Correlated Forward Rates and Swaption Pricing

Summary

The document explains why pricing a portfolio of swaptions by simulating several forward rates independently can be inconsistent. In the example, a swap rate depends on multiple forward rates, while under the relevant forward measure only one rate can retain lognormal dynamics. Simply assigning lognormal dynamics to all component rates therefore does not produce the assumed model.

An alternative is to model the swap rates directly, specify a plausible correlation structure, and simulate them jointly. Correlation affects the portfolio price, so its treatment should be informed by market data. The discussion is conceptual and gives no calibration method, empirical evidence, or specific correlation estimates; its example concerns a particular set of swap tenors and does not establish a universal model choice.

Key ideas

• A swap rate is built from multiple forward rates over the swap's tenor.
• Under a given forward measure, the described setup permits only one forward rate to have lognormal dynamics.
• Modeling several forward rates as independent lognormal processes can therefore be inconsistent with the measure assumptions.
• Directly modeling swap rates allows their joint correlation to be specified and simulated.
• Correlation assumptions affect the resulting swaption portfolio price and should reflect market data.

Tags

# Forward rates diffusion


# 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 ?

## Answer by Probilitator (score 1)

https://quant.stackexchange.com/a/11440

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.