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3.4. CHOICE OF THE WEIGHTS 111
Then the sequence {Q δ k
αk
} k≥1 where Q δ k
αk
is a solution of problem (2.27) with data C δ k
M , prior
P and regularization parameter α k chosen according to the discrepancy principle, has a weakly
convergent subsequence. The limit of every such subsequence of {Q δ k
αk
} k≥1 is a MELSS with
data C M and prior P .
The alternative principle of the preceding section does not carry over to non-attainable
problems as easily and is not discussed here.
3.3.3 Computing the noise level
If the bid and ask prices are known, the noise level δ and an estimate of option prices C δ M can
be computed directly using
CM(T, δ K) := Cbid M (T, K) + Cask (T, K)
, ∀ T, K,
2
δ := ‖Cbid M
− Cask M
Since for all i, the true option prices satisfy C M (T i , K i ) ∈ (C bid
C M ‖ w ≤ δ.
2
‖ w
.
M , Cask M
), we clearly have ‖Cδ M −
If the bid and ask prices are unavailable, one can assess the order of magnitude of δ directly
from C δ M , using the following heuristic argument. Suppose that the exact data C M is attainable
and that the errors in different option prices are independent. Then it is reasonable to assume
that the main part of error, present in C δ M
will lead to violations of arbitrage constraints on
option prices (e.g. convexity). Since the least squares solution Q + is an arbitrage-free model,
these violations of no-arbitrage constraints will contribute to the discrepancy ‖C Q+ − C δ M ‖2 w ≡
ε δ (0), as shown in Figure 3.3. Therefore, ε δ (0) will have the same order of magnitude as δ 2
and one can approximately take δ ∼ √ ε δ (0). Since the noise level δ is only used to choose
the regularization parameter α, we do not need to know it with high precision and this rough
estimate is sufficient.
3.4 Choice of the weights of option prices
The relative weights w i of option prices in the pricing error term (2.3) should reflect our confidence
in individual data points, which is determined by the liquidity of a given option. This