Valuation of option on amortized IR swap

Question

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).

Answer 1

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 of the amortization, the correlation effect can be large or small. For example, a 1yr option on a swap whose notional amortizes 50% after year 3 and 50% after year 4, will be very closely approximated by taking the average of the 1yr-3yr and 1yr-4yr European swaption volatilities. However a 5 year option on a swap which amortizes by 5% every year from year 6 to year 26, will need to take into account the correlation surface. This correlation effect should reduce the implied volatility of the amortizing swap rate, relative to the naive calculation, since it is similar to an option on a basket. Quantifying this would require some study of the intracurve correlation structure.