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One can write for the payoff of an swaption $$\sum_i\tau_i P_{i+1}(S_{\alpha,\beta}(T_\alpha)-K)^+ $$ and therefore the pricing equation follows Joshi's explainations. To derive the above equation use that the swap rate is given by $$S_{\alpha,\beta} = \sum_i \frac{\tau_iP_{i+1}}{\sum_i\tau_iP_{i+1}}F^i, $$ where $F^i$ are the corresponding forward rates. ...

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