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Quantitative Finance · Glossaire

Qu'est-ce que « Cross-gamma » ?

Definition 3.13 Rates, Credit, XVA and Risk · Chapitre 3 — Rates Risk

The cross-gamma of a position between inputs kk and ll is Γkl=∂2V/∂qk∂ql\Gamma_{kl} = \partial^2V/\partial q_k\partial q_l for k≠lk\ne l, per square basis point; with the diagonal terms it forms the gamma matrix, and the second-order P&L is ∑kΔkδqk+12∑k,lΓklδqkδql\sum_k\Delta_k\delta q_k + \tfrac12\sum_{k,l}\Gamma_{kl}\delta q_k\delta q_l.

Exemples

Example 3.14 (The second-order term of the hook)

On the move of 21 October 2022 the book’s ladder gives a first-order loss of exactly USD 3 000 000 (by construction); full revaluation gives −3 077 970-3\,077\,970. The gamma matrix, computed by central differences, gives 12 δq⊤Γ δq=−78 189\tfrac12\,\delta q^\top\Gamma\,\delta q = -78\,189, which accounts for the difference to within USD 220. The book’s parallel gamma is −377-377 dollars per square basis point: it is short convexity, mostly through its thirty-year payers.

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