### Multi-factor interest rate models

There are at least three things that make interest rate derivative modelling really hard. It has all the problems of equity derivative modelling, including dealing with the volatility surface,--and data for out of the money implieds isn't often plentiful. It concerns a curve rather than a single underlying, and there are various obvious constraints, like non-negative (forward) Libors. Lastly, there are two calibration instruments, caps and swaptions, which often seem not wholly compatible.

Which problems people have chosen to work on is always insightful. The game of theory building, at least in academia, rewards complete less descriptive models more highly than insightful model fragments. Thus there has been a lot of work on models which use ever more refined processes for the short rate, and quite a lot on stochastic models of the full curve under a few (typically at most 3) sources of uncertainty. Stochastic and to a lesser extent local volatility models are catching up fast, so we are starting to see Libor Market Models - that is models where the variables are the forward Libors under a suitable numeraire - with stochastic volatility.

But it strikes me that this may be an example of where setting up the rules so you have to have a model that actually prices something before you can publish is holding us back. We do not need yet another Brownian style coupled spot process and vol process model, Markovian or otherwise. What we need is some fresh thinking about the sources of uncertainty in interest rates beyond one variable for level, the second for tilt and the third for curvature.

For instance, just to throw something in, suppose we have stochastic variables for each caplet where the next maturity caplet mean reverts to the level of the last one, with progressively increasing mean reversion constants.

Thus we start with a standard CIR model of the short rate

dr = (e - g r) dt + s r^a dW

Where a is often 1/2, and generalise this to processes for each forward Libor r_i

dr_i = (f_i - g_i r_i) dt + s_i r_i^a dW_i

Here the terms g_i are constant; f_i = e_i r_{i-1} for constant e_i, so that the i+1 th Libor reverts to a constant times the i th Libor; and dW_i is a random walk. We would expect that the vols s_i decrease as i increases.

One of the nice things about this kind of setting is that the correlation structure between the forward Libors is built in through the mean reversion, rather than through some (rather artificial to my mind) condition like dW_i dW_j = rho_ij dt. In fact, this might well perform reasonably well with dW_i dW_j = 0 for i not equal to j.

I would much rather read about one of those, even if the authors hadn't figured out how to price Bermudan swaptions with it yet, than yet another variation on (the admittedly rather useful but conceptually rather barren) LMM.

Which problems people have chosen to work on is always insightful. The game of theory building, at least in academia, rewards complete less descriptive models more highly than insightful model fragments. Thus there has been a lot of work on models which use ever more refined processes for the short rate, and quite a lot on stochastic models of the full curve under a few (typically at most 3) sources of uncertainty. Stochastic and to a lesser extent local volatility models are catching up fast, so we are starting to see Libor Market Models - that is models where the variables are the forward Libors under a suitable numeraire - with stochastic volatility.

But it strikes me that this may be an example of where setting up the rules so you have to have a model that actually prices something before you can publish is holding us back. We do not need yet another Brownian style coupled spot process and vol process model, Markovian or otherwise. What we need is some fresh thinking about the sources of uncertainty in interest rates beyond one variable for level, the second for tilt and the third for curvature.

For instance, just to throw something in, suppose we have stochastic variables for each caplet where the next maturity caplet mean reverts to the level of the last one, with progressively increasing mean reversion constants.

Thus we start with a standard CIR model of the short rate

Where a is often 1/2, and generalise this to processes for each forward Libor r_i

Here the terms g_i are constant; f_i = e_i r_{i-1} for constant e_i, so that the i+1 th Libor reverts to a constant times the i th Libor; and dW_i is a random walk. We would expect that the vols s_i decrease as i increases.

One of the nice things about this kind of setting is that the correlation structure between the forward Libors is built in through the mean reversion, rather than through some (rather artificial to my mind) condition like dW_i dW_j = rho_ij dt. In fact, this might well perform reasonably well with dW_i dW_j = 0 for i not equal to j.

I would much rather read about one of those, even if the authors hadn't figured out how to price Bermudan swaptions with it yet, than yet another variation on (the admittedly rather useful but conceptually rather barren) LMM.

Labels: Financial Models, Markets, Random Process

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