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Quantitative Finance · Glosarium

Apa itu Forward-rate correlation?

Definition 8.11 Rates, Credit, XVA and Risk · Bab 8 — Forward-Rate and Market Models

The forward-rate correlation ρij\rho_{ij} of a market model is the instantaneous correlation of the Brownian motions driving FiF_i and FjF_j. It is usually parametrised, for example ρij=e−β∣Ti−Tj∣\rho_{ij} = e^{-\beta|T_i-T_j|}, and reduced to two or three factors by keeping the leading eigenvectors of the matrix (the principal components of chapter 3), which speeds simulation and removes noise.

Correlation of the first annual forward with the others under the exponential parametrisation _ij=e- |T_i-T_j| for three values of . Data: the chapter’s tutorial.
Figure 8.4. Correlation of the first annual forward with the others under the exponential parametrisation ρij=e−β∣Ti−Tj∣\rho_{ij}=e^{-\beta|T_i-T_j|} for three values of β\beta. Data: the chapter’s tutorial.
Five-year-expiry swaption volatilities from two market models that reprice the same caplets. Decorrelating the forwards lowers the volatility of longer swap rates; Rebonato’s formula (lines) tracks Monte Carlo (marks). Data: the chapter’s tutorial.
Figure 8.5. Five-year-expiry swaption volatilities from two market models that reprice the same caplets. Decorrelating the forwards lowers the volatility of longer swap rates; Rebonato’s formula (lines) tracks Monte Carlo (marks). Data: the chapter’s tutorial.

Contoh

Example 8.12 (Same caplets, different swaptions)

Calibrate the model to the same caplets with β=0.02\beta=0.02 (forwards nearly perfectly correlated) and with β=0.40\beta=0.40 (the one-year and nine-year forwards correlated at 0.04). The swaption expiring in five years into one year is priced at 25.0% of Black volatility by both (it is one forward); into five years, at 22.1% and 17.3% (Figure 8.5). On EUR 100 million, at the money (forward 2.933%, annuity 4.102), that is EUR 2.35 million against 1.84 million: 22% apart for the same caplets. Monte Carlo confirms Rebonato’s formula to within its standard errors.

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