Equities, commodities, currencies, indices — the underlying assets — together with forwards, futures and the no-arbitrage glue that ties them together.
Wilmott opens the book by stating its scope sharply: it is about derivative pricing, the mathematics behind contracts whose value depends on something else — a stock, a currency, an interest rate, an electricity price.
"Quantitative finance is the application of mathematical methods to the modelling of financial markets and to the valuation of derivative securities."
— Wilmott, Ch. 1Chapter 1 doesn't yet do any of that. It introduces the things derivatives are written on — the equities, commodities, currencies and indices that play the role of the underlying — and the one piece of arithmetic the rest of the book never lives without: the time value of money. By the end of the chapter you'll already have seen the first genuine result of quantitative finance: the no-arbitrage forward price.
This deck stays close to Wilmott's section numbering. The interactive forward-pricing calculator on slide 08 is the same example the book uses to motivate no-arbitrage; it's a knob you can turn rather than a numerical worked example.
An equity is a share of ownership in a company. The owner has two sources of return — capital appreciation (if the price rises) and dividends (if the company pays them).
A scheduled cash payment per share. When the dividend is paid, the share's value drops by approximately the cash amount — the company has paid out cash and is worth less by that amount.
If a stock at $S$ pays a known dividend $D$:
$$S^+ = S^- - D \quad \text{at the ex-dividend date.}$$
An $n$-for-$1$ split multiplies the share count by $n$ and divides each share price by $n$. The market cap is unchanged; the only purpose is to keep the share price in a "tradable" range.
$$S \to S/n, \quad N \to nN.$$
Both dividends and splits are known events at known times — the modelling problem is the share price itself, which is not. That is the subject of Chapters 4 and 5.
$S$ — spot price of the underlying. $S_t$ — spot at time $t$. $r$ — the continuously compounded risk-free interest rate. $D$ — a dividend (cash or, eventually, a yield). $T$ — maturity. The book uses these consistently, and so will every deck in this series.
Three more underlyings round out the set Wilmott treats:
Oil, gold, copper, wheat. Unlike an equity they often cost to hold — storage, insurance — and sometimes pay a convenience yield for being available physically. Both effects later show up in the forward price.
A foreign currency at exchange rate $S$ earns the foreign risk-free rate $r_f$. Holding GBP at home rate $r$ vs. holding USD that earns $r_f$ is the seed of covered interest parity.
An index (S&P 500, FTSE 100) is a weighted basket of stocks. It pays a stream of dividends and is conveniently modelled as paying a continuous dividend yield $q$ — the index analogue of $r_f$.
Wilmott's important observation: all four — stocks-with-yield, commodities-with-carry, foreign currencies, indices — behave like one another for pricing purposes. The carry, the foreign rate, and the dividend yield all play the role of "income from holding the asset," and the forward formula has one shape that accommodates all of them.
A pound today is worth more than a pound in a year because you can put today's pound in a riskless bank account and have more than a pound by next year. This trivial observation is the engine of everything that follows.
Three flavours of the same idea:
| Compounding | Value of £1 after time $T$ at rate $r$ |
|---|---|
| simple | $1 + rT$ |
| discrete, $m$ times per year | $\left(1 + r/m\right)^{mT}$ |
| continuous | $e^{rT}$ |
The book uses continuous compounding exclusively from Chapter 4 onward, because the resulting algebra is much cleaner — $e^{rT}$ is multiplicative across time, $\log$ is linear, and the SDE for the share price (later) has $r$ sitting next to $\mu$ in a simple drift term.
The present value of a cashflow $C$ to be received at time $T$ is
$$\mathrm{PV}(C) = C\, e^{-rT}.$$
If you're rolling daily, weekly, or monthly the discrete formula is exactly right; but the cleanest description of a portfolio that is being rebalanced at every instant — which is what Black–Scholes hedging assumes — is the $e^{rt}$ growth factor. From now on, $r$ means the continuously compounded rate unless otherwise stated.
A bond is a contract that pays a stream of known cashflows at known dates. The simplest is the zero-coupon bond: a single payment of £1 at maturity $T$. Its price today is
$$Z(t, T) = e^{-r(T-t)}.$$
That's just the discount factor of the previous slide written as a tradable instrument. A general bond is then a sum of zeros:
$$B = \sum_i C_i\, e^{-r\, t_i}.$$
"Real" bonds (TIPS in the US, index-linked gilts in the UK) pay coupons indexed to a price index; their cashflows track inflation, so the discount rate that matters for them is the real rate, not the nominal.
Decks 11 and 12 take the rate $r$ apart. In this chapter, $r$ is a single number; in fixed income, it depends on the maturity ($r(T)$, the yield curve) and eventually on time and chance ($r_t$, a stochastic process). The basic discounting picture survives unchanged.
The next two contracts are the simplest derivatives in existence and they don't even need a model.
An agreement struck today to buy (or sell) an asset $S$ at a fixed price $F$ on a future date $T$. No money changes hands today.
At maturity the long pays $F$, receives the asset:
$$\text{payoff to long at }T = S_T - F.$$
An exchange-traded, margined, daily-settled version of the same thing. Economically equivalent when interest rates are non-random (Wilmott proves this in §6.15).
Standardised contract sizes, central clearing, daily mark-to-market.
The famous question is: what is the fair value of $F$? The answer requires no probability, no stochastic process, no random walk. It requires only the principle of no arbitrage.
Consider two ways to own one share of $S$ at time $T$:
Borrow $S_0$ at rate $r$, buy the share today, hold it until $T$.
Net position: $S_T - S_0 e^{rT}$.
Sign a forward to buy at $F$ on date $T$. Put nothing in today.
Net position: $S_T - F$.
Both strategies cost zero today and deliver the same asset at $T$. If the no-arbitrage principle is to hold — if you can't make a riskless profit out of two equivalent positions — the two payoffs must agree:
$$\boxed{\;F = S_0\, e^{rT}\;}$$
When the asset pays a continuous yield $q$ (foreign currency interest, index dividend yield, commodity convenience) the cost-of-carry argument generalises in the obvious way:
$$F = S_0\, e^{(r-q)T}.$$
Priced a derivative without writing down a single probability. This is the spirit of the rest of the book: derivative prices are pinned down not by what you think the market will do, but by what would happen to anyone foolish enough to offer a price that allowed a free lunch.
Slide the spot, the rate, the yield and the maturity. The forward price $F = S_0\,e^{(r-q)T}$ updates live. The bar at the bottom shows the arbitrage profit if the market posts a quote $F^*$ different from the fair value.
If $F^* > F$ sell the forward, borrow $S_0 e^{-qT}$ to buy the spot (and reinvest the yield); locked-in profit at $T$ is $F^* - F$. If $F^* < F$ run the trade in reverse. The chart shows the two payoff legs and their net — flat, risk-free, and non-zero whenever the market quote is wrong.
The same formula in three different costumes:
| Underlying | "Yield" $q$ | Forward price |
|---|---|---|
| Non-dividend stock | $0$ | $F = S_0\, e^{rT}$ |
| Dividend-paying stock / index | continuous yield $q$ | $F = S_0\, e^{(r-q)T}$ |
| Foreign currency | foreign rate $r_f$ | $F = S_0\, e^{(r-r_f)T}$ |
| Storable commodity | $-u$ ($u$ = storage cost) | $F = S_0\, e^{(r+u)T}$ |
| Consumption commodity | convenience yield $y$ | $F \le S_0\, e^{(r-y)T}$ |
For consumption commodities, the no-arbitrage argument is one-sided — you can't easily short physical wheat — so the formula becomes an inequality. The same single relation thus covers Chapters 1's worth of products on three different exchanges.
Futures are daily settled: the contract is repriced and cash flows between the counterparties each evening. If interest rates are deterministic this washes out exactly and the futures price equals the forward price. When rates are stochastic (Deck 12) a small convexity adjustment appears.
Chapter 1 hands the next chapters three things:
Equities, FX, commodities, indices, bonds — the universe of underlyings on which everything else is written.
Continuous compounding and the discount factor $e^{-rT}$. Every present-value calculation in the book starts here.
The principle that two strategies with the same payoff must have the same price. Black–Scholes (Deck 06) is this idea, applied at every instant.
Discount factor: $e^{-rT}$. Forward price: $F = S_0\, e^{(r-q)T}$. No-arbitrage: if two portfolios have the same payoff at $T$, they have the same price now. Convention: $r$ is continuously compounded.