Exam Revision Path

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.

Eight payoff diagrams cover almost every exam strategy question. **Long and short futures** produce straight 45-degree lines through the entry price. **Long calls and puts** produce **hockey sticks** with limited loss and unlimited gain. **Short calls and puts** produce **inverted hockey sticks** with limited gain and unlimited or large loss. Combinations build the **straddle** (V-shape long, inverted V short) and **spreads**. Being able to sketch each shape from the formula is essential for exam success.

Six topic clusters recur across derivatives exams: **put-call parity arbitrage**, **two-step binomial trees**, **Black-Scholes calculation and interpretation**, **margin call mechanics**, **basis risk in cross-hedges**, and **swap comparative advantage**. Mastering these six families covers roughly **70 to 80 percent** of typical exam marks. Each cluster has a predictable trap, so revising the traps is as important as revising the formulas.

**Put-call parity** is a no-arbitrage condition that pins the prices of European call and put options with identical strike and expiry to the current stock price and the present value of the strike. The relation is $C + K e^{-rT} = P + S_0$, first formalised by Stoll (1969) and extended by Merton (1973). Any deviation from this equality allows a **riskless profit**, so in liquid markets violations are fleeting. The relation applies strictly to **European options**. For American options on non-dividend-paying stocks, only an inequality holds.

A **two-step binomial tree** divides the option's life into two equal periods of length $\Delta t$, generating three terminal stock prices $S_0 u^2$, $S_0 u d$, and $S_0 d^2$. Pricing proceeds by **backward induction**. Find the option value at the two intermediate nodes, then discount one more step back to today. Adding steps refines the approximation and, in the Cox–Ross–Rubinstein parameterisation, the tree converges to **Black–Scholes** as the number of steps grows.

The **Black-Scholes-Merton (BSM) model** gives a closed-form price for European options on a non-dividend paying stock. The call formula is $C = S_0 N(d_1) - K e^{-rT} N(d_2)$. The model rests on five assumptions, continuous trading with no transaction costs, lognormal stock prices, no dividends in the basic form, constant volatility $\sigma$, and a constant risk-free rate $r$ with continuous compounding. Black and Scholes (1973) and Merton (1973) showed that, under these assumptions, a self-financing portfolio of stock and bond can replicate the option, so its price is unique and free of any risk premium.

A **margin call** is the broker's demand for additional collateral when accumulated losses pull a futures account below the **maintenance margin** threshold. The trader must restore the balance to the **initial margin** level, not merely to the maintenance line, usually within one business day. Failure to meet the call gives the broker the right to **close out** (liquidate) the position at the prevailing market price, with any residual loss still owed by the trader.

**Basis risk** is the uncertainty in the basis $b_2 = S_2 - F_2$ at the close date of a hedge. A short hedger receives an effective price of $F_1 + b_2$, and a long hedger pays $F_1 + b_2$. Because $b_2$ is unknown when the hedge is opened, the realised outcome differs from the intended lock by the **basis change**. Three drivers of basis risk are (1) closing the hedge before delivery, (2) using a futures contract on a **different but correlated asset** (cross hedge), and (3) timing mismatches between physical delivery and contract maturity.

The textbook rationale for an interest rate swap is **comparative advantage**. One borrower may face better terms in fixed-rate funding while another faces relatively better terms in floating. Each borrows where its advantage is largest and the two parties swap payment streams. The **total gain** equals the **quality spread differential** (QSD), the absolute difference between the two parties' fixed-rate gap and floating-rate gap. Hull (2022) §7.4 notes the argument has been criticised because true comparative advantage in efficient capital markets should not survive long-term arbitrage.