

C + K e^{-rT} = P + S_0 - D

. Used to derive synthetics and detect arbitrage.

Black-Scholes call

C = S_0 N(d_1) - K e^{-rT} N(d_2)

Closed-form for European call on non-dividend stock.

N(\cdot)

is the cumulative standard normal.

BSM $d_1$ and $d_2$

d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}}, \quad d_2 = d_1 - \sigma \sqrt{T}

N(d_1)

is the option delta.

N(d_2)

is the risk-neutral probability that the call finishes in the money.

Risk-neutral probability (binomial)

p = \frac{e^{r \Delta t} - d}{u - d}

Hull (2022), Ch. 13. Used to discount risk-neutral expected payoff. No-arbitrage requires

d < e^{r \Delta t} < u

Minimum-variance hedge ratio

h^* = \rho \frac{\sigma_S}{\sigma_F}

\rho

is the correlation between

\Delta S

and

\Delta F

. Reduces hedged variance by a factor of

(1 - \rho^2)

. See Hull (2022), §3.4.

Number of futures to adjust portfolio beta

N^* = (\beta^* - \beta) \frac{V_P}{V_F}

\beta^*

is target beta,

\beta

is current portfolio beta.

V_P

is portfolio value,

V_F

is one futures contract value (futures price times multiplier).

Australian bank bill price (yield basis)

P = \frac{FV}{1 + r \cdot d/365}

FV

is face value,

r

is the bank bill yield as a decimal,

d

is days to maturity. Australian money market convention uses a 365-day year, unlike the US (360-day) convention.

Worked examples

Scenario

Self-test before the exam. Cover this list, write down each formula from memory, and check.

Solution

Eight or more correct indicates strong preparation. Five to seven correct means revisit the weak families. Below five means run focused revision on the missing formulas, especially BSM and the hedge ratio.

Scenario

Quick application. A 6-month forward on a non-dividend stock at $S_0 =$ A$100 with $r = 5\%$ .

Solution

Apply $F_0 = S_0 e^{rT} = 100 e^{0.025} = 102.53$ . The exam might also give a continuous dividend yield $q = 2\%$ , in which case $F_0 = 100 e^{(0.05 - 0.02) \times 0.5} = 100 e^{0.015} = 101.51$ . Choosing the right variant of $F_0$ is the test, not the arithmetic.

Common mistakes

✗Knowing formulas is enough. Formulas without context create exam traps. Always know when and why to apply each formula, especially the difference between European and American, continuous and discrete compounding, and dividend treatment.
✗All money markets use a 360-day year. Australian bank bills use a $365$-day year, while US and European money markets typically use 360 days. Misreading the convention turns a correct formula into a wrong number.
✗ $N(d_2)$ is the actual probability of the call finishing in the money. It is the risk-neutral probability, computed using $r$ rather than the expected return $\mu$ . The real-world probability uses $\mu$ in place of $r$ and is generally different.

Revision bullets

•Forward price: $F_0 = S_0 e^{rT}$ with carry adjustments
•Call payoff $\max(S_T - K, 0)$ , put payoff $\max(K - S_T, 0)$
•Parity: $C + K e^{-rT} = P + S_0$
•BSM: $C = S_0 N(d_1) - K e^{-rT} N(d_2)$
•Risk-neutral $p = (e^{r\Delta t} - d)/(u - d)$
•Min-variance hedge $h^* = \rho \sigma_S / \sigma_F$
•Bank bills use 365-day Australian convention

Quick check

In the Black-Scholes formula $C = S_0 N(d_1) - K e^{-rT} N(d_2)$ , the term $N(d_2)$ represents:

A 90-day Australian bank bill with face value A$1,000,000 trades at a yield of $4.5\%$. Using the standard money-market convention, the price is:

Connected topics

Payoff Shapes Exam Hotspots

In learning paths

Exam Revision Path

Sources

Hull (2022), full text
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Anchor textbook covering every formula listed here. Chapters 5, 11, 13, and 15 are the most relevant.
Black & Scholes (1973)
Black, F. and Scholes, M. The Pricing of Options and Corporate Liabilities. Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-654.
Original derivation of the Black-Scholes formula. Required reading for the BSM formula and $d_1$, $d_2$.
Merton (1973), Theory of Rational Option Pricing
Merton, Robert C. Theory of Rational Option Pricing. Bell Journal of Economics, Vol. 4, No. 1, 1973, pp. 141-183.
Provides the no-arbitrage foundation for put-call parity and option pricing under continuous trading.
RBA, Money market conventions
Reserve Bank of Australia. Australian Money Market Conventions. RBA Domestic Markets Department, accessed 2026.
Confirms the 365-day year basis for Australian bank bill yields and prices.

How to cite this page

Dr. Phil's Quant Lab. (2026). Common Formulas. Derivatives Atlas. https://phucnguyenvan.com/concept/revision-formulas

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