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Mathematical issues with volatility modellingMarek Musiela BNP Paribas

25th May 2005

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Contents

1. Mixture models

2. Uncertain volatility models

3. Stochastic volatility models

4. Ergodicity

5. Selfsimilarity

6. What is mean reversion?

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Motivation There is an ongoing discussion on the benefits and

drawbacks of using different approaches to capture the market smile.

Mixture and uncertain volatility models offer simplicity of calibration and pricing of certain products but are criticised for not giving the dynamics of the stochastic volatility.

The classical stochastic volatility models are more difficult to implement and do not capture certain features of the volatility markets ( flat butterfly in FX or short term cap and swaption vols ).

References: D. Brigo, F. Mercurio and F. Rapisadra (2004), D. Gatarek (2003), V. Piterbarg (2003)

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Mixture models Deterministic function representing time dependent

volatility

The classical Black-Scholes model with deterministic volatility

Probability distribution over different volatility functions, namely, a measure on the space of deterministic volatility functions.

00

,

SS

dWStdtrSdS tttt

0

2: dttcadlagRR

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Mixture models The process driving the stock dynamics is a Brownian

motion under the measure

The measure on the space of deterministic volatility functions is

The mixture model is given by the product measure

The product measure is defined on the appropriate product probability space equipped with the relevant product sigma algebra and the ‘marginal’ Brownian filtration

1P

2P

21 PPP

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Mixture models For each deterministic volatility the associated density

of the underlying asset at time t is

The asset density of the mixture model is defined by

The above expectation is calculated with respect to the product measure. It depends only on the norm of the volatility function

t

t

t

tt

duu

rtS

stsf

0

22

22

02 2

1ln

2

1exp

2

1,

tsEftsf ,,

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Mixture models The density can be also written in the form

The above can be seen as an equation for

with the density on the left determined from the option prices.

vPvF

vdFvrtS

s

vvtsf

tt

t

2

2

00

2

2

2

1ln

2

1exp

2

1,

2

t

F

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Uncertain volatility models For the time being there is no filtration giving the

dynamics of the volatility.

The volatility can be viewed an infinite dimensional random vector taking values in the space of the deterministic volatility functions.

The mixture model coincides with the uncertain volatility model defined by

In the above dynamics the volatility is an infinite dimensional random vector which is independent of the Brownian motion W. For example, it can take only finite number of values or may be constant as a function of t

tttt dWStdtrSdS

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Uncertain volatility models The stochastic integral is well defined if one takes the

filtration at 0 to be given by the sigma algebra generated by the volatility process.

The solution is of course given by

t t

ut duudWurtSS0 0

20 2

1exp

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Uncertain volatility models It can also be written in the following form

For each t, the random variables

are independent.

t

tt

t

u

t

tttt

duudWu

rtSS

0

220

2

0 2

1exp

,t

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Uncertain volatility models The density of the asset price in the uncertain volatility

model coincides with the density of the mixture model.

The only quantity that needs to be specified is the distribution of

Obviously we have

t

tduu

0

22

v

vrtKSNKe

v

vrtKSNSvtKSc

vdFvtKScKSE

rt

tt

2//ln

2//ln,,,

,,,

0

000

0 0 2

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Stochastic volatility models Popular models with stochastic volatility assume a mean

reverting stochastic volatility process.

There is no formal definition of this concept.

The intuition behind is the same as in the context of interest rates.

When the volatility is too low it will tend to rise, when it is too high it will tend to fall.

The classical examples of such process are OU (Stein & Stein) and CIR (Heston)

Both of them are ergodic, when started from the steady state distribution, and hence they satisfy

11lim 2

0

2 Eduut

t

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Stochastic volatility models Clearly, in such models the distribution of

depends very strongly on t.

Indeed the ergodic theorem shows that the limit of the averaged squared volatility is constant and hence has zero variance

This is inconsistent with the butterfly prices which do not depend very strongly on maturities.

Indeed, the butterfly price is directly related to the distribution of

t

duut 0

21

duut

t

021

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Uncertain volatility models Consider the simplest uncertain volatility model in

which the volatility vector is constant, i.e., for all t

Clearly, in this case we have for all t

Such a model fits the butterfly by construction but does not produce dynamics of the smile, and hence may not be a good model for exotics, in particular, those which depend strongly on the forward volatility.

t

20

21 Lawduut

Lawt

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Alternative models One could try to construct the volatility process with

the following scaling property

for all c>0. Taking c=1/t we see that the butterfly prices are flat in maturity.

The above property hold if and only if

This in turn is a particular case of selfsimilarity

tcttduucLawtduuLaw

0

2

0

2 0,0,

0,0, 22 ttLawtctLaw

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Selfsimilarity The process X is said to be selfsimilar if for any a>0

there exists b>0 such that

Brownian motion is selfsimilar with

Constant process, i.e.,

is selfsimilar with b=1. This corresponds to an uncertain volatility model.

0,0, ttbXLawtatXLaw

ab

0, tXtX

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Selfsimilarity (Lamperti1962) If the process X is nontrivial, stochastically continuous

at t=0 and selfsimilar, then there exists a unique nonnegative constant H such that

Moreover, H=0 if and only if X(t)=X(0) almost surely for every t>0.

If X is H-ss and H>0, then X(0)=0 almost surely.

Is it possible to construct 0-ss processes that is not trivial?

Hab

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0-Selfsimilarity (Kono 1984)

Let Y be a strictly stationary process defined on R, Z a random variable independent of the process Y. Define

Then

implying that X is 0-ss. However X(0) is not equal to zero.

Recall that a process is strictly stationary if its finite dimensional distributions are invariant on shifts.

To construct an example take an Ornstein-Uhlenbeck process on R for Y and define X by the procedure above.

Clearly, if we take Z Gaussian, the process remains Gaussian but its covariance function changes. The rate of de-correlation is polynomial and not exponential.

ZXttYtX 0,0,log

ZttXLawZtatXLaw ,0,,0,

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0-Selfsimilarity and forward volatility The classical uncertain volatility model does not

generate smile dynamics.

The transition probabilities generating the smile dynamics are the transition probabilities of a time transformed strictly stationary process.

Such a model can clearly be used for pricing exotics.

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H-Selfsimilarity and Lamperti transform If Y is a strictly stationary process on R and if for

some H>0, we let

then X is H-ss. Conversely, if X is H-ss and if we let

then Y is strictly stationary.

00,0,log XttYttX H

RteXetY ttH ,

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What is mean reversion? Mean reversion expresses an intuitive idea of a

variable that remains in a range. We think that when the value is too low it will tend to rise, when it is too high it will tend to fall.

Typical processes used to describe such behaviour are ergodic.

They do not capture well the flat butterfly structure.

Time transformed strictly stationary process do.

Such process still capture the intuitive idea behind the mean reversion.

They also generate reasonable smile dynamics.