# Valuation of option on amortized IR swap

I'm currently valuing swaptions using an implied volatility surface and Black's formula. This formula is given by

$$A (S\Phi(d_+) - K \Phi(d_-))$$

where $$d_{\pm} = \frac{\log\left(S/K\right) \pm \frac{1}{2}\sigma_{\text{impl}}^2 T_{\text{exc}}}{\sigma_{\text{impl}} \sqrt{T_{\text{exc}}}}\\ A = N\sum_{i}\alpha_i P(0, T_i)\\ S = \frac{P(0, T_{\text{exc}}) - P(0, T_{\text{final}})}{\sum_{i}\alpha_i P(0, T_i)}$$

The different symbols represent:

• $A$ = annuity
• $N$ = notional
• $T_{\text{start}}$ = exercise date of the swaption
• $T_{\text{final}}$ = date of final payment of the underlying swap
• $S$ = swap rate of the underlying swap
• $K$ = strike / accrual rate of fixed leg of underlying swap
• $\alpha_i$ = accrual length of each leg
• $P(0, T)$ = discount factor.

Finally, the volatility $\sigma_{\text{impl}}$ is provided by market data of swaption prices. For this I use some interpolation technique. This interpolation technique uses the exercise date of the swaption and the tenor of the underlying swap (tenor = time between first and final payment). For simplicity I only consider the ATM volatility (which is still different for swaptions with different maturities / tenors).

I want to use this formula (or something similar) to price options on swaps with an amortizing profile. For an amortizing swap the notional decreases over time (i.e. for each leg we have a different notional). You can define a swap rate by

$$S^{\text{amort}} = \frac{\sum_i N_i \alpha_i P(0, T_i)F(0, T_{i-1}, T_i)}{\sum_i \alpha_i N_i P(0, T_i)}$$ and similarly for the annuity. Here $F$ are the forward rates.

But now the problem is that there is no volatility surface for swaptions on amortizing swaps. So what implied volatility should I use? Ideally I would like to extract the volatility from the implied volatility surface that I already have (based on non-amortizing swaptions).