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# Swap market model

To price the Swaption we use that
\[ \text{d} y_t[\bar{S}]=\frac{p(t,S_0)-p(t,S_n)}{\sum_{i=1}^n(S_i-S_{i-1})p(t,S_i)}=\frac{p(t,S_0)-p(t,S_n)}{p(t,\bar{S})}.\]
We have that $\sum_{i=1}^n(S_i-S_{i-1})p(t,S_i)$ is a positive sum of traded assets. It can therefore by used as a numeraire. Let $\mathbb{Q}^{\bar{S}}$ be the corresponding numariare measure, usually called the SWAP-measure. Under $\mathbb{Q}^{\bar{S}}$ we have that the SWAP-rate $y_t[\bar{S}]$ is the ratio of a traded assets and the numeraire. So it should
be a $\mathbb{Q}^{\bar{S}}$-martingale.

So we assume the following $\mathbb{Q}^{\bar{S}}$-dynamics: \[ \text{d} y_t[\bar{S}]= y_t[\bar{S}]\sigma(t,\bar{S})\text{d} W^{\mathbb{Q}^{\bar{S}}}_t,~0\leq t \leq S_0\] where $W^{\mathbb{Q}^{\bar{S}}}_t$ is $\mathbb{Q}^{\bar{S}}$ Brownian motion. The SWAP market model is not consistent with the LIBOR market model.

Having LIBOR rates as geometric Brownian motions (log normal distribution) will not make the SWAP rate a geometric Brownian motion under the SWAP measure.

So if we are to price caps, floors and SWAPtions at the same time we have to choose which framework to use.

However if we just want to price Swaptions we can use the Swap market model without any problem (see Björk 27.11-27.12).

So we assume the following $\mathbb{Q}^{\bar{S}}$-dynamics: \[ \text{d} y_t[\bar{S}]= y_t[\bar{S}]\sigma(t,\bar{S})\text{d} W^{\mathbb{Q}^{\bar{S}}}_t,~0\leq t \leq S_0\] where $W^{\mathbb{Q}^{\bar{S}}}_t$ is $\mathbb{Q}^{\bar{S}}$ Brownian motion. The SWAP market model is not consistent with the LIBOR market model.

Having LIBOR rates as geometric Brownian motions (log normal distribution) will not make the SWAP rate a geometric Brownian motion under the SWAP measure.

So if we are to price caps, floors and SWAPtions at the same time we have to choose which framework to use.

However if we just want to price Swaptions we can use the Swap market model without any problem (see Björk 27.11-27.12).

Questions: Magnus Wiktorsson

Last update: 2014 Aug 13 15:22:37. Validate: HTML CSS

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