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Quantitative Finance · Begrippenlijst

Wat is Rebonato’s formula?

Definition 8.10 Rates, Credit, XVA and Risk · Hoofdstuk 8 — Forward-Rate and Market Models

Rebonato’s formula approximates the Black volatility of the swaption expiring at TaT_a on Sa,bS_{a,b} in the market model by freezing the weights and the forwards at today’s values:

σa,b2 Ta≈1Sa,b(0)2∑i,j=ab−1wi(0)wj(0)Fi(0)Fj(0) ρij∫0Taσi(t)σj(t) dt.\sigma_{a,b}^2\,T_a \approx \frac1{S_{a,b}(0)^2}\sum_{i,j=a}^{b-1} w_i(0)w_j(0)F_i(0)F_j(0)\,\rho_{ij}\int_0^{T_a}\sigma_i(t)\sigma_j(t)\,dt .

Voorbeelden

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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