ConvexityAdjustment

Closed-form convexity adjustments — the corrections that turn a rate quoted or observed in the "wrong" measure into its forward value.

Eurodollar futures: the futures rate exceeds the forward rate because a futures contract is marked to market daily; the Hull-White one-factor adjustment reduces to sigma^2 t1 (t1 + 2 tau) / 2 as a -> 0 (the rigorous futures-rate convexity — Hull's textbook sigma^2 t1 t2 / 2 drops one sub-term).
LIBOR in arrears: a rate that pays on its own fixing date rather than one accrual period later is worth more than its forward, by tau L0^2 (e^{sigma^2 T} - 1) / (1 + tau L0) (exact for a lognormal forward).
CMS: a swap rate paid once rather than as an annuity carries a convexity adjustment -0.5 y0^2 sigma^2 T G''(y0)/G'(y0), with G the flat-yield annuity function.

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Methods

static FuturesConvexity eurodollarFutures(double meanReversion, double sigma, double t1, double t2)

Hull-White one-factor convexity adjustment for the rate covering [t1, t2] implied by a Eurodollar future expiring at t1:

CA = (1/(t2-t1)) * (sigma^2/(2a)) * B(t1,t2) * [ B(t1,t2)(1 - e^{-2 a t1}) + a B(0,t1)^2 ]

the rigorous E^Q[(1/P(t1,t2) - 1)/tau] - forwardRate to first order, which tends to sigma^2 t1 (t1 + 2 tau) / 2 as a -> 0. (The 1/P(0,t2) scaling — a fraction of a percent — is dropped to keep the signature curve-free.)

meanReversion: Hull-White a (>= 0; near-zero gives Ho-Lee)
sigma: absolute (normal) short-rate volatility
t1: futures expiry / rate start, years
t2: rate end, years

static Adjustment inArrears(double forward, double blackVol, double accrual, double fixingTime)

LIBOR-in-arrears adjustment: a forward rate forward that fixes at fixingTime and pays then (rather than at fixingTime + accrual) has expected value forward + L0^2 sigma^2 tau T / (1 + tau L0) under the pay-date measure.

forward: the forward rate L0
blackVol: the lognormal (Black) volatility of the forward rate
accrual: the accrual fraction tau
fixingTime: the fixing time T in years

static Adjustment cms(double forwardSwapRate, double blackVol, double expiry, int paymentsPerYear, int numberOfPayments)

CMS convexity adjustment (the convexity term only) for a swap rate observed at expiry and paid once: -0.5 y0^2 sigma^2 T G''(y0)/G'(y0), with the flat-yield annuity G(y) = sum_{i=1}^{n} (1/m) / (1 + y/m)^i.

forwardSwapRate: y0
blackVol: lognormal vol of the swap rate
expiry: observation time T in years
paymentsPerYear: m (e.g. 2 for semi-annual)
numberOfPayments: n (tenor in years times m)