Path Dependent Options - Which choice of model?

Question

Can someone please help elaborate/clarify the below statements? I've heard about them from people but would like to know some more detail behind these statements..

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1) SABR is not useful in pricing path dependent options like bermudan swaptions, because it only models the terminal distribution of forward rates.

(does that just mean the model is SIMPLY not descriptive enough, because it does not have a mean reversion parameter to describe conditional expected forward rates?)

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2) Local Vol models are useful to price path dependent options because it provides the conditional (local) volatilities at each projected node on the tree, this is aside from the fact that the model is easier to calibrate and hit vanilla market prices.

(again since a local vol model is just a simpler version of a stochastic vol model, can't a stoch vol model like SABR also build up a lattice/tree of projected rates used to price them?)

Thanks

Davide

Answer 1

The SABR framework is really two things

• A stochastic vol model of forward rates, which is most certainly amenable to Monte Carlo simulation
• Reasonably accurate high-speed approximations of the terminal distribution and therefore european swaption prices

There's no problem (in theory) applying Monte Carlo to a SABR model: you just need to simulate the two-dimensional process

$$ dF = \sigma F^\beta dZ_1 \\ d\sigma = \alpha \sigma dZ_2 \\ <dZ_1,dZ_2> = \rho $$

with the usual caveats about bias in Euler integration, etc.

The main trouble with this in practice is that users of SABR employ different values of $\alpha, \beta, \rho$ at different time horizons, calibrated to observed swaption prices for the corresponding tenor. That's not a problem for vanilla options, but for path-dependent cases you no longer know what parameters to use at tenors beyond the shortest horizon.

Let's say you have such a set of calibrations $$\alpha_{\tau_i}, \beta_{\tau_i}, \rho_{\tau_i}$$ for $i=1,\dots,N$ and some path dependent option of tenor $\tau_N$. When your path simulation is at time $t=\tau_1$ should it be using $\alpha_{\tau_1}$ or $\alpha_{\tau_N}$?

The former makes the path dynamics consistent with market information observed around that time, but will then disagree about the terminal distribution for the latter, meaning european swaptions will be mispriced. The latter is inconsistent with the market information at $t$.

What you would like is something consistent with all market observations at every tenor. That way all european swaption prices will be recovered, and yet path dependent options can be priced as well. A local vol model is the (stochastically) simplest way to achieve that.

Bermudan swaptions are usually priced on grids rather than via Monte Carlo simulations but the consistency principles remain the same.

It is possible to choose a single set of $\alpha, \beta, \rho$ to describe the market as closely as possible, which then makes consistent simulations and grids feasible, but you will find that it does not match market prices all that well.