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6 Conclusions (or, Limitations of Quadratic Variation)

6 Conclusions (or, Limitations of Quadratic Variation)

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10.6 CONCLUSIONS (OR, LIMITATIONS OF QUADRATIC VARIATION)



317



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

intrinsic limitations.



Chapter 11



Local-Volatility Models:

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).



11.1



General Considerations on Stochastic-Volatility

Models



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



(11.1)



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

1 Numerical



work carried out by Dr James Pfeffer is gratefully acknowledged.



319



320



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



(11.2)



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



(11.3)



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



(11.4)



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



11.2



321



Special Cases of Restricted-Stochastic-Volatility

Models



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)



(11.5)



A further simplification is possible if the function f (s) is assumed to be a simple

power law:

f (S) = S β



(11.6)



and in this case

β



dSt = rt St dt + St s(t) dzt



(11.7)



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.



11.3



The Dupire, Rubinstein and Derman-and-Kani

Approaches



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.



322



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

satisfaction.

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.



11.4



Green’s Functions (Arrow–Debreu Prices) in the DK

Construction



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



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