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6 Conclusions (or, Limitations of Quadratic Variation)
10.6 CONCLUSIONS (OR, LIMITATIONS OF QUADRATIC VARIATION)
independent of the future level of the stock price, a suitably chosen deterministic-futuresmile setting would produce the same prices. We shall look at this question in Chapter 13,
and in more detail in Chapter 17.
Admittedly, this way of looking at models can appear unusual: traditional approaches
are normally presented in terms of the realism of the underlying assumptions for the
evolution of the underlying, analytical tractability, ease of practical implementation, etc.
Indeed, I will examine several modelling approaches following this traditional blueprint in
the following chapters. This type of model-by-model analysis, however, tends to obscure
the fact that, for all their apparent diversity, different models share some very fundamental
common features. I believe, therefore, that examining a model through the looking-glass
provided by the analysis outlined above can provide a powerful conceptual insight into the
structure of a given modelling approach, about some qualitative features of its associated
smiles, about what ﬁnancial features it can naturally account for, and about its possible
the Derman-and-Kani Approach
In the ﬁnal section of the previous chapter I mentioned that stochastic-volatility and
jump–diffusion models are among the most commonly used to model and account for
smiles. In the next three chapters I will analyse in some detail the features of stochasticvolatility models. I will focus ﬁrst on an important subclass, i.e. on the so-called ‘restricted-volatility’ or local-volatility models, and examine in detail one of their most popular
numerical implementations (the Derman-and-Kani/Dupire method).1 See Chapters 11 and
12. I will then move on to more general stochastic-volatility approaches (Chapter 13). The
next two chapters will introduce a discontinuous component to the process (Chapter 14
for jump–diffusion and Chapter 16 for variance–gamma processes).
General Considerations on Stochastic-Volatility
If we require that the process for the underlying should be everywhere continuous, we
are basically left with variations on the diffusion theme, i.e. with a description of the
stochastic evolution of the price process of the form
dSt = µ(St , t) dt + σ (St , t) dzt
In Equation (11.1), St indicates the value of the price or rate at time t, µ(St , t) dt its drift
(in the risk-adjusted or in the real world, as appropriate) and σ (St , t) its volatility.
As far as the volatility term, σ (St , t), is concerned, there are several modelling alternatives. In order of increasing complexity, the volatility can be described by a deterministic
function of time, by a deterministic function of time and of the underlying stock price, or
by a stochastic process driven by a variable other than the stock price. This process for
work carried out by Dr James Pfeffer is gratefully acknowledged.
CHAPTER 11 LOCAL-VOLATILITY MODELS
the volatility, in turn, can be of a continuous nature, or discontinuous (see Naik (1993)
for an interesting discussion).
If one does not allow for the possibility of jumps in the volatility, a natural description
of its time evolution can be given by a process of the form
dσ (St , t) = µσ (S, σ, t) dt + v(S, σ, t) dwt
In Equation (11.2) µσ (S, σ, t) and v(S, σ, t) denote the drift and the volatility of the
diffusion coefﬁcient σ , and the Brownian processes z(t) and w(t) are correlated in such
a way that
E[dzt dwt ] = ρ dt
Once again, in order to keep the notation as simple as possible, we have assumed that
a single Brownian shock, dw(t), affects the volatility; the extension to the multi-factor
case does not present conceptual difﬁculties (but does have an impact on the number of
instruments needed to complete the market).
Note that, in general, there can be two distinct sources of stochastic behaviour for the
volatility σ : the ﬁrst stems from the functional dependence of σ on the underlying S,
which itself is a stochastic quantity. The second is due to the fact that the volatility is
allowed to be shocked by a second Brownian motion, w(t), only imperfectly correlated,
if at all, with dz(t), as shown in Equation (11.3). Models displaying stochasticity in the
volatility originating both from the functional dependence on S and from a separate (possibly Brownian) process will be referred to in what follows as ‘fully-stochastic-volatility
models’. Models for which the volatility is stochastic only because of their dependence
on the underlying will be described as ‘restricted-stochastic-volatility models’. The distinction is important, because, as discussed in Chapter 13, a risk-neutral valuation cannot
in general be used to obtain a unique option price for fully-stochastic-volatility models,
while it can do so in the restricted-stochastic-volatility case. (Since ‘restricted-volatility
models’ is quite a mouthful, when there is no possibility of ambiguity in what follows I will often refer to them as ‘local-volatility models’.) The ‘restricted’ setting is
probably the most general set-up that goes beyond the case of a purely deterministic
(time-dependent) volatility, and still allows unique pricing by risk-neutral valuation without introducing other hedging instruments apart from the underlying itself (i.e. other
options). Therefore with a local-volatility diffusion the risk-neutral evolution for the stock
price is given by
dSt = rt dt + σ (St , t) dzt
It is important to point out that, while stochastic, the volatility implied by Equation (11.4)
displays a perfect functional dependence on the (random) realization of the underlying.
In other words, at a given future point in time, the value of the volatility is uniquely
determined by the value attained at that point in time by the stock price (or rate) S. One
should therefore not forget that the model described by Equation (11.4), while richer than
a purely-deterministic-volatility approach, still contains very strong restrictions about the
possible values that can be assumed by the volatility.
11.3 THE DUPIRE, RUBINSTEIN AND DERMAN-AND-KANI APPROACHES
Special Cases of Restricted-Stochastic-Volatility
Despite the fact that the assumptions behind the stochastic process for the underlying S
described by Equation (11.4) are already very strong, further simpliﬁcations are sometimes
made. The ﬁrst level of simpliﬁcation is to assume that the function σ (St , t) should be
separable, i.e. of the form
σ (St , t) = f (St )s(t)
A further simpliﬁcation is possible if the function f (s) is assumed to be a simple
f (S) = S β
and in this case
dSt = rt St dt + St s(t) dzt
The reason behind introducing these two simpliﬁcations are different: with the ﬁrst
approach (Equation (11.5)), one is motivated by the ﬁnancial desire to disentangle in
a transparent way the dependence of the volatility on time from the dependence on the
underlying. The motivation for the second approximation (Equation (11.6)) is partly ﬁnancial and partly computational. From the computational point of view, for β = 0, 12 or
1 simple explicit closed-form solutions exist. More complex general solutions in terms
of modiﬁed Bessel functions are also available for an arbitrary coefﬁcient β (see, for
example, Reiner (1998)), although these entail inﬁnite sums of gamma functions. Despite
its more restrictive nature, decomposition (11.7) does have some ﬁnancial justiﬁcation.
Indeed, in what follows we will discuss several important cases where this speciﬁcation might actually provide a plausible description of ﬁnancial reality. For the moment,
however, we will retain the degree of generality afforded by Equation (11.4). For future
reference, it is useful to keep in mind that the class of models described by Equation (11.7)
is often referred to as constant-elasticity-of-variance (CEV) models.
The Dupire, Rubinstein and Derman-and-Kani
Dupire (1993, 1994), Rubinstein (1994) and Derman and Kani (1998) (DK in what follows) (see also Derman et al. (1996)) provide tree-based algorithms to extract the function
σ (St , t) from today’s quoted prices of a series of plain-vanilla options of different strikes
and maturities, under the assumption that the process for the stock price, S, is described
by Equation (11.4). Despite the differences in the numerical implementations, the three
approaches share the same conceptual foundations. I will therefore deal in detail with
what is probably the most common of the three approaches, i.e. the DK, leaving the
extension to the other cases as the proverbial exercise for the reader.
CHAPTER 11 LOCAL-VOLATILITY MODELS
Strictly speaking, Rubinstein’s, Dupire’s and DK’s constructions ‘only’ require that
the user should provide the prices of options for the discrete maturities and strikes that
correspond to the nodes of their computational lattices. Since, however, in order to obtain
results of acceptable numerical quality a very ﬁne space–time mesh is needed, in what
follows I will always assume that a continuous price (or implied-volatility) function vs
maturity and strike has already been obtained by some numerical means. This task is far
from trivial (see Chapter 9), but I will assume that it has been carried out to the trader’s
It is also important to point out that Rubinstein’s, Dupire’s and DK’s approaches provide, at the same time, a ‘pricing engine’ (the calibrated tree), and the local-volatility
surface (i.e. the function σ (St , t)). These two ‘pricing ingredients’, however, are conceptually completely distinct, and an efﬁcient pricing methodology need not provide the most
accurate tool to extract the local volatility (or vice versa). I will argue in what follows that
this is indeed the case, and that a different procedure to the one proposed by DK can provide a more efﬁcient pricing mechanism once the function σ (St , t) has been obtained. See
Chapter 12. Before attempting to de-couple the extraction of the local-volatility function
from the pricing engine, however, the DK procedure will be analysed in detail.
Since their approach depends more fundamentally than the ‘usual’ bi- or trinomial tree
construction on the Green’s function formalism, this topic is examined in the following
section in the discrete-time, discrete-space framework best suited to the DK approach.
Green’s Functions (Arrow–Debreu Prices) in the DK
Let us place ourselves in a universe where trading only takes place at discrete time
intervals, and where only a ﬁnite number of states at each possible time-step are reachable.
We impose no restrictions on the trading frequency and the number of states, other than
that they both have to be ﬁnite. As for the characterization of a state at a given time, we
will assume that it is fully and uniquely deﬁned by the realization of the price at that point
in time. Therefore, speaking of state (j, k) (with the ﬁrst index referring to the time slice
and the second to the state) is exactly equivalent to speaking of the stock price having
attained value Sk a time tj .
Given this set of prices and times, let us construct a recombining trinomial lattice
with nodes located corresponding to the trading times and to the possible prices. See
Figures 11.1 and 11.2 below. We will also assume that we have already obtained the set
of all the probabilities connecting each parent node at time j with its three ‘offsprings’
at time j + 1. The construction of recombining trees is greatly facilitated by the so-called
Green’s functions (also sometimes referred to as Arrow–Debreu prices). Their use in
the context of tree construction and calibration was pioneered by Jamshidian (1991) for
interest-rate models. Given their importance, their main properties are reviewed below.
11.4.1 Deﬁnition and Main Properties of Arrow–Debreu Prices
Let us make the assumption that a future (time-t) state of the world is fully characterized
by the time-t realization of the stock price. We are therefore dealing with a system
described by a single Markovian state variable. Let us identify one such future state of