Pricing Swaptions from Black Implied Volatility Quotes

# Swaption : Bloomberg Black implied volatility quotes and pricing in the Black model


# Swaption : Bloomberg Black implied volatility quotes and pricing in the Black model












I used a lot Bloomberg's VCUB for data, but never used its integrated swaption pricer "Quick Pricer for Swaptions", nor Bloomberg's "full" swaption pricer from "SWPM -OV".

I am retrospectively quite puzzled by how they quote the swaptions implied volatilities as well as by the results of their Quick Pricer for Swaptions. These doubts puzzle me in turn about how the markets themselves quote swaption implied volatilities. All notations used below are defined in the end "Notations" section. (Especially the two "black" prices, I guess they are correct, even if I now have doubts about the cash-settled one, about the rightful presence of the $P_{0,T_{\textrm{exp}}}^{OIS}$ bit.)

Concerning the results of their Quick Pricer for Swaptions :

This is for EUR swaptions (they are still cash-settled in VCUB, and will apparently be physically-settled as of mid june according to Bloomberg) as of 20190603 (3rd june) the volatility being indeed a Black volatility in %. I am pricing a 1Y into 10Y ATM payer (I would have to pay the fixed rate) swaption.

Applying Black Formula (for cash-settled swaption) from the notation section I find that the black bit is equal to 0.0026425037403560968, and that the cash-settled annuity is equal to 9.01629985437 (for a spot forwar start rate equal to 2.2089%). Doing the product I find 0.3222, and multiplying this by the zero coupon bit $P_{0,T_{\textrm{exp}}}^{OIS}$ (equal to 1.0166666666666666) won't make it for sure near the 2.36 from the picture.

Notations.

$$ \mathscr{N}(z) \equiv \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z} e^{-\frac{t^2}{2}} dt $$

$$ d\left(k,s,\nu,\varepsilon\right) \equiv \frac{\ln\left(\frac{s}{k}\right) + \varepsilon\frac{\nu^2}{2}}{\nu} $$

$$ \textrm{Black}\left(k,s,\nu,\varepsilon\right) \equiv \varepsilon\left( s \mathscr{N}\left( \varepsilon d\left(k,s,\nu,1\right)\right) - k \mathscr{N}\left( \varepsilon d\left(k,s,\nu,-1\right)\right) \right) $$

For a fixed leg schedule $\{T_0, T_1, \ldots, T_n\}$ of a forward swap starting at the swaption's expiry, one notes $\delta_i = \delta_{T_{i-1}; T_i}$ the year fraction represented by $[T_{i-1},T_i]$ (and calculated for the basis of the fixed leg of the forward swap), and :

$$ A_t^{\left[ T_0,\ldots,T_n \right]} \equiv \sum_{i=1}^n \delta_i P_{t,T_i}^{\textrm{OIS}} $$ the usual (physically-settled) annuity

and

$$ C^{\left[ T_0,\ldots,T_n \right]}(T,x) \equiv \sum_{i=1}^n \frac{\delta_i}{\left( 1 + \delta_i x \right)^{\delta_{T,T_i}}} $$ the cash-settled annuity, where $T \mapsto P_{0,T}^{\textrm{OIS}}$ denotes the OIS discount curve.

For a given strike $K$, option expiry $T_{\textrm{exp}}$ and implied volatility $\hat{\sigma}_{T_{\textrm{exp}}, K}$ for physically settled swaption the corresponding price of a physically-settled swaption is :

$$ \pi_0^{p.s.,\textrm{Mkt}} = A_{0}^{\left[ T_0,\ldots,T_n \right]} \times \textrm{Black}\left(K,s_0,\widehat{\sigma}_{T_{\textrm{exp}},K}\sqrt{T_{\textrm{exp}}},\varepsilon\right) $$

If the implied volatility is for cash-settled swaptions then the corresponding price of a cash-settled swaption is :

$$ \pi_0^{c.s.,\textrm{Mkt}} = P_{0,T_{\textrm{exp}}}^{OIS} \times C^{\left[ T_0,\ldots,T_n \right]}\left( T_{\textrm{exp}}, s_{0} \right) \times \textrm{Black}\left(K,s_{0},\widehat{\sigma}_{T_{\textrm{exp}},K}\sqrt{T_{\textrm{exp}}},\varepsilon\right) $$ (In both previous formulas $s_0$ is the forward swap rate at the pricing date.)