Skip to Content
DocumentationAnalyticsVanilla Rate Swaptions

Vanilla Rate Swaptions


Vanilla rate swaptions on Aladdin can be valued using Normal (Bachelier), Black or SABR models. The default overnight model is the Normal model, and we focus on that here.

Let VRTPV_{RTP} represent a payer swaption price, then we have

VRTP=Am((F−X)(1−N(h))+σn(h)t)V_{RTP} = \frac{A}{m}\left((F-X)(1-N(h))+\sigma n(h) \sqrt{t}\right)

where

F=Forward swap rateX=Striket=Time to expiryh=F−XσtN(h)=St Normal CDFn(h)=St Normal PDFA=Sum of swap discount factorsm=Fixed leg pay frequency\begin{gathered} F = \text{Forward swap rate} \\ X = \text {Strike} \\ t = \text{Time to expiry} \\ h=\frac{F-X}{\sigma \sqrt{t}} \\ N(h) = \text{St Normal CDF} \\ n(h) = \text{St Normal PDF} \\ A = \text{Sum of swap discount factors} \\ m = \text{Fixed leg pay frequency} \end{gathered}

Although not present in the pricing formula, let SS denote the price of the underlying forward swap.

💡

Observation 1

For swaption delta, rather than the single rate FF, it is computed with respect to the (tradeable) underlying forward swap with price SS

Magnitude of delta value

On Aladdin the delta is defined with respect to the price of the underlying swap, SS. Since VV depends on SS implicitly, we can use the chain rule

Δ=∂VRTP∂S=∂VRTP∂F∂F∂S\Delta = \frac{\partial V_{RTP}}{\partial S} = \frac{\partial V_{RTP}}{\partial F} \frac{\partial F}{\partial S}

and this is very closely approximated as

Δ≈DV01VDV01S\Delta \approx \frac{\text{DV01}_{V}}{\text{DV01}_{S}}

Sign of delta value

Back in swap price space we can write

∂VRTP=Δ∂S=Δ(S+−S−)\begin{align*} \partial V_{RTP} & = \Delta \partial S \\ & = \Delta (S_{+}-S_{-}) \end{align*}

From this, we can see this swap price delta is the change in swaption price for a unit increase in the swap price.

💡

Observation 2

The price of receive fixed swap is pp and the price of a pay fixed swap is pp. There is only one price. The value of a receive fixed swap is Np/100Np/100 and the value of a pay fixed swap is −Np/100-Np/100.

Aladdin treats price as a fundamental metric and uses a one price, many values data model. This is because position quantity is aggregated up from trades, and value is derived from price combined with positions. This is part of the broader Aladdin data model.

Because of this, the price of the underlying swap is essentially the price of a receive fixed swap.

For the put (RTP) option, the price of the swaption will decrease when the price of the underlying receive fixed swap increases, hence the delta is negative. For the call (RTR) option, the price of the swaption will increase when the price of the underlying receive fixed swap increases, hence the delta is positive.

Example

Last updated on
Derivatives