If I treat the 2 legs as bonds, and I want to calculate the present value somewhere between 2 payment date, should I calculate accrued interest?
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When you sum up all the discounted cash flows, you effectively a "dirty price" (i.e., inclusive of "accrued interest"). This quantity is used for marking to market or unwinding. You don't need to worry about "accrued interest" separately.
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$\begingroup$ Does that mean in a swap, the accrual interest for both legs are cancelled? $\endgroup$ Commented Apr 16, 2016 at 13:34
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$\begingroup$ @StudentT They won't cancel out completely, since the flt/fixed rates, day count conventions, payment frequency are all different. $\endgroup$– HelinCommented Apr 16, 2016 at 16:30
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$\begingroup$ Could you give an explicit formula? As far as I can see, the price at time $T_0 < t < T_1$ should be the sum of prices of a forward starting swap (i.e. the swap with the first resent date $T_1 > t$) and $P(t,T_1)\tau(T_0, t)(L(T_0, T_1) - K)$ where this last term is the PV of the accrued interest since the last resent date $T_0$ to be paid at time $T_1$. $\endgroup$ Commented Oct 1, 2020 at 8:58