# 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 ?

• If you already have Bloomberg then the easiest is if you just create a swaption via SWPM -OV 1Yx10Y USD <GO>, save it and then retrieve it in a week‘s time via the deal ID. SWPM will then take the vol surface and curves at that point in time to price the swaption. – oronimbus Jun 5 at 20:51
• Well yes, but no. Yes I know about it, but no because we want to value the swaption in our systems ... – 11house Jun 6 at 6:05

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 $$t_1$$.

• What is $s_t$ ? – 11house Jun 5 at 20:36
• It is the forward swap rate (for the underlying forward swap of the swaption) at time $t$, I edited my answer. – Olorin Jun 5 at 20:37
• Ok, thank you. How do you calculate it from the markets ? (At t=0 it is quoted ok, but for t>0 ?) – 11house Jun 6 at 6:06