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3

As the swap rate is not tradable, the delta hedge ratio with respect to the spot swap rate is not really useful. However, note that \begin{align*} V_0 &= \sum_{i=\alpha+1}^{\beta}\tau_i P(0, T_i)\big[S_{\alpha, \beta}(0)N(d_1) - k N(d_2) \big]\\ &= \sum_{i=\alpha+1}^{\beta}\tau_i P(0, T_i) S_{\alpha, \beta}(0)N(d_1) - N(d_2) k ...


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You are asking about the term structure of lognormal implied volatilities for European swaptions, which is a two dimensional function (expiration and tenor). First expiration: typically (but not always), implied volatilities are increasing in the 0 to 6 month sector, because the immediate future is often more predictable than the medium term. At some ...


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Pricing via characteristic functions arises naturally in models that involve Levy processes. Therefore I can see how Black's formula for swaptions can be generalized for Levy dynamics: As in Black's model take the annuity as numeraire, and define the relevant measure $Q$ Black assumes that under this measure the swap rate is martingale GBM, that is to say ...


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I have traded swaptions for many years. The answer is that it is not possible to calculate exactly the implied volatility for a European option on an amortizing swap from the matrix of non-amortizing swaption volatilities. This is because there is a dependence on the correlation structure in addition to the volatility structure. Depending on the nature ...


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from a practitioner perspective, i can say there's no such thing as a 0 year swap (obviously). The shortest tenor that you could trade would be a contract on one month LIBOR or more likely 3 month LIBOR. Then the instrument you are asking about is a 5 year expiration caplet (payoff in 5 years = max (0, LIBOR- strike).)


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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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