Differences between main classes of interest pricing derivatives models

Question

There seems to be 3 main classes of interest rate pricing models: 1) Short rate models, 2) Heath Jarrow models and 3) Libor Market Model. My book doesnt seem to explain why we need all these different models, when they are appriopriate for use and what the advantages and disadvantages of these models are?

Could someone please summarise the difference between the various interest rate pricing models and their advantages and disadvantages?

Answer 1

I am not sure if you can classify it like that. Mind you, I never wrote a book. I'll write what I know below and you can decide if the classification makes sense or not.

1 ) STIR: as the term indicates - short term like Eurodollar frequently modelled with Black or Bachelier (normal) model. HW1F is also a short rate model.

2 ) HJM is a framework (M is not model but Morton): CIR, HW-1F and HW-2F can be shown to be part of it @Gordon shows this for HW1f

It should be clear now, why I do not agree with the postulated classification.

Advantages vs disadvantages are similar to any other derivative model. If you price something simple, do not overcomplicate things. Price a vanilla European call in FX with Garman Kohlhagen or Black. A complex chooser TARF with SLV. Ideally, if calibration works, you price vanilla instruments consistently but it is computationally a completely unnecessary task. In terms of Greeks it becomes even more problematic if you use complex models for vanilla instruments.

In FI, anything including capped/floored floaters or digitals should be done by using vanilla models in my opinion (normal, black, and via replication if needed).

3 ) LMM is the IR equivalent of what the SLV model is for FX - market standard generally (at least very widely used).

Benefits?

• it simulates genuinely quoted market rates of interest (LIBOR) as opposed to unobservable, instantaneous rates of interest (HJM framework)
• calibrates to the skew observed in OTM swaptions: especially shifted LMM (short rate models cannot although strike dependent IVOL is backed out via Black - again a layover in classification)
• creates realistic correlation structure
• ...

Uses? complex products like (Bermudan) CMS spread options/caps/floors, step up callables etc.

Downside? It is complex; to implement but also to use. It is not like Black (or even SLV) where you mainly input your terms (K,t,..) and you get your price. You need to select the appropriate calibration instruments based on the structure you price. Typical choices range from Swaptions (full spectrum, upper triangle etc), caps or caplets, CMSSO (single look and multilook) if you have access to quotes, ...

Edit:

Generally, I would suspect that book should go through the model and the processes. If not, it is probably just an intro that explains the intuition and the book is a bit too sketchy.

HJM:

• unobservable rate (not desirable): LMM simulates genuine market rates
• non-markovian makes estimation of model parameters difficult, also here
• since it is a framework, lots of models fit into it
• @ir7's comments: it is multi factor in it's generic framework (HW1F obviously is not).

Why I wrote it is like SLV?

• First, what is the SLV? It combines LV (not really a model, just uses vanilla surface to get a grid) with SV (in a nutshell, BSM with a separate stochastic process for vol, hence multiple dynamic factors).
• Shortcomings SLV tries to address?: BS does not price exotic option well. LV calibrates nicely to vanilla but the calibrated leverage surface is typically observed to flatten with maturity which means the forward volatility smile will be less convex than on the initial pricing date and you will not be pricing deals properly which are primarily sensitive to forward volatility skew and smile (cliquets and co). SV prices for barriers and touches tend to be overvalued by SV (undervalued by LV). In any case, Once calibrated to the vanilla market, LV and SV offer no extra flexibility in matching the dynamics of implied volatility.
• Why SLV solves this? Appropriate calibration of the mixing parameters will allow you to closely match market quotes for exotic products.
• Is it used? Yes, de facto the market standard for all exotic FX option pricing (there are many flavours and ways to combine these, also how to construct local vol, stochastic vol etc, so this is very sketchy and there are better and worse models).

(Shifted) LMM

• Vanilla models (Black, Normal) and short term interest rate models will not price exotic products properly.
• What it does: attempts to match ATM/OTM-swaption/cap/caplet volatility and CMS spread vol/correlation/option at calibration.
• Why? This is needed to properly price derivatives depending on this market data. If you ignore important aspects (skew, correlation etc) it will still price, just classic GIGO.

I like this tweet. If you use Local Vol Monte Carlo, you will get a price for essentially every underlying that can be priced and any product you can define a payoff for. You will get expected values from the MC runs, compute cashflows and discount. However, you can also model something by simply assuming next days price will be equal to todays price. As silly as this sounds, that is actually the way to forecast a random walk and works remarkably well in some circumstances (see the section with Kenneth Rogoff and Richard Meese). Also, as mentioned by @noob2, standard HJM is non-markovian, which may explain why it is not as popular as the LMM.

So LMM achieves what SLV is doing. Try to model the underlying processes in a way that is consistent with market prices. Both are the most widely used models for pricing exotic options in their respective asset classes. SLV or LMM is not like BSM though. The latter is simple and there is no variation (ignoring american option pricing and term structure BSM but my point is that everyone uses the same formula). The complexity of SLV and LMM allow for numerous ways to implement these.

Now, one could argue that it's a chicken or the egg dilemma but markets price products in a way that whoever sells them does not constantly lose money. Complex structures have complex problems that define what the "fair" price should be.

LMM is really a set of models combined together: correlation (frequently using a method proposed by Rebonato), the vol model and shift model. So it's a complex system of coupled stochastic differential equations where Brownian motions exhibit a forward-forward correlation structure.

In a super short summary. In my opinion, everything goes back to Bachelier. He just never got the credit he would deserve in my opinion. No matter what model, if you turn all the additional features off, you essentially end up with his ideas. The reason more complex models exist is to overcome shortcomings.

You assume the forward (swap rate) obeys the stochastic differential equation of geometric Brownian motion. Black assumes volatility is independent of the strike (usually not true, which is why you have a vol skew/smile). CEV and SABR formulate ways to interpolate the vol smile (which is needed to calibrate). For example, looking at SABR (not using LMM as it is so much more nuanced and complex), setting vol parameter $\nu$ and correlation coefficient $\rho $ to zero, it reduces to CEV. If $\beta = 1$ in CEV it becomes regular Black SDE. Not using lognormal but normal -> Bachelier.

A quick attempt to explain some LMM dynamics. LMM tries to best match given smiles (market quotes) using shifted-lognormal skew (skew between Black and Normal, again Bachelier). $\alpha = 0%$ implies Black skew and $\alpha =\infty$ represents the Normal skew.

Bottom line, one book will never explain LMM, maybe The SABR/LIBOR Market Model: Pricing, Calibration and Hedging for Complex Interest‐Rate Derivatives answers a lot of questions. However, I use that model, but would never be able to implement this. Generally, these models are constantly enhanced and refined and it takes a large number of very smart people (predominantly, if not entirely with a non finance backgrounds like physics, computer science and the like). Maybe there is someone in this forum who actually is contributing to developments who is willing to outline how much time they spend(t) setting up LMM.

Answer 2

Just an addendum to the above answers and comments:

• The main decision is whether to use single or multiple factor dynamics.
• LMM models term forward rates. HJM models instantaneous forward rates.
• The main disadvantage of HJM, high-dimensional stochastic process as underlying, was overcome by Cheyette, back in 1994, by restricting the general HJM model to a subset of models with a specific parametrization of the volatility function. See for example Ingo Beyna's book Interest Rate Derivatives Valuation, Calibration and Sensitivity Analysis for a comprehensive review (and recent tractability improvements). Multifactor Gaussian Markovian HJM models (including linear ones) are still used.
• For current needs, risk free rates (RFR), see Mercurio and Lyashenko's Looking Forward to Backward-Looking Rates: A Modeling Framework for Term Rates Replacing LIBOR, which is an extension of LMM model providing additional information about the rate dynamics (extended HJM, locally-equivalent Cheyette model) not accessible in the LMM.