# Forward rates diffusion

I used a simple market model (Black 76) to price an american swaption.

It's a formula similar to B&S, with another numeraire and forward rate as underlying.

I used the SDE: $$dF = \sigma * F dW$$

Now I want to price a contract wich is the sum of 4 swaptions with different Tenors, so I have to simulate 4 forward rates.

Should I take into account the correlation between the forward rate or can I simply simulate the rates independently ?

• @Imorin why would you need to simulate at all - Black's formula offers a closed form solution !! for swaption prices – Probilitator May 25 '14 at 8:52
• Thus the price of your contract would just be the sum of the four swaption prices calculated via Black's formula. You might need to get different vols in case of different maturities. – Probilitator May 25 '14 at 8:54
• I use simulation to price an american swaption – lcrmorin May 25 '14 at 14:46
• do use finite differences or monte-carlo least squares or a tree ? – Probilitator May 25 '14 at 15:52
• you should perhaps complement you question by mentioning that you are pricing american-style swaption. Otherwise most people we react as I did – Probilitator May 25 '14 at 15:53

The filtration is hardly the problem.

Let's say you want to price a 1x4 and a 2x3 years swaption. Thus you model three forward rates $L(t,T_1,T_2), L(t,T_2, T_3), L(t,T_3,T_4)$

The swaprate $S_{\alpha,\beta}(t)$ depends on the forward rates $L_i(t,T_{i-1},T_i)$ with $i \in (\alpha+1, \dots, \beta)$

Thus the price of the 1x4 swaption given by $P(0,T_1)E^1[(S_{1,4}(T_1)-K)^+]$. Under the $Q^1$ measure only one rate can have the log-normal dynamics !! Thus you can't just model the three forward rates as log-normal !!!!

What you can do however is to directly model the swap rates $S_{1,4}, S_{2,4}$. You can assume log-normal dynamics for both and simulate them with whatever correlation structure you think is plausible. In this case you can also just add the expectations.

Whether you model the rates as correlated or not depends on your market data !! Obviously the correlation structure will have an effect on the price.

• I may have mixed L and S... – lcrmorin May 26 '14 at 20:24
• I don't understand why there is a difference between modelling S and L. Why Under the Q1 measure only one (forward ?) rate can have the log-normal dynamics ? – lcrmorin May 26 '14 at 20:29
• the swap rate is different quantity than the forward rate (but they coincide in case of a one period swap) Perhaps this can help: citeseerx.ist.psu.edu/viewdoc/… – Probilitator May 27 '14 at 7:08
• The difference is clear now, but not the reason why $S_{i,j}$ could all be log normal but not the $L_{i,j}$ – lcrmorin May 27 '14 at 20:39
• both can - but with the $S_ij$ each swap-rate could be modelled as being log normal. You can however model only one forward rate as log normal if you want to price swaption by simulating forward rates – Probilitator May 27 '14 at 20:42