Adjusting Caplet and Floorlet Pricing for Different Day-Count Fractions

# Cap & Floor pricing using distinct year fractions for floating and fixed rates


# Cap & Floor pricing using distinct year fractions for floating and fixed rates












The usual Black (lognormal) formula to price IBOR-linked caplets and floorlets (see e.g. Brigo & Mercurio textbook) reads $$ \textit{cf}(t;T_{i-1},T_i,K,\omega) = N P_d(t,T_i)\tau(T_{i-1},T_i) \times\textit{Black}\left[F_{x,i}(t),K,v_x(t;T_{i-1}),\omega\right], $$ with somewhat obvious notation (see below). This pricing formula makes the assumption that the year fraction $\tau(T_{i-1},T_i)$ associated to the floating and fixed legs of the underlying FRA (Forward Rate Agreement) is the same. Actually, this is not true for real market FRAs, where the the fixed leg has daycount 30/360 and the floating leg has daycount act/360 (at least in the EUR case, see e.g. LSEG/Refinitiv page ICAPEURO2).

Now the question is: how to modify the formula above to correctly take into account the appropriate market conventions? One possible solution could be to rescale the strike $K$ with the ratio between the fixed and floating leg year fractions, denoted by $\tau_K(T_{i-1},T_i)$ and $\tau_x(T_{i-1},T_i)$, respectively, as $$ K \rightarrow K' = K\frac{\tau_K(T_{i-1},T_i)}{\tau_x(T_{i-1},T_i)}. $$

This approach is correct from a mathematical point of view (if you disagree, please explain). Is it correct also from a trading point of view?

Notation: in the previous formula $N$ is the nominal amount, $t$ is the valuation date, $\left[T_{i-1},T_i\right]$ is the future interest rate period, $\tau(T_{i-1},T_i)$ is the year fraction, $K$ is the strike, $\omega=\pm1$ distinguishes between caplets and floorlets, $P_d(t,T_i)$ is the discount factor (taken e.g. from the EUR OIS curve), $F_{x,i}(t)$ is the forward rate (taken e.g. from the EURIBOR6M curve), $v_x(t;T_{i-1})$ is its lognormal variance for the period $\left[t,T_{i-1}\right]$ and $$\textit{Black}\left[F,K,v,\omega\right] = \omega \left\{F\Phi\left(\omega d^{+}\right)-K\Phi\left(\omega d^{-}\right)\right\},\\ d^{\pm} =\frac{\ln\frac{F}{K}\pm\frac{v}{2}}{\sqrt{v}},\\ v(t,T) = \int_t^T\sigma(u)^2\,du,\quad \sigma(t,T) := \sqrt{\frac{v(t,T)}{\tau(t,T)}} ,\\ \Phi(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-\frac{1}{2}y^2}\,dy,\\ dF_i(t) = F_i(t)\sigma(t)dW^{Q_d^{T_i}}(t) $$ is the usual Black formula, where $Q_d^{T_i}$ is the $T_i-$ forward measure associated to the numeraire $P_d(t,T_i)$.