Revaluing a Swaption as Its Expiry Approaches Using the Volatility Cube

# Swaption valuation across time using vcub


# Swaption valuation across time using vcub












On Bloomberg one has access to the rates vol cube with the VCUB function. For a given currency, today, one sees Black implied volatilities for swaptions of various expiries and strikes, for forward swaps of various tenors. Suppose I see 80% of implied volatility for a 1y10y payer physically settled swaption. This means that the corresponding market price is $$\textrm{forward swap annuity} \times \textrm{Black}(s_0,80\%,\ldots)$$ where $\textrm{Black}$ is the Black function and $s_0$ is today's 1Y forward 10Y swap value and the $s_0$ quote is available on the market (on Bloomberg).

Now imagine I come the next week (in 7 days), and I want to value the same swaption in the Black model again. How can I do this using the vol cube in 7 days ?

## Answer by Olórin (score 4, accepted)

https://quant.stackexchange.com/a/45958

Forget for a moment that your option is delivering the immediate entrance in a swap (if the swaption is physically settled) or the cash amount of the swap (if the swaption is cash-settled), as your question doesn't depend on this fact, and take a "general" 1Y option.

Your today's (date $t_0$) cube loses the "swap tenor dimension" and becomes a today's implied volatility surface, on which you read (through Black-Scholes function) the price of your option through implied volatity for 1Y expiry and given strike.

In 1W (date $t_0$+1W) your option will be an option on the same underlying (work it out in the swaption case) with same strike $K$ but expiry "1Y minus 1W" (date $t_1$). So to value your option 1W after, you need to know the implied volatility at $(t_1, K)$. And it this one is note quoted, you'll have to resort probably to interpolation, or even extrapolation.

To make it simple, the time $t$ price of the option is

$$\pi_t (T,K) = \textrm{Black}\left( \hat{\sigma}_t (T,K), T-t, K, s_t \right)$$

where $\hat{\sigma}_t (T,K)$ is the time $t$ implied volatility for expiry $T$ and strike $K$ (and swap tenor $10$Y) and where $s_t$ is the forward swap rate (for the underlying forward swap of the swaption) at time $t$.

As I said the fact that $\hat{\sigma}_{t_0} (T,K)$ is quoted (i.e. is directly readable on VCUB) doesn't imply that $\hat{\sigma}_{t_1} (T,K)$ will be, hence you'll probably have to resort to some interpolation to get $\hat{\sigma}_{t_1} (T,K)$ from values observables in VCUB at $t_1$.