7 Mean Reversion in Short-Rate Lattices: Recombining vs Bushy Trees
Tải bản đầy đủ - 0trang
618
CHAPTER 18 MEAN REVERSION IN INTEREST-RATE MODELS
up/up
up
up/down
down/up
down
down/down
Figure 18.3 A bushy-tree construction for the case of decreasing volatility.
up
up/up
up/down
down/up
down/down
down
Figure 18.4 The corresponding BDT construction for the same decreasing volatility.
the user. In particular, this ‘extra’ drift can at most be time dependent (the term µ(t)
in Equation (18.16), algorithmically accounted for by the construction in Figure 18.7),
and, therefore, cannot introduce ‘true’ mean reversion (since this would require the state
variable, r).
From these considerations one can see from yet another angle why a mean reversion
(of sorts) can only occur with the BDT algorithm if the volatility is time dependent (and,
more speciﬁcally, decaying). Note that the negative aspects introduced by the procedure
are more insidious than the usual limitations of one-factor or low-dimensionality models:
the binomial recombining-lattice geometry introduces an inextricable link between its
18.7 RECOMBINING VS BUSHY TREES
619
up/up
down/up
up
down
up/down
down/down
Figure 18.5 A bushy-tree construction for the case of increasing volatility.
up/up
up
down/up
up/down
down
down/down
Figure 18.6
The corresponding BDT construction for the same increasing volatility.
reversion speed and the (logarithmic) derivative of the short rate volatility. The timedecaying volatility needed in the BDT model in order to ‘contain’ an excessive dispersion
of rates does succeed in obtaining an unconditional distribution of rates consistent with
the one implied by the cap market. However, in the absence of mean reversion, this
can only be obtained by means of a lower and lower future volatility. The future term
structure of volatilities predicted by the BDT model is therefore radically different from
the term structure of volatilities observed today, and in the future will become lower
and lower and ﬂatter and ﬂatter. The future vega re-hedging costs predicted by the BDT
model are therefore unlikely to be encountered in real life. Furthermore, day after day
the model will have to be re-calibrated to account for a future implied volatility surface
620
CHAPTER 18 MEAN REVERSION IN INTEREST-RATE MODELS
up/up
up
down
down/up
up/down
down/down
Figure 18.7 Same as Figure 18.6 but with a purely time-dependent (negative) drift (the term θ (t)
in Equation (18.18)).
that systematically fails to come true. Given the discussion in Sections 1.1.4 and 1.3.2 of
Chapter 1, these are highly undesirable features.
18.8
Extension to More General Interest-Rate Models
The previous section highlighted that the limitations of the BDT model directly stem
from the algorithmic prescription for its construction and from the binomial choice for
the geometry of the tree in particular. This, however, need not be the case. For instance, a
procedure based on a three-branching (trinomial) lattice could be used. The construction
and the calibration become more cumbersome, but a very signiﬁcant advantage is reaped.
In computational terms, this can be seen as gaining one degree of freedom (e.g. one extra
probability, or the location of the third node); in ﬁnancial and economic terms, as the
ability to specify a ‘true’ (i.e. volatility-independent) mean reversion.
As we saw in the equity/FX case, mean reversion in the real world has no direct bearing
on option pricing, since the transformation from the real-world to the pricing measure
effects a drift transformation that ‘scrambles’ the real-world dynamics. In the interest-rate
case, however, one is typically evolving non-traded quantities. No-arbitrage considerations
therefore do not directly dictate that these non-traded state variables should have a drift
equal to the riskless rate. When one is dealing with interest rates, one is therefore at
liberty to start with very general functional forms for the speciﬁcation of the drift of the
(non-traded) short rate in the risk-neutral world. As we saw in Section 18.2, this will
then affect the volatility of the traded asset prices. In particular, this speciﬁcation could
incorporate a mean-reverting feature. Despite the fact, however, that one can introduce
a mean-reverting term into the risk-neutral dynamics of the short rate, it is not a priori
obvious why it should be advantageous to do so.
18.8 EXTENSION TO MORE GENERAL INTEREST-RATE MODELS
621
In qualitative terms, the explanation goes as follows. If part of the burden to avoid
‘excessive’ dispersion of the rates is shouldered by a mean-reverting term (rather than
by a time-decaying volatility, as in the BDT case), then the volatility can be constant,
or mildly varying, and the market prices of caplets can still be recovered by the model.
This observation, however, raises another question: why should we want to require that
the volatility of the short rate be constant, or mildly varying? Or, more generally: since
our information about the real-world mean reversion does not imply anything about the
drift of the risk-adjusted dynamics, how does one ‘guess’ the nature of the risk-adjusted
mean reversion?
The key to a satisfactory answer to both questions lies in the observation that the
transformation from the real-world to the pricing measure affects (via Girsanov’s theorem) the drift but not the volatility of the stochastic quantities of interest. Therefore,
in a deterministic-volatility setting, our calibrated model will imply that whatever future
volatility structure we are left with in the pricing measure after recovering the caplet
market prices, this will be the same as the one that we should expect in the real world.
Because of this, whatever property we calculate in the risk-adjusted world, as long as
purely related to and derived from the measure-invariant volatilities, can be directly compared with the corresponding real-world quantity.4 One such quantity is the evolution of
the term structure of volatilities. On the basis of the empirical observations mentioned
in Chapter 7, we may believe that term structures of volatilities by and large preserve
their shape over time. If this is the case, one possible (and useful) criterion to choose
an appropriate mean-reverting dynamics for the driving factors (the short rate or the forward rates) could therefore be the following: choose the parameters of the risk-adjusted
mean-reverting process in such a way that the volatility functions needed to price the
market-traded options produce (reasonably) time-homogeneous evolutions for the term
structure of volatilities. In other words, make sure that the future trading universe looks
acceptably similar to the universe we observe today.
The ultimate justiﬁcation for this possible criterion is the empirical observation that
term structures of volatilities remain approximately self-similar over time (see, in this
regard, the discussion in Chapter 19). Therefore, unless we have speciﬁc views to the
contrary, it is reasonable to assume that they will (at least approximately) retain their
structural features also in the future. Needless to say, if the trader thought that the future
will not be like the past, she should use a model that reﬂects, in the evolution of the
relevant volatilities, this particular belief. This issue is tackled in Chapter 19, where I
argue that one of the main reasons why the modern (LIBOR market model) approach is
so much preferable to the ‘traditional’ (short-rate-based) framework is because the former
gives the trader a direct way of controlling the evolution of those quantities (such as the
term structure of volatilities) that have a direct impact on future re-hedging costs.
It is worthwhile pointing out that imposing time homogeneity for some ﬁnancial quantities can have wider applicability than the purely interest-rate case. Similar conclusions
can be drawn, in fact, in the case of the implied volatility smile. I have stressed in the
4 Note that this is not true if we believed that the correct dynamics of the yield curve should be expressed
in terms of a stochastic-volatility diffusion. Indeed, some of the techniques presented in Part IV of this book
deal with the question of how this comparison across measures should be carried out when the volatility is
assumed to be stochastic.
622
CHAPTER 18 MEAN REVERSION IN INTEREST-RATE MODELS
equity/FX case that reproducing today’s smile is only a (relatively small) part of the overall task of a model. As I discuss in Chapter 1, just as important is the model’s prediction
of the future smile.
Note also that, if enough liquid instruments were traded in the market, and exactly
priced by a model, so as to capture not only the prices embedded in today’s smile or
in today’s cap volatility term structure, but also of the evolution of these quantities,
then the trader’s job would truly be one of pure arbitrage. In the case of interest-rate
options I will argue in Chapter 19 that the missing liquid instruments are serial options.
In the equity/FX world, forward-setting options of the type discussed in Section 3.8 of
Chapter 3 would be required in addition to plain-vanilla calls and puts. If that were
the case, the exotic trader’s task would be considerably simpler (or, perhaps, her job
would disappear): she would not have to worry whether the market is ‘implying’ an
unreasonable future evolution for the smile or for the term structure of caplet volatility.
She could simply engage in the transactions suggested by her model (calibrated to all the
products), virtually resting assured that she could ‘lock-in’ any price discrepancy. ‘True’
model arbitrages, are, however, exceedingly rare, and, as the name suggests, they will
only prove to be true arbitrages if the world indeed evolves as the model predicts.5 So,
ultimately, the justiﬁcation for having to specify correctly the future evolution of smile
surfaces or term structures of volatilities comes down to the intrinsic incompleteness of
option markets.
In the light of this discussion, one can leave the short-rate models with the observation
that, apart from all their other shortcomings, they provide the user with a rather blunt
instrument when it comes to making the model produce a pre-chosen future evolution for
the term structure of volatilities. Some models, as we have seen, just cannot accomplish
this goal. Others (like the Black-and-Karasinski or the Hull-and-White) have the potential
to deliver this result, but offer a very indirect way of producing the desired effect. I have
always felt that driving the term structure of volatilities from the short-rate end is like
pushing, rather than pulling, a rope. The next chapters will show how, by means of a
LIBOR-market-model-type of approach, one can establish a more constructive interaction
with a rope.
18.9
Appendix I: Evaluation of the Variance of the
Logarithm of the Instantaneous Short Rate
From Equation (18.41) one can write
d ln r(t) + a(t) ln r(t) = a(t)b(t) dt + σ (t) dz(t)
(18.46)
This implies that
t
exp −
t
a(s) ds d ln r(t) exp
0
a(s) ds
0
= a(t)b(t) dt + σ (t) dz(t)
5 This
should be contrasted with the model-independent arbitrages discussed in Chapter 17.
(18.47)
18.9 VARIANCE OF THE LOGARITHM OF THE INSTANTANEOUS SHORT RATE 623
T
But the quantity ln r(T ) exp
a(s) ds can be written as
0
T
ln r(T )
T
a(s) ds = ln r(0) +
0
t
a(t)b(t) exp
0
T
+
a(s) ds] dt
0
t
σ (t) exp
0
a(s) ds dz(t)
(18.48)
0
and, therefore
T
ln r(T ) = exp −
a(s) ds ln r(0)
0
T
+ exp −
0
0
a(s) ds dt
0
T
a(s) ds
t
a(t)b(t) exp −
0
T
+ exp −
T
a(s) ds
t
σ (t) exp
0
a(s) ds dz(t)
0
Remembering that, for any deterministic function f (t),
t
var
t
f (u) dz(u) =
0
f (s)2 dt
(18.49)
0
it then follows that
var[ln r(T )] = E[(ln r(T ))2 ] − (E[ln r(T )])2
T
= exp −2
0
T
a(s) ds
0
t
σ (t)2 exp 2
0
a(s) ds dt
(18.50)