what interest rate models to use - buy side versus sell side
TRANSCRIPT
Electronic copy available at: http://ssrn.com/abstract=1723924
What Interest Rate Models to Use? Buy side versus Sell side
Sanjay K. Nawalkha* University of Massachusetts, Amherst
and
Riccardo Rebonato**
RBS and Oxford University
Abstract Does the selection of a specific interest rate model to use for pricing, hedging, and risk‐return analysis depend upon whether the user is a buy‐side institution or a sell‐side dealer bank? Sanjay Nawalkha and Riccardo Rebonato debate this question in this paper and provide some insightful conclusions. Responding to Nawalkha’s [2010] critique of the LMM‐SABR model, Rebonato argues that the LMM‐SABR model is currently the best available model for the sell‐side dealer banks for pricing and hedging large portfolios of complex interest rate derivatives within tight time constraints. Nawalkha in his rejoinder argues that the LMM‐SABR model is not so useful for the buy‐side institutions, and these institutions must use time‐homogeneous fundamental and single‐plus interest rate models (e.g., such as affine and quadratic term structure models) for risk‐return analysis under the physical measure, as this cannot be done using the time‐inhomogeneous double‐plus and triple‐plus versions of the LMM‐SABR model. *Corresponding author: Professor of Finance, Department of Finance and Operations Management, Isenberg School of Management, University of Massachusetts, Amherst, MA 01003.
**Quantitative Analytics Group, GBM, Royal Bank of Scotland; Oxford University, OCIAM, and Imperial College Business School, London. This draft – January 2011
Electronic copy available at: http://ssrn.com/abstract=1723924
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Part I. A Defense of the LMM‐SABR model Riccardo Rebonato
The LIBOR market model (LMM) was introduced in the mid 1990s (see, eg., Brace,
Gaterk and Musiela (1996)) to allow consistent pricing of plain‐vanilla and complex interest‐
rate products. The Black (1976) model was the then‐universally‐accepted market standard for
plain‐vanilla products, and the LMM model won its ‘market’ sobriquet exactly for its ability to
replicate the prices produced by the Black model.
In the early 2000s pronounced ‘smiles and smirks’ began to appear in plain‐vanilla
market, which could not be accounted for either by the Black of the LMM models. The SABR
model (see, eg, Hagan et al, 2002) became accepted as the market standard to account for
these features of the ‘implied’ volatility surface, fulfilling for plain‐vanilla options the role of
the 1990s Black model. What was missing was a model to price complex derivatives
consistently with the new market standard, the SABR model. Rebonato and others (see e.g.,
Rebonato [2007], Rebonato and White [2010], Rebonato, McKay and White [2009]), provided
one such extension with the LMM‐SABR model, which attempts to bridge the same gap
between plain‐vanilla and complex options that the LMM did in extending the Black model.
Nawalkha [2010] has presented a critical review of the LMM and of the LMM‐SABR
model. In this review, he looks at these classes of models from the perspective of the “triple
plus models” described in the taxonomy introduced in Nawalkha, Beliaeva, and Soto [2007,
2010]. In his review article, Nawalkha also analyzes in detail the version of the LMM‐SABR
model referred to above. In doing so, he raises two different strands of criticisms:1 i) that the
LMM‐SABR model lacks time‐homogeneity (stationarity), because of forward‐rate‐specific
factors k(Ti) that multiply the time‐homogeneous forward‐rate volatility function h(Ti ‐ t); ii)
the assumption of zero‐drift for the volatility process (in its own measure) is questionable, has
no economic justification and can lead to explosive behavior.
1To keep this note as short as possible, the same notation is here employed as in Nawalkha [2010] and Nawalkha, Beliaeva and Soto [2007, 2010], to which thereafter is referred for precise definitions of the symbols.
2
There is validity to both objections. However, in order to address these points fairly, it
is essential to keep in mind what the LMM‐SABR model (or, for that matter, all term‐structure
models designed to be used for the pricing of complex derivatives) are meant to deliver, and
which conditions they must fulfill in order to be able to do so. Let me start from the latter.
Complex‐derivatives models are technological tools mainly used to price and hedge structured
products. They firmly belong in the ‘engineering’ — as opposed to the ‘physics’ — camp of
asset pricing. They serve an important function, as they allow financial institutions to manage
the basis risk (more about the term ‘basis’ later) arising from very large books, made up of
very non‐standard deals (i.e., deals with different strikes, maturities, and other contractual
features). The computational bottleneck in the risk management of these complex books is
the evaluation of the hedge ratios — the deltas, of course, but also the vegas, volgas, vannas,
etc. Just the calculation of the ‘bucketed swaption vega’ will require hundreds of Monte Carlo
re‐evaluations per (customized) deal. Whenever the computational time required to extract
the risk metrics for a book that may contain thousands of deals exceeds 24 hours, the game is,
literally, over.
Also, pricing models need to be re‐calibrated daily to market prices of caplets and
swaptions. The practice is as theoretically dubious as universally adopted in the industry.2
Unless accurate analytic solutions or approximations exist, the trader and the risk manager
would have to make use of a Monte Carlo simulation each hedging instrument and for each
trial in a brute‐force numerical search that may require hundreds of iterations. This starts the
clock for another race against time. The outcome of this race is a fait accompli: unless the
model allows fast and stable calibration it cannot be used for the industry purposes, no matter
how ‘attractive.’
The trading desks and the risk management departments of financial institutions
therefore require models that calibrate and price the complex deal and the associated plain‐
vanilla hedges within tight time constraints. Such computationally frugal models must, by
necessity, cut many computational corners. However, to the extent that the modeling
2A case could be made as to why this theoretically inconsistent technique may not be quite as silly as it can prima facie appear, but this would entail too long a detour. See, however, the discussion in Rebonato [2004].
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foundations underlying the pricing of the complex product and of the plain‐vanilla hedges
(caps and swaption) are very close, the trader can rely on (or hope for) a high degree of error
cancellation. The price of the structured product may be ‘wrong’, but if the price of the
hedging portfolio is similarly ‘wrong’ (as opposed to ‘wrong in a different way’) not all is lost3.
The industry practice of exotic‐option risk management is fully predicated on this cancellation
of errors.
If the pricing models for the plain‐vanilla and the complex products are very similar,
the complex desk can relatively simply and successfully hedge the first‐order exposure (say, to
the overall level of volatility, or the steepness of the risk reversal), and will be left with the
task of managing the residual ‘basis’ risk (often correlation risk, or time‐dependence‐of‐
volatility risk) between the imperfectly replicating portfolio and the focus complex product.
From this ‘engineering’ point of view, the questions to be asked when assessing a complex
pricing model are therefore: how successful is the model in achieving a large cancellation of
errors? And how manageable is the residual basis risk?
We can now look at the LMM‐SABR implementation by Rebonato [2007], and
Rebonato, White, and McKay [2009] in this light. It squarely falls in the ‘engineering’ camp:4 it
tries to fulfill a useful technological function by allowing better feasible risk management of
complex books than other competing models can offer. When judged by this yardstick,
empirical work by Rebonato, Pogudin, and White [2009] suggests that its cross‐hedging
properties of radically different portions of the swaption matrix are surprisingly good and
robust. It would therefore appear that the basis risk alluded to above can be handled
successfully (at least in the market conditions and for the products explored in the study
above) despite the corners that have been cut.
Moving to the specific (and valid) objections raised by Nawalkha [2010], we agree that
the zero‐drift assumption for the volatility process is unpalatable and economically
3 By ‘similarly wrong’ we mean that the same type of modelling imperfection should affect the complex product and the hedges. To the extent that the portfolio of hedges closely mimics the complex product, the trader can count (or hope) for cancellation of errors. 4 See, e.g., the Introduction in Rebonato, McKay and White [2009]: “...We present a ‘philosophy’ of option pricing that takes into account both the realities of the industry needs (e.g., the need to calibrate as accurately as possible to the plain‐vanilla reference hedging instruments, the need to obtain prices and hedges in reasonable time) while reproducing a realistic future evolution of the smile surface (our ‘financial reality’)...”
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unjustifiable. This criticism, however, must be laid at the door of the SABR model — which, for
better or for worse, has become the market standard, as Nawalkha [2010] acknowledges —
not of the LMM‐SABR model. We dislike the zero‐drift assumption as much as he does.
However, there is no guarantee that a ‘better’ complex model would in practice produce
better hedging performance (read: a better cancellation of errors) if the ‘incorrect’ SABR
model for pricing the plain‐vanilla hedging options remains the market standard. We can’t
wait for this market standard to improve (i.e., for someone to come up with a simple and
accurate approximation for the SABR process that incorporates mean‐reversion of volatility).
When this happens, we already have a couple of ideas as to how a new LMM‐SABR model
could be formulated, with analytical calibration and fast hedges. In the meantime, however,
we don’t want to add another tombstone to the cemetery of theoretically‐better‐justifiable
models that the industry will ignore5 — and we fear that we, with many other and better
researchers, have already taken up considerable space in this crowded plot.
As for the time‐inhomogeneity criticism, we emphatically try to avoid the curse of
“triple‐plus models, [which] are calibrated to market prices by allowing time‐inhomogeneous
volatilities, and by changing numerous model inputs period by period”.6 As Nawalkha [2010]
acknowledges, our strategy is explicitly designed to find the calibration that is most
compatible with time homogeneity and stationary parameters.7 As for the offending ‘fudge
factors’, k(Ti), these should only be used when the trader needs to price the plain‐vanilla
hedging instruments exactly, as she does in the case of a competitive bid. In relative‐value
5 Some of the pre‐requisite for industry acceptance are ease, quickness and stability of calibration, speed of pricing for real‐live quotes and for hedging, ability to give an intuitive ‘interpretation’ to the model parameters, market consensus about the modelling approach, etc. Not surprisingly, it is rare for a theoretically justifiable model to display all, most, or even a few of these features. 6Nawalkha, Beliaeva, and Soto [2007, 2010] introduce a new taxonomy of term structure models (TSMs) which classifies all TSMs as fundamental models, single‐plus models, double‐plus models, and triple plus models. Table 1 gives the assumptions made in each class of models. Time‐homogeneous models assume time‐homogeneous drifts and volatilities for the interest rate processes and bond price processes. Fundamental models and single‐plus models are time‐homogeneous models. Double‐plus and triple‐plus models are time‐inhomogeneous models. Double‐plus models allow one source of time‐inhomogeneity to fit the initial bond prices. Triple‐plus models allow two sources of time‐inhomogeneity to fit both the initial bond prices and the prices of a chosen set of plain‐vanilla derivatives. 7An obvious consequence of non‐time‐homogeneous calibrations is that, day after day, the model parameters do not just ‘oscillate’, but systematically drift. This is the hallmark of a mis‐specified (as opposed to simplified) model, and this exactly what we try to avoid, as discussed in Rebonato [2004].
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analysis or in comparing, say, caplets and swaptions, these ad hoc factors should not be used
— and we have not used them either in the hedging study mentioned above, or in a recent
investigation of the coherence of the swaption and caplet markets (Rebonato and Pogudin
[2010]).8
Since they have no economic meaning (as Nawalkha [2010] correctly points out) why
use them at all? Once again, because if we did not, we would have no hope that our model
would be adopted by the financial industry: rightly or wrongly, no trader will use a model that
does not price correctly the instruments she will place in her replicating portfolio. We do not
believe that this practice is necessarily always wise, but, again, Nawalkha’s [2010] valid
criticism should be leveled elsewhere.
In short, with our version of the LMM‐SABR model we have tried to provide the
financial industry with a technology that can make the pricing and hedging of interest‐rate
structured products easier and safer. As these complex products, if poorly handled, can pose
severe risks, we think that there is value in such an offering. This value should not be
overstated, and we know that there are better models that describe more realistically the
dynamics of the underlying forward rates and of their volatilities. Incorporating these more
attractive and realistic features into an industry pricing model would, however, would make it
unusable and probably less effective unless the model for the underlying plain‐vanilla options
changes in the first place. The ‘much better’ modeling approach could, in other words, easily
turn out to be an enemy of the ‘modestly better’ one.
8Nawalkha [2010] comments on the degree of inhomogeneity obtained in actual calibration, by noticing, that “7% difference in implied volatility is explained away by the function k(Ti)”. (Nawalkha [2010], page 101). This is correct, but, for the avoidance of doubt, it should be pointed out that this means 7% of the level of volatility, not 7% in volatility points. So, if a time‐homogeneous volatility were to attain a value of 20% in one year’s time, the values produced by the model are 20.6% and 19.2%, not 27% and 13%. These values could be looked at in the light of a typical bid‐offer spread of 1 percentage vega.
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Part II. Uses and Abuses of Interest Rate Models
Sanjay K. Nawalkha
In part I of this article, Rebonato has provided an insightful response to my critique of
the LIBOR and LMM‐SABR market models (see Nawalkha [2010]), and the gist of his response
can be summarized as follows:
i. The LMM‐SABR models are technological tools used mainly to price and hedge
structured interest rate products, and so they firmly belong to the engineering side as
opposed to physics camp of asset pricing. They allow large financial institutions to
manage the basis risk arising from very large books consisting of complex non‐standard
structured products.
ii. The main computational bottlenecks for the management of basis risk for the trading
and risk management departments of these institutions is associated with evaluating
the risk metrics of the hedge ratios for thousands of deals within 24 hours at most.
iii. The industry practice of risk management is predicated on the cancellation of errors
between the structured products and the hedging portfolios.
iv. As long as SABR model remains the industry standard, one must use zero‐drift
assumption for the volatility processes under their own measure using the LMM‐SABR
model, because the industry will most likely ignore better models that are
computationally more burdensome.
v. The “fudge factors” k(Ti), calibrate forward rate volatilities (see equation (32) in
Nawalkha [2010]) to market prices of plain vanilla derivatives, while remaining as close
as possible to a desirable time‐homogeneous calibration. They unfortunately bring
about not fully time‐homogeneous volatilities. These are needed because traders will
not use models that do not exactly fit the prices of plain vanilla interest rate options
used in the replicating portfolios for hedging the structured interest rate products.
While I agree with the above responses, note that these responses justify the use of
the LMM‐SABR model only by the sell‐side dealer banks. More specifically, though the
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cancellation of hedging errors under the risk‐neutral measure allows sell‐side dealer banks to
use the LMM‐SABR model even if it is “wrong,” such cancellation does not occur for most buy‐
side institutions who do not maintain zero exposure to interest rate risk by the very nature of
their business model. I have divided my rejoinder to Rebonato’s response (to my original
critique of the LIBOR and LMM‐SABR market models given in Nawalkha [2010]) in three
sections. The first section reaffirms the main points made by Rebonato in the first part of this
paper. The second section makes a strong case for why the LMM‐SABR model is not a useful
model for most buy‐side institutions (such as, fixed income mutual funds or hedge funds,
pension funds, insurance companies, commercial banks, corporate borrowers, municipal
borrowers, and others). Finally, the last section recommends time‐homogeneous fundamental
and single‐plus interest rate models for buy‐side institutions, and time‐inhomogeneous
double‐plus and triple‐plus interest rate models (such as the LMM‐SABR model) for sell‐side
dealer banks.
A. The LMM‐SABR Market Model: Useful as a “Trading and Hedging” Model for the Sell‐side Dealer Banks. The global notional value of interest rate derivatives was a little more than $450 trillion
as of June 2010, out of which the majority constituted interest rate swaps (77.91%), followed
by FRAs (12.45%) and interest rate options including swaptions, caps/floors, etc. (10.64%).9
Though notional values are not meaningful in the derivative markets for assessing the
systemic risk exposure of that market, they are indicative of the volume of trading in specific
derivative instruments, and the magnitudes of trading profits for the major dealer institutions
that trade these derivatives. Clearly, the LMM‐SABR model is not used to price the big
majority of the notional value of plain vanilla interest rate derivatives, as the swap curve and
the prices of caps/swaptions are taken as “given” and the model is calibrated daily to these
“given” prices to obtain the risk‐neutral parameters. To the extent the model is wrong, it
results in basis risk, but with enough hedging mechanisms in place even this basis risk is not a
problem, as errors on the long side cancel out with the errors on the short side.
9 See BIS Quarterly Review, December 2010, and the published notional amounts on the following link http://www.bis.org/statistics/otcder/dt1920a.pdf.
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To understand this simply, consider using the Black and Scholes [1973] model for
pricing equity options when volatility is stochastic. Even though the model is wrong, a trader
could potentially use the Black and Scholes model by creating an implied volatility surface (for
various strikes and expirations), and then not only delta hedge, but also Vega hedge against
various types of shape changes of the implied volatility surface using say a third degree
polynomial. This would require many more long and short positions, but if done properly can
make the trader almost immune from daily changes in the implied volatility surface. If the
trader has enough informational advantage and pricing power to charge a spread between the
long side and the short side positions, it will not matter whether the Black and Scholes model
is wrong due to the assumption of constant volatility, because hedging has been done using
multiple options across strikes and maturities to cancel out the hedging errors.
The main point that Rebonato makes in the first part of this article, and the one I agree
with is that from a trader’s perspective, the rightness and wrongness of a model is not always
the primary concern when trying to make profits. The primary concern is whether a model
can be executed in real time, which is where the fast computation power of the LMM‐SABR
model has an advantage. Of course, with a “more correct” model, higher trading profits could
be generated with a lower notional value in positions, since positions will then not be used so
much for cancelling out hedging errors, but simply to lock in profits. But this would be
impractical if the more correct model takes forever to execute. Hence, the trade‐off is
between loss in trading profits due to the extra time taken to run a more correct model versus
loss in trading profits due to the extra long and short positions taken simply to cancel out the
hedging errors.
An example of valuing and hedging correctly using the wrong model can be given in
context of calibrating the LMM‐SABR model to an excited state of very high volatilities as
observed in a few months after the Lehman collapse in September 2008. Since the LMM‐SABR
model does not allow mean reversion in the volatilities, it would assume that high volatilities
will continue, forever (as pointed out by RMW [2009]). However, since the LMM‐SABR uses
prices as given, by definition it values caps and swaptions correctly. If enough hedging is done
with respect to the Vegas related to the changes in the shape of the implied volatility surface,
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then even if this model is wrong, the portfolio hedging errors obtained by applying this model
will cancel out. However, in those instances in which “almost” perfect hedging cannot be done
due to very large sizes of positions required and/or lack of liquidity, then basis risk resulting
from using the “wrong” model may undo the traders’ spreads. In these instances, the sell‐side
dealer banks may increase their spreads to deal with the basis risk if they have the pricing
power to do so. If they cannot increase the spreads, then they must evaluate the risk‐return
trade‐offs more carefully through simulations based on information from the physical
measure. However, doing this is typically not possible using the LMM‐SABR model for the
reasons noted in the following section.
B. The LMM‐SABR Market Model: Not so Useful for the Buy‐side Borrowers and Investors
The virtues of the LMM‐SABR model which allow it to perfectly fit the observed prices
of bonds and derivatives and obtain realistic hedge ratios under the risk‐neutral measure, are
of not much use to most sophisticated institutional users of interest rate derivatives and
structured products, if these virtues do not allow a meaningful risk‐return analysis under the
physical measure. Sophisticated borrowers and investors from the buy‐side (e.g., fixed
income mutual funds and hedge funds, pension funds, insurance companies, commercial
banks, large corporations, etc.,) with large Treasury departments often maintain a non‐zero
exposure to interest rate risk, as fully hedging interest rate risk is neither desirable nor
possible due to the structure of their assets and liabilities. These institutions are not in the
business of making money by “trading” interest rate derivatives while maintaining zero
exposures. This is the major difference between sell‐side dealer banks and the buy‐side
borrowers and investors. Hence, the buy‐side users of interest rate derivatives need to
perform risk return analysis under the physical measure to understand the risk and return
trade‐offs.
Unfortunately, by its very design, the LMM‐SABR model cannot be used for this
purpose. The structure of this model has time‐dependencies required for it to be perfectly
calibrated to the existing forward rate curve and the prices of the chosen set of plain‐vanilla
derivatives. Further, the LMM‐SABR model omits some realistic features of the evolution of
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state variables such as very high mean reversion in volatilities observed empirically. Due to
the presence of time‐inhomogeneous variables and complete absence of mean reversion in
forward rate volatilities under their own measures , this model cannot be econometrically
estimated using time‐series statistical methods to obtain the market prices of interest rate risk
and volatility risk. In absence of market prices of risk, the users cannot make any conditional
predictions about the risk and return trade‐offs under the physical measure. Hence, the buy‐
side users are practically clue less as to whether the additional risk they take on (hedge) using
interest rate derivatives or structured products, justifies the increase (decrease) in the return
over a finite horizon. For example, a pension fund manager who invests in a structured note
with an embedded cap cannot infer whether the additional risk she takes on by writing the
cap in order to get a slightly higher coupon is either “too little” or “too much,” using the LMM‐
SABR model. Answering such questions requires a time‐homogeneous model with realistic
assumptions about the volatility processes, which are not modeled using the LMM‐SABR
model.
To consider some implications of using this model for the buy‐side, consider the
famous loss incurred by Proctor & Gamble (P & G) on a structured interest rate swap it did
with Bankers’ Trust in the autumn of 1993.10 To save a few basis points on its cost of
financing, P & G wrote a very expensive spread option to Bankers’ Trust as a part of the
structured swap. When interest rates increased by only less than a percent, P & G incurred a
huge loss of $157 Million, representing more than 75% of the notional value of the swap.
Though such highly leveraged, severely mispriced transactions are no longer the norm, the
basic principle that underlies how dealer banks sell structured interest rate products remains
the same. Very often, the buy‐side institution either makes slightly higher return or reduces
its cost of borrowing, in return for writing some type of interest rate option. The dealer bank
may use the LMM‐SABR model to hedge its long position in options bought cheaply from the
buy‐side users (as in the case of P & G), and sells plain vanilla options such as caps and
swaptions in the OTC derivatives market maintaining a zero exposure, while pocketing the
difference between the values of the long side and short side of its option positions. In many
10 See Das [2006].
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other instances, the buy‐side institution either makes slightly lower return or increases its cost
of borrowing, in return for buying some type of protection against interest rate risk. However,
in either case, the sell‐side bank could care less if it misprices the options on either side,
provided that it misprices them in a consistent way, such that the pricing errors cancel out
with hedging. This is why the sell‐side traders do not worry so much about the physical
process consistent with the LMM‐SABR model, and typically do not know them because it is
virtually impossible to do any time‐series econometric analysis using this model.
Yet, the buy‐side institutions such as P & G need a model that not only fairly prices the
options they buy or write under the risk‐neutral measure, but also simulates realistic scenarios
under the physical measure for doing risk‐return analysis. This is because pricing errors do not
cancel out for these buy‐side institutions. Unlike the sell‐side dealer banks, they are not
trading with zero exposures using long and short positions on both sides of the market. By the
very nature of their asset liability structure, the buy‐side institutions are exposed to some
level of interest rate risk.
Though the late Nobel Laureate Merton Miller famously quipped that Procter was the
widow, and Gamble was the orphan, the real widows and orphans are not the major
corporations, hedge funds, and other sophisticated buy‐side institutions, but the municipal
borrowers of numerous cities and counties, and investors at many foundations and charities,
who often fall prey to the sales pitches of the interest rate structured products desks of the
major dealer banks. The Treasures of many of these financially unsophisticated institutions
are political appointees, who often discover their mistakes after being surprised by large
losses in the structured securities they purchase from dealer banks.11
How can both the sophisticated buy‐side borrowers and investors (such as hedge
funds, pension funds, insurance companies, non‐dealer commercial banks, savings banks,
corporations, etc.), and unsophisticated buy‐side borrowers and investors (such as, cities,
11 The poster child of these naïve investors is, of course, Robert Citron, the Treasurer of the Orange County, CA,
whose investments in structured notes such as inverse floaters with Merrill Lynch and other dealer banks, led to
a record loss of $1.7 billion, when interest rates rose sharply in 1994. Das [2006] and Partnoy [1999, 2009] detail
many abuses in this industry in their popular best sellers.
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counties, foundations, charities, wealthy retirees, etc.) protect themselves from the avalanche
of interest rate derivatives and structured products, and use them more cautiously to serve
their financial goals? Though both the Bank For International Settlements (BIS) and the
International Swaps and Derivatives Association, Inc. (ISDA) provide the global notional value
of interest rate derivatives (more than $450 trillion as of June 201012), there is no known
published figure on the global notional value of structured products that are not classified as
derivatives, but have implicit derivatives type positions.13 The next section addresses the
above question and recommends a range of time‐homogenous interest rate models for
valuing and understanding the risk‐return characteristics of interest rate derivatives and
structured products. These models are useful for both the sophisticated and unsophisticated
users from the buy‐side, unlike the LMM‐SABR model, which is a trading/hedging model
beneficial only for the sell‐side dealer banks.
C. Time‐Homogeneous Interest Rate Models for the Buy‐side
As argued in the previous section, due to their non‐zero risk exposures, buy‐side
borrowers and investors always inhabit the physical universe, in which fundamental risk‐
return analysis is the key. In contrast, due to their zero risk exposures due to matching of the
long side with short side positions, sell‐side dealer banks inhabit the risk‐neutral universe, in
which with enough hedging, the pricing errors cancel out in the physical universe. Hence, buy‐
side practitioners and sell‐side banks need different types of interest rate models. The buy‐side
needs interest rate models which allow fundamental analysis of interest rates under the
physical measure, while sell‐side banks that need models that fit the observed prices of bonds
and interest rate derivatives, under the risk neutral measure.
The type of interest rate models for the buy‐side and sell‐side institutions can be
classified using the new taxonomy of term structure models given by Nawalkha, Beliaeva, and
Soto [2007, 2010]. The new taxonomy divides all interest rate models into time‐
12See BIS Quarterly Review, December 2010, and the published notional amounts on the following link http://www.bis.org/statistics/otcder/dt1920a.pdf. 13 For example, an inverse floater is classified as a “security,” and not a “derivative,” but it implicitly equals a long position in a fixed rate bond, plus a long position in a pay‐floating, receive‐fixed swap, and a long position in an out‐of‐money interest rate cap.
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homogeneous models14 and time‐inhomogeneous models, which are further sub‐classified as
follows:
A. Time‐Homogeneous Models (Buy‐side models)
1. Fundamental models
2. Single‐Plus models
B. Time‐Inhomogeneous Models (Sell‐side models)
3. Double‐plus models
4. Triple‐plus models
Table 1 gives the assumptions made in each class of models. Time homogeneous
models assume time stationary drifts and volatilities for interest rate processes. Single‐plus
models have an additional degree of freedom over fundamental models, since valuation under
these models does not explicitly depend on market prices of risk (MPR), and so more general
forms of MPR functions are allowed for physical processes than assumed under the
fundamental models. For example, Nawalkha, Beliaeva, and Soto [2007, 2010] show how to
allow single‐plus extensions of various classes of affine and quadratic models, which allow
more state variables and non‐linearities in the MPR functions in order to fit the moments of
the physical processes of interest rates with additional flexibility, while maintaining the same
analytical apparatus for valuing interest rate derivatives, as in the fundamental models. The
empirical estimates of risk‐neutral parameters are different under the fundamental models
and the single‐plus models, so derivative prices obtained using single‐plus models would be
different, despite the fact that analytical apparatus for valuation is identical under both classes
of models.
Some classes of multiple factor fundamental models are given as the affine models
(see Dai and Singleton [2000, 2002, 2003], Grasseli and Tebaldi [2007], Joslin [2007], Joslin,
Singleton, and Zhu [2010], Collin‐Dufresne, Goldstein, and Jones [2008], Chib and Ergashev
[2009], Juneia [2010], Filipovic and Mayerhofer [2009], Le, Singleton, and Dai [2010], and
others) and quadratic models (see Ahn, Dittmar, and Gallant [2002], Lieppold and Wu [2003,
2007], Chen, Filipovic, and Poor [2004], Cheng and Scaillet [2007], Jiang and Yan [2009],
14Time‐homogeneous models assume time‐homogenous drift and volatility functions for the short rate process, and a time‐homogeneous volatility function for the bond price process.
14
Leblon and Moraux [2009], Le, Singleton, and Dai [2009], Realdon [2006], and others). Single
plus extensions of some of these affine and quadratic models are given by Nawalkha, Beliaeva,
and Soto [2007, 2010].
In general, the sell‐side banks typically do not use time‐homogeneous models such as
the fundamental models and single plus models for the following reasons:
1. These models cannot fit the initially observed bond prices.
2. These models cannot fit the correlation structure of interest rate changes.
3. These models do not match the observed prices of plain vanilla interest rate
derivatives, such as caps and swaptions.
4. These models require time‐series estimation using generalized method of moments
(GMM), efficient method of moments (EMM), etc., using historical interest rate data,
which is time consuming for traders.
5. Using more factors can be time consuming for pricing interest rate derivatives.
From the perspective of buy‐side borrowers and institutions, all of the above
criticisms can be addressed by using higher dimensional time‐homogeneous models. It is true
that a single‐factor or two‐factor time‐homogeneous model cannot fit all of the initially
observed bond prices, however using three to five factors allows these many state variables to
be fitted to the initially observed term structure of interest rates, matching the given prices
very closely every period.15 Similarly, though a two‐factor affine or quadratic model cannot
reproduce realistic correlation structure (as noted by Rebonato and Cooper [1995]), a three to
five factor affine or quadratic model can replicate the correlation structure of changes in the
interest rates.
Since these models typically do not use interest rate derivatives data for estimation
(instead they use historical interest rate data), they do not always match the prices of plain
vanilla interest rate derivatives, such as caps and swaptions. However, from the perspective of
buy‐side, this should not be viewed as a criticism. Buy‐side borrowers and investors are
concerned more about the physical risk‐return characteristics of the interest rate derivatives,
since the derivative positions are held for intermediate to long run, and the institution
15 Even the widely used term structure estimation models, such as Nelson and Siegel [1987] only use three to four parameters to estimate the entire yield curve.
15
typically maintains non‐zero interest rate risk exposure. So if an interest rate derivative is
priced cheaply (expensively) based on the time‐homogenous model, then buy‐side institution
is more likely to buy (sell) it, if its portfolio requirements need a long (short) position in that
derivative based on the risk and return profile of the entire interest rate dependent portfolio.
Otherwise, the institution may ignore the derivative. The institution is interested in knowing
whether the derivative is priced too cheaply or too expensively by a realistic model that can
simulate the risk and return trade off under the physical measure.
As an example consider the case a major internationally active money manager, who
was recently reported to have written inflation floors, betting that United States will not suffer
a decade of deflation like the one that crippled Japan starting in the 1990s. This fund manager
received $8.1 Billion in premiums for writing these derivatives, which exposes this fund to the
risk of deflation materializing in the next decade.16 In all likelihood, the analysis by this fund
manager must be based upon a careful evaluation of risk and return trade‐off under the
physical measure, as this fund is not likely to trade these derivatives in the very short run. An
extension or a modified version of the LMM‐SABR type time‐inhomogeneous model calibrated
to the existing prices of plain vanilla interest rate derivatives would not have been of much
use to this fund manager for determining if the inflation floors were priced too cheaply or too
expensively. The pricing errors from using a wrong time‐inhomogeneous model would not
cancel out for this fund, as it is not fully hedged on both sides of trade on deflation risk. This
fund would benefit more by using a time‐homogenous fundamental or single‐plus model that
can price the inflation floors under the risk‐neutral measure, as well as simulate the behavior
of inflation rates and other relevant state variables under the physical measure, to evaluate
the risk‐return trade off. Hence, the fact that time homogeneous models cannot match the
observed prices of chosen set of plain vanilla interest rate derivatives (i.e., reason # 3 given
above for why sell‐side will not use time‐homogeneous models) is a virtue of these models,
16 Bloomberg News, September 15, 2010, “Pimco Makes $8.1 Billion Bet Against `Lost Decade' of Deflation.”
16
and not a limitation from the buy‐side perspective, assuming of course, that the model
captures the behavior of interest rates, somewhat accurately.
The fourth reason why sell‐side firms do not use time‐homogeneous models is because
these models are hard to estimate using non‐linear regression techniques such as GMM,
EMM, etc., using historical data. However, note that borrowers and investors ‐ like Procter &
Gamble in 1993 or the fund manager that wrote inflation floors in 2010 ‐ do not trade these
derivatives daily. These institutions could spend more than a few days on estimating the
models, employ interest rate risk consultants, or whatever is needed to make the right
decision based on the risk return trade‐off under the physical measure. After all they are
exposing themselves to potentially hundreds of millions in gains and losses, and some
investment in estimating a time homogeneous model is well worth the cost. Unfortunately,
many buy‐side institutions do not use some of the best fundamental and single‐plus interest
rate models available for doing such econometric analysis, which allow simulating the interest
rate processes over time under the physical measure.
Finally, the fifth reason why sell‐side banks may not use time‐homogeneous models is
because these models are considered harder to implement for pricing derivatives. However,
the recent advances in computational techniques, such as the Fourier inversion method (see
Duffie, Pan, and Singleton [2000] and Chacko and Das [2002]) and the cumulant expansion
method (see Collin‐Dufresne and Goldstein [2001]) allow very fast computation of derivative
prices, regardless of the number of factors involved. Nawalkha, Beliaeva, and Soto [2007]
derive general analytical solutions of interest rate derivatives under certain classes of
fundamental affine and quadratic models (i.e., AM(N) and Q3(N) models), and their single plus
extensions (i.e., AM(N)+ and Q3(N)+ models) with an arbitrary number of factors, based on the
Fourier inversion method for pricing caps, and the cumulant expansion method for pricing
swaptions. They also show the mathematical connection between these two methods, which
allows further gain in computational efficiency. Using these extremely fast methods, solutions
of plain‐vanilla interest rate derivatives can be obtained within seconds even when using a
three to five factor model. Further, both the Fourier inversion method and the cumulant
expansion method allow obtaining the probability density functions of interest rates in quasi
17
closed forms, which can be used for fast Monte Carlo simulations for pricing exotic options.
Hence, the advances in computational methods in the past decade allow efficient pricing of
interest rate derivatives using the time‐homogeneous fundamental and single‐plus models for
both the buy‐side and sell‐side institutions.
Of course, the time homogenous models cannot serve the sell‐side banks because
these models cannot be calibrated to the observed prices of plain vanilla interest rate
derivatives, and no trader would use such models to hedge structured products or to price
exotic products. For this reason, and due to the time consuming econometric estimation issue
(i.e., reasons # 3 and # 4 given earlier), sell‐side banks will keep using double‐plus and triple‐
plus models such as the LMM‐SABR model. As shown in Table 1, double plus models allow
time‐inhomogeneous drift of the short rate process, while triple plus models allow both time‐
inhomogeneous drifts and time‐inhomogeneous term structure of bond volatilities (or forward
rate volatilities). This is not something that needs to be corrected necessarily. If sell‐side
firms cannot trade without a model that can exactly match the plain vanilla derivative prices,
and do so in a computationally efficient manner, they should continue to use double‐plus and
triple‐plus models for making trading profits. But due to their different needs and constraints,
buy‐side institutions and sell‐side banks should use the models that are suitable for them.
Conclusions This paper evaluates a range of time‐homogeneous fundamental and single‐plus
interest rate models, and time‐inhomogeneous double‐plus and triple‐plus interest rate
models from the perspective of buy‐side institutions and sell‐side dealer banks. Responding
to Nawalkha’s [2010] critique of the LMM‐SABR model, Rebonato argues that the LMM‐SABR
model is currently the best available model for the sell‐side dealer banks for pricing and
hedging large portfolios of complex interest rate derivatives within tight time constraints.
Nawalkha in his rejoinder argues that the LMM‐SABR model is not so useful for the buy‐side
institutions, and these institutions must use time‐homogeneous fundamental and single‐plus
interest rate models (e.g., such as affine and quadratic term structure models) for risk‐return
analysis under the physical measure, as this cannot be done using the time‐inhomogeneous
18
double‐plus and triple‐plus versions of the LMM‐SABR model. The arguments presented in
this paper are relevant for both the buy‐side institutions and sell‐side dealer banks that
actively participate in the $450 trillion notional value interest rate derivative market.
In general, buy‐side institutions and sell‐side banks need different models because of
the nature of their business. The main challenge for the buy‐side institutions is to structure
portfolios and manage risk optimally over intermediate to long term, so they need time‐
homogeneous models that can obtain intrinsic (or fair) values of securities and allow risk‐
return analysis under the physical measure. On the other hand, the main challenge for the
sell‐side dealer banks is to have models that are calibrated consistently in the different
interest rate derivative markets, while hedging their short term trading positions. Though in
this paper we have argued that buy‐side and sell‐side may need different models in the
interest rate derivatives markets, we wouldn’t be surprised if different models are needed by
the two sides even in the credit and equity derivatives markets for similar reasons given in this
paper.
19
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Table 1. A New Taxonomy of Interest Rate Models.
Model MPR Risk‐neutral drift of short rate process
Term Structure of bond volatilities or forward
rate volatilities
Fundamental Explicit Time Homogeneous Time Homogeneous
Single Plus Not Explicit Time Homogeneous Time Homogeneous
Double Plus Not Explicit Time Inhomogeneous Time Homogeneous
Triple Plus Not Explicit Time Inhomogeneous Time Inhomogeneous