3.1.1 What Is a Derivative? From Zero

Basic Prerequisites: None - high-school algebra only. This is the most accessible page in Pillar 3.

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1. Intuition & Practical Objective

This page answers one question with no derivatives knowledge assumed: what actually is a derivative, and why does it have a price? The objective is the mental picture - a derivative is a contract that pays a specified function of some other price, and the only reason its price is knowable is that you can manufacture that function yourself out of traded assets.

Start from the dumbest possible question. A stock has a price because it is a claim on future cash flows. A derivative has no cash flows of its own - a forward contract on gold literally promises “at time $T$ I pay you $S_T-K$ and nothing happens until then.” So why isn’t it worth zero? Because buying the contract and then holding the right inventory is the same thing as buying the inventory now. If you want the gold today, you can buy it outright for $S_0$; if you instead buy a forward and put $S_0$ in the bank, you get the gold at $T$ for a known cost. The two routes must cost the same, or money is free. That is the entire field in one paragraph.

The three shapes, in words:

1. Forward / futures - a symmetric promise. Both sides are obliged to trade at $K$. Long receives $S_T-K$, short receives $K-S_T$. One wins exactly what the other loses. It costs nothing to enter, because an at-market forward has zero expected value and zero initial cash flow.
2. Option - an asymmetric right. The buyer may trade at $K$; the seller must if asked. Because the buyer’s downside is truncated at the premium paid, and the seller bears the tail, the buyer pays an up-front premium. This asymmetry is the entire reason option pricing is hard and forward pricing is easy.
3. Swap - a strip of forwards. Two parties exchange a fixed cash flow for a floating one, period after period. Nothing new is needed to value it; it is just many forwards glued together.

The one-sentence essence. “A derivative is a contract; its price is the cost of the cheapest portfolio of traded assets that pays exactly the same thing - and because that cost does not depend on anyone’s forecast, the derivative trades at it.”

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2. Mathematical Ground Truth & Derivations

2.1 Forward payoffs (Hull Ch 1.3)

Entering a forward costs nothing, so its payoff is its profit and loss:

$$\text{long forward: }{S}_T-K,\text{short forward: }K-S_T.$$

Compare with an option, where the buyer pays a premium $c$ up front:

$$\text{long call net P\&L: }\max (S_T-K,0)-c.$$

The $\max (\cdot ,0)$ is the kink that makes options non-linear. A forward payoff is a straight line through zero; a call payoff is flat at zero and then a $45°$ line. Everything interesting follows from that kink.

2.2 Why the forward price is $S_0{e}^{rT}$ (Hull eq. 5.1)

Suppose a non-dividend stock trades at $S_0$, the risk-free rate is $r$, and the forward delivers in $T$ years. Consider two strategies:

• Cash-and-carry: borrow $S_0$, buy the stock, hold it, sell it forward at $F_0$.
• Do nothing (and settle the forward at market).

The cash-and-carry strategy costs $0$ today and pays $S_T-F_0$ at time $T$ after repaying $S_0{e}^{rT}$. For no arbitrage the terminal payoff must be zero:

$$\boxed{F_0=S_0{e}^{rT}}\text{(no income; Hull eq. 5.1)}.$$

So the forward price is not a forecast - it is the cost of carrying the asset to $T$. If it trades anywhere else, the difference is a free lunch (the reverse cash-and-carry, or a short sale for the other direction).

2.3 The value of a forward after it is struck (Hull eq. 5.4/5.5)

Once the market moves, an old forward becomes an asset or a liability. The standard argument: contract to buy at $K$ today is equivalent to entering a new forward at $F_0$ plus the certainty of receiving $F_0-K$, so

$$f=(F_0-K)e^{-rT}=S_0-K{e}^{-rT}\text{(no income)}.$$

This is the whole use of forwards in hedging: the contract’s mark-to-market is the present value of the price move, not the price move itself. (For a futures contract the same total accrues, but it arrives in daily instalments - Hull Ch 2 - because futures are marked to market every day.)

2.4 The asymmetry that creates the option premium

$$\begin{array}{rlrl}\text{forward long: }& S_T-K& & \text{(can go arbitrarily negative)}\\ \text{call long: }& \max (S_T-K,0)-c& & \text{(floor at }-c)\\ \text{put long: }& \max (K-S_T,0)-c& & \text{(floor at }-c)\end{array}$$

Truncating the loss transfers the left tail from buyer to seller. That transfer has value, and that value is the option price. Put–call parity (§ next folder page) will make this razor-sharp: a call plus a discounted strike is identical to a put plus the stock, so the two premiums are linked by the same no-arbitrage force that gave $F_0=S_0{e}^{rT}$.

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3. Computational Implementation - payoffs are the whole story

Standard library only. This runs the three ideas above: the symmetric forward payoff, the fair forward price, the marked-to-market value of an old forward, and the truncated option payoff.

Read the last three lines: the forward loses $8$ when $S_T=55$; the call loses only the $3$ premium. That truncation is what the buyer pays for. Notice also that the forward’s fair price $63.0763$ comes entirely from the carry cost $S_0{e}^{rT}$, with no view on gold at all.

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4. Failure Modes & First-Principles Breakdowns

1. “A derivative is a bet on direction.” A forward is not a view - it is a contract priced off the carry, and its value is a present value. Two traders with opposite forecasts agree on $F_0$. If you find yourself pricing a forward with a forecast, stop.
2. Treating the payoff as the value. A forward entered at $K=F_0$ has zero value, but a forward struck at $K\ne F_0$ has value $(F_0-K)e^{-rT}$. Discounting the payoff is mandatory; the “payoff = P&L” shortcut applies only because entry cost is zero and settlement is at $T$.
3. Forgetting the daily settlement. For a futures contract the same total gain accrues, but in daily cash flows - and those flows earn or cost interest. This is why futures and forward prices differ when rates are stochastic (Hull §5.8, and the convexity adjustment in Hull Ch 6).
4. Assuming option and forward obligations are symmetric. A short option has unbounded loss potential; a short forward’s loss is linear in $S_T$. The margin and risk treatment of the two is fundamentally different (Hull Ch 2 vs Ch 10.7).
5. Ignoring income. A dividend-paying stock or an index changes the forward to $F_0=S_0{e}^{(r-q)T}$; a foreign currency to $S_0{e}^{(r-r_f)T}$. Using $S_0{e}^{rT}$ for all of them is the single most common beginner error and the first item on the failure checklist in 05 · Failure Modes.

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5. References

• Hull, Options, Futures, and Other Derivatives
• Shreve, Stochastic Calculus for Finance I
• Haug, The Complete Guide to Option Pricing Formulas

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6. Connected Graph Bridges

• Next: 02 · Options Mechanics & Payoffs · Index Hub
• Forward topic-page: No-Arbitrage Foundations & Binomial Trees (the discrete seed of this replication idea)
• Sibling topic-folders: Black–Scholes–Merton · Interest Rate & Term Structure (forwards on rates, FRAs and swaps)