SABR Calibration Consistency and Path-Dependent Swaption Pricing
Summary
The document clarifies why SABR’s usefulness for path-dependent interest rate options depends on how its parameters are calibrated. SABR is a stochastic volatility model that can be simulated, while its common fast approximations target terminal distributions and European swaption prices. The difficulty is not that simulation is impossible, but that practitioners often fit different SABR parameters to options with different expiries.
For a Bermudan swaption or another path-dependent claim, using parameters fitted at the final expiry can imply short-horizon dynamics that conflict with market prices for near-term options. Using short-expiry parameters throughout can instead miss the longer-expiry terminal distribution. The response presents local volatility as a relatively simple way to construct dynamics consistent with vanilla prices across expiries, while noting Bermudans are often priced on grids. A single SABR parameter set can support consistent paths, but may fit observed market prices poorly. Model choice therefore involves a tradeoff between path consistency and calibration quality.
Key ideas
- SABR can be simulated, but its standard approximations focus on terminal distributions and European option prices.
- Expiry-specific SABR calibrations can imply inconsistent dynamics across a simulated path.
- Path-dependent pricing requires dynamics that remain coherent with market information at different horizons.
- Local volatility offers a relatively simple way to match vanilla prices across expiries, though Bermudans are commonly priced on grids.
- A single SABR parameter set can improve path consistency at the cost of weaker market fit.
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# Path Dependent Options - Which choice of model?
# Path Dependent Options - Which choice of model?
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 by Brian B (score 4, accepted)
https://quant.stackexchange.com/a/22404
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.
## Answer by kolbe (score 0)
https://quant.stackexchange.com/a/71932
You aren't wrong that SABR can be used in the same way, but SABR is really bad a modeling a full path. For example, go ahead and calibrate alpha-v-rho to market prices across different expirations of the same security. You'll see that the values are wildly different. I haven't done this experiment in a while, but IIRC in SPX, you'll find a vol of vol of like 3 for a one week option, but 0.5 for a one year option.
Why is this bad? well, because the process that generates values for the one week options should be the same as (or at least similar to) the first week of the process for the yearly options. So, if you're trying to price a derivative that's a year out using the calibration from the 1-year fit, then the dynamics represented in the first week of the path will be wildly inconsistent with the reality that the one week options are implying.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.