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Absolute error

Measure the size of a pricing miss in its original units, keep its direction separate, and distinguish a reference comparison from uncertainty.

Absolute error is the nonnegative distance between an approximation and a specified reference value. For a quant comparing two valuations, it answers a concrete question: how many dollars apart are they?

If xx is the reference and x^\hat{x} the approximation, define

Δ=x^−x,Eabs=∣x^−x∣.\Delta=\hat{x}-x, \qquad E_{\mathrm{abs}}=\lvert\hat{x}-x\rvert.
Here Δ\Delta is the signed error, with the convention approximation minus reference. A negative value means an underestimate. Absolute error removes that sign, so it cannot tell you which valuation is higher. It is zero exactly when the two values agree.

A price miss in dollars

Suppose a synthetic instrument has a stipulated model reference value of USD 100.00, while a faster approximation returns USD 99.80. Then
Δ=99.80−100.00=−0.20 USD,Eabs=0.20 USD.\Delta=99.80-100.00=-0.20\ \text{USD}, \qquad E_{\mathrm{abs}}=0.20\ \text{USD}.

The approximation is twenty cents below the reference. An approximation of USD 100.20 has the same absolute error, but a positive signed error. Keeping both measures prevents a report of “0.20 error” from hiding the direction of a pricing discrepancy.

Synthetic USD values 99.8 and 100.2 sit equally far from reference 100.0. Signed errors are minus 0.2 and plus 0.2; both absolute errors are USD 0.2.
Synthetic USD values 99.8 and 100.2 sit equally far from reference 100.0. Signed errors are minus 0.2 and plus 0.2; both absolute errors are USD 0.2.
Original synthetic comparison. Distance is nonnegative; direction is retained only by the signed error. Values and distances are in USD.
Absolute error keeps the units of the quantity being compared. Express the same prices in cents and the absolute error becomes 20 cents, with no change in the economic discrepancy. Comparing an error in dollars with one in volatility points requires a decision about scale and purpose; the raw numbers alone do not rank their importance.
To put the price miss on a proportional scale, use relative error:
Erel=Eabs∣x∣=0.20100.00=0.002=0.2%.E_{\mathrm{rel}}=\frac{E_{\mathrm{abs}}}{\lvert x\rvert} =\frac{0.20}{100.00}=0.002=0.2\%.

Relative error is dimensionless. Its denominator is the magnitude of the reference, including when the reference is negative. At a zero reference it is undefined: a computed USD 0.20 against USD 0 still has absolute error USD 0.20, but there is no corresponding percentage error under this definition. Even exact agreement at zero does not make 0/00/0 a valid relative error.

Name the reference before judging the error

A closed-form model value, a carefully computed benchmark and an observed market quote answer different questions. The calculation above measures disagreement with the stipulated model reference. It does not establish that the instrument's true market value is known to within twenty cents. If the benchmark itself is approximate, report its precision or error bound separately.
Likewise, an observed absolute error is not a standard error, a confidence interval or an error bound. Those describe uncertainty or guarantees under additional assumptions. A small difference between two algorithms can also reflect a shared mistake.
For how input sensitivity and computation affect a valuation, continue to numerical stability. The SelfMadeQuant lesson Numerical Conditioning and Stability connects these distinctions to hedge ratios and covariance calculations; its preview is public and the full lesson requires membership.

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  • numerical methods
  • error analysis
  • pricing
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