3.1.5 Failure Modes & Real-World Practice

Basic Prerequisites: 04 · No-Arbitrage & Bounds.

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1. Intuition & Practical Objective

Every relationship on the previous pages assumes you can trade the exact underlying, costlessly, without limit, and settle on the model’s schedule. Real markets break all four. The objective of this page is to name the breakages precisely and to quantify them, so that “the model says X” is never the end of the conversation - the residual is.

The four frictions, in one line each:

1. Basis risk - you hedge with a related instrument, not the asset, so the hedge is imperfect. This is the single largest practical failure of hedging (Hull Ch 3).
2. Liquidity & margin - a hedge you cannot fund or roll is not a hedge; margining and collateral are parts of the payoff (Hull Ch 2, 9).
3. Contract-specification traps - multipliers, delivery options, day-counts and dividend/split adjustments silently change the number.
4. Model risk - the no-arbitrage identities are safe, but the moment you parameterise (pick a $\sigma$, a carry, a correlation) you inherit estimation error.

The one-sentence essence. “Textbook no-arbitrage assumes perfect substitutes and frictionless trading; practice replaces exact hedging with minimum-variance hedging and measures the leftover - the basis - instead of assuming it away.”

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2. Mathematical Ground Truth & Derivations

2.1 Basis and the effective hedged price (Hull Ch 3.3)

$$\text{Basis}=S-F(\text{asset price}-\text{futures price of the contract used}).$$

For a short hedge (you own the asset and will sell it), the effective price received is

$$S_2+F_1-F_2=F_1+b_2,b_2=S_2-F_2\text{ (the basis at close).}$$

So the hedge locks in the initial futures price plus the terminal basis - not the initial spot. If the basis were zero at close, the hedge is perfect; the leftover risk is uncertainty in $b_2$, i.e. basis risk. Choose the delivery month as close as possible to, but later than, the hedge horizon to minimize it (Hull §3.3).

2.2 Minimum-variance hedge ratio and hedge effectiveness (Hull eq. 3.1–3.2)

Regress $\Delta S=a+b\Delta F+\epsilon$. The variance-minimising slope is $\rho {\sigma }_S/{\sigma }_F$, so

$$\boxed{h^{\ast }=\rho \frac{{\sigma }_S}{{\sigma }_F}}{N}^{\ast }=h^{\ast }\frac{Q_A}{Q_F}.$$

Hedge effectiveness $=$ the fraction of variance eliminated $=$ the regression $R^2={\rho }^2$. Residual variance

$$\mathrm{Var}(\Delta S-h^{\ast }\Delta F)=(1-{\rho }^2){\sigma }_S^2.$$

This is the quantitative version of “a cross-hedge is never exact”: even a $\rho =0.928$ hedge (Hull’s airline example) removes only ${\rho }^2=86\%$ of the variance. The daily-settlement version regresses percentage one-day changes, $h^{\ast }=\tilde{\rho }{\tilde{\sigma }}_S/{\tilde{\sigma }}_F$, $N^{\ast }=h^{\ast }{V}_A/V_F$, and tailing divides $N^{\ast }$ by $(1+r)$ for the interest effect.

2.3 Structure as a failure mode (Hull Ch 2, 6)

• Daily settlement / convexity. Futures settle daily; forwards once at $T$. Equal prices only when $r$ is constant. For interest-rate futures a closed-form convexity adjustment separates them (Hull Ch 6.3, Ch 30).
• Delivery options. The short party of a Treasury-bond futures chooses when (wild card), which (cheapest-to-deliver), and in what form to deliver; each option lowers the futures price (Hull Ch 6.2).
• Day-count. Actual/360 vs Actual/365 distorts accrued interest by $\approx 1.4\%$; applied across a book it is material.

2.4 Where the identities still hold

Crucially, put–call parity and the bounds do not need the model - they survive every friction if the hedges remain feasible. What dies under frictions is (i) the exactness of the hedge, (ii) the uniqueness of the price (incomplete markets), and (iii) the assumption that you can capture a violation. A “violation” narrower than the transaction cost is not a trade.

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3. Computational Implementation - basis risk in numbers

Standard library only (seeded RNG for reproducibility). It estimates the minimum-variance hedge ratio from simulated correlated price changes, confirms it matches $\rho {\sigma }_S/{\sigma }_F$, reports the hedge effectiveness ${\rho }^2$ (the fraction of variance not removed is the irreducible basis risk), computes the contract count, and shows the tailing adjustment.

The empirical slope $0.7808$ reproduces the theoretical $0.7798$ (Hull’s airline cross-hedge, $\approx 0.78$), the contract count $37.13\to 37$ reproduces Hull’s worked example, and the $13.88\%$ of variance left over ($1-{\rho }^2$) is the basis risk that no amount of hedging removes - it is a property of the instrument, not the analysis.

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4. Failure Modes & First-Principles Breakdowns

1. Basis risk is irreducible with a cross-hedge. $h^{\ast }$ minimises variance, it does not eliminate it: residual $=(1-{\rho }^2){\sigma }_S^2$. Choosing the wrong contract (delivery month before the horizon) can make the basis, and the risk, larger.
2. Liquidity and margin are not frictions to ignore - they are constraints. A hedge of $37$ futures contracts is only executable if the market can absorb it without moving, and only fundable if the margin account is funded. Illiquidity turns a “riskless” arbitrage into a loss; unfunded margin forces liquidation at the worst moment (Hull Ch 2, 9).
3. Contract-specification traps. The multiplier ($100\times$), the delivery options in bond futures, and the day-count convention each silently change the cash flows. The wild-card and cheapest-to-deliver options lower the futures price - a bid you must account for (Hull Ch 6.2).
4. Model risk under parameterisation. Bounds and parity are robust; the inputs are not. A single $\sigma$ cannot fit all strikes (the smile), the carry $b$ must match the instrument, and correlations used for cross-hedging are unstable. Every parameter is a place to be wrong.
5. Leverage and the asymmetry of short options. A naked short call has unbounded loss and a margin call that scales with the underlying. Under-margining plus leverage is the mechanism behind every headline derivatives blow-up.
6. Dividend/split adjustment. Unadjusted splits and mishandled cash dividends make both the payoff and the hedge ratio wrong by a factor - a pure specification error, not a market one.

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5. References

• Hull, Options, Futures, and Other Derivatives
• Hull, Risk Management and Financial Institutions
• Shreve, Stochastic Calculus for Finance I

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6. Connected Graph Bridges

• Back: 04 · No-Arbitrage & Bounds · Index Hub
• Next: 06 · Advanced Extensions
• Deeper failure analysis: Binomial · Failure Modes · BSM · Failure Modes
• Structure and counterparty: Counterparty Risk & XVA