Information-Based Asset Pricing
Information-Based Asset Pricing
Dorje C. Brody∗, Lane P. Hughston†, and Andrea Macrina†
∗Blackett Laboratory, Imperial College, London SW7 2BZ, UK
†Department of Mathematics, King’s College London,
The Strand, London WC2R 2LS, UK
Abstract. A new framework for asset price dynamics is introduced in which the
concept of noisy information about future cash flows is used to derive the correspond-
ing price processes. In this framework an asset is defined by its cash-flow structure.
Each cash flow is modelled by a random variable that can be expressed as a function
of a collection of independent random variables called market factors. With each
such “X-factor” we associate a market information process, the values of which we
assume are accessible to market participants. Each information process consists of
a sum of two terms; one contains true information about the value of the associated
market factor, and the other represents “noise”. The noise term is modelled by an
independent Brownian bridge that spans the interval from the present to the time at
which the value of the factor is revealed. The market filtration is assumed to be that
generated by the aggregate of the independent information processes. The price of
an asset is given by the expectation of the discounted cash flows in the risk-neutral
measure, conditional on the information provided by the market filtration. In the
case where the cash flows are the dividend payments associated with equities, an
explicit model is obtained for the share-price process. Dividend growth is taken into
account by introducing appropriate structure on the market factors. The prices of
options on dividend-paying assets are derived. Remarkably, the resulting formula for
the price of a European-style call option is of the Black-Scholes-Merton type. We
consider both the case where the rate at which information is revealed to the market
is constant, and the case where the information rate varies in time. Option pricing
formulae are obtained for both cases. The information-based framework generates a
natural explanation for the origin of stochastic volatility in financial markets, with-
out the need for specifying on an ad hoc basis the dynamics of the volatility.
Key words: Asset pricing; partial information; stochastic volatility; correlation; div-
idend growth; Brownian bridge; nonlinear filtering; market microstructure
Working paper. Original version: December 5, 2005. This version: October 22,
2018.
Email: dorje@imperial.ac.uk, lane.hughston@kcl.ac.uk, andrea.macrina@kcl.ac.uk
I. INTRODUCTION
In derivative pricing, the starting point is usually the specification of a model for the
price process of the underlying asset. Such models tend to be of an ad hoc nature. For
example, in the Black-Scholes-Merton-Merton theory, the underlying asset has a geometric
http://arxiv.org/abs/0704.1976v1
Brownian motion as its price process. More generally, the economy is often modelled by a
probability space equipped with the filtration generated by a multi-dimensional Brownian
motion, and it is assumed that asset prices are adapted to this filtration. This example is
of course the “standard” model within which a great deal of financial engineering has been
carried out. The basic problem with the standard model (and the same applies to various
generalisations thereof) is that the market filtration is fixed, and no comment is offered on
the issue of “where it comes from”. In other words, the filtration, which represents the
revelation of information to market participants, is modelled first, in an ad hoc manner, and
then it is assumed that the asset price processes are adapted to it. But no indication is given
about the nature of this “information”, and it is not obvious, a priori, why the Brownian
filtration, for example, should be regarded as providing information rather than noise.
In a complete market there is a sense in which the Brownian filtration provides no ir-
relevant information. That is to say, in a complete market based on a Brownian filtration
the asset price movements reflect the information content of the filtration. Nevertheless,
the notion that the market filtration should be “prespecified” is an unsatisfactory one in
financial modelling. The intuition behind the “prespecified-filtration” approach is that the
filtration represents the unfolding in time of a succession of random events that “influence”
the markets, causing prices to change. For example, bad weather in South America results
in a decrease in the supply of coffee beans and hence an increase in the price of coffee.
Or, say, a spate of bad derivative deals causes a drop in confidence in banks, and hence a
downgrade in earnings projections, and thus a drop in their prices. The idea is that one
“abstractifies” these influences in the form of a prespecified background filtration to which
price processes are adapted. What is unsatisfactory about this is that little structure is given
to the filtration: price movements behave as though they were spontaneous. In reality, we
expect the price-formation process to exhibit more structure.
It would be out of place in the present context to attempt an account of the process
of price formation. Nevertheless, we can improve on the “prespecified” approach. In that
spirit we proceed as follows. We note that price changes arise from two sources. The
first is that resulting from changes in agent preferences—that is to say, changes in the
pricing kernel. Movements in the pricing kernel are associated with (a) changes in investor
attitudes towards risk, and (b) changes in investor “impatience”, the subjective discounting
of future cash flows. Equally important are changes in price resulting from the revelation of
information about the future cash flows derivable from a given asset. When a market agent
decides to buy or sell an asset, the decision is made in accordance with the information
available to the agent concerning the likely future cash flows associated with the asset. A
change in the information available to the agent about a future cash flow will typically have
an effect on the price at which they are willing to buy or sell, even if the agent’s preferences
remain unchanged. Consider the situation where one is thinking of purchasing an item at
a price that seems attractive. But then, one reads an article pointing out an undesirable
feature of the product. After reflection, one decides that the price is too high, given the
deficiencies that one is now aware of. As a result, one decides not to buy, not at that price,
and eventually—because other individuals will have read the same report—the price drops.
The movement of the price of an asset should, therefore, be regarded as an emergent
phenomenon. To put the matter another way, the price process of an asset should be viewed
as the output of (rather than an input into) the decisions made relating to possible trans-
actions in the asset, and these decisions should be understood as being induced primarily
by the flow of information to market participants. Taking into account this observation we
propose in this paper a new framework for asset pricing based on modelling of the flow of
market information. The information is that concerning the values of the future cash flows
associated with the given assets. For example, if the asset represents a share in a firm that
will make a single distribution at some agreed date, then there is a single cash flow. If the
asset is a credit-risky discount bond, then the future cash flow is the payout of the bond
at maturity. In each case, based on the information available relating to the likely payouts
of the given financial instrument, market participants determine estimates for the value of
the right to the impending cash flows. These estimates lead to the decisions concerning
transactions that trigger movements in the price.
In this paper we present a class of models capturing the essence of the scenario described
above. In building the framework we have several criteria in mind that we would like to see
satisfied. The first of these is that our model for the flow of market information should be
intuitively appealing, and should allow for a reasonably sophisticated account of aggregate
investor behaviour. At the same time, the model should be simple enough to allow one to
derive explicit expressions for the asset price processes thus induced, in a suitably rich range
of examples, as well as for various associated derivative price processes. The framework
should also be flexible enough to allow for the modelling of assets having complex cash-flow
structures. Furthermore, it should be suitable for practical implementation. The framework
should be mathematically sound, and manifestly arbitrage-free. In what follows we show
how our modelling framework goes a long way towards satisfying these criteria.
The role of information in financial modelling has long been appreciated, particularly in
the theory of market microstructure (see, e.g., Back [1], Back and Baruch [2], O’Hara [20],
and references cited therein). The present framework is perhaps most closely related to the
line of investigation represented, e.g., in Cetin et al. [5], Duffie and Lando [8], Giesecke [10],
Giesecke and Goldberg [11], Guo et al. [13], and Jarrow and Protter [14]. The work in this
paper, in particular, develops that described in Brody et al. [3] and Macrina [19], where
preliminary accounts of some of this material appear (see also Rutkowski and Yu [22]).
The paper is organised as follows. In Sections II, III, and IV we illustrate the framework
for information-based pricing by considering the scenario in which there is a single random
cash flow. An elementary model for market information is presented, based on the specifica-
tion of a process composed of two parts: a “signal” component containing true information
about the upcoming cash flow, and an independent “noise” component which we model in a
specific way. A closed-form expression for the asset price is obtained in terms of the market
information available at the time the price is being specified. This result is summarised in
Proposition 1. In Section V we show that the resulting asset price process is driven by a
Brownian motion, an expression for which can be obtained in terms of the market informa-
tion process: this construction indicates in explicit terms the sense in which the price process
can be viewed as an “emergent” phenomenon. In Section VI we show that the value of a
European-style call option, in the case of an asset with a single cash flow, admits a simple
formula analogous to that of the Black-Scholes-Merton model. In Section VII we derive
pricing formulae for the situation when the random variable associated with the single cash
flow has an exponential distribution or, more generally, a gamma distribution.
The extension of the framework to assets associated with multiple cash flows is established
in Section VIII. We show that once the relevant cash flows are decomposed in terms of a
collection of independent market factors, then a closed-form expression for the asset price
associated with a complex cash-flow structure can be obtained. In Section IX we show how
the standard geometric Brownian motion model can be derived in an information-based
setting. This remarkable result motivates the specific choice of information process given in
equation (4). In Section X we present a simple model for dividend growth. In Section XI
we show that by allowing distinct assets to share one or more common market factors in the
determination of one or more of their respective cash flows we obtain a natural correlation
structure for the associated asset price processes. This method for introducing correlation
in asset price movements contrasts with the ad hoc approach adopted in most financial
modelling. In Section XII we demonstrate that if two or more market factors affect the
future cash flows of an asset, then the corresponding price process will exhibit unhedgeable
stochastic volatility. This result is noteworthy since even for the class of relatively simple
models considered here it is possible to identify a candidate for the origin of stochasticity in
price volatility, as well as the form it should take, which is given in Proposition 2.
In the remaining sections of the paper we consider the case where the rate at which
the information concerning the true value of an impending cash flow is revealed is time
dependent. The introduction of a time-dependent information rate adds additional flexibility
to the modelling framework, and opens the door to the possibility of calibrating the model
to the market prices of options. We consider the single-factor case first, and obtain a closed-
form expression for the conditional expectation of the cash flow. The result is stated first
in Section XIII as Proposition 3, and the derivation is then given in the two sections that
follow. In Section XIV we introduce a new measure appropriate for the consideration of a
Brownian bridge with a random drift, which is used in Section XV to obtain an expression
for the conditional probability density of the random cash flow. The consistency of the
resulting price process is established in Section XVI. We show, in particular, that, for the
given information process, if we re-initialise the model at some specified future time, the
dynamics of the model moving forward from that time can be represented by a suitably
re-initialised information process. The statement of this result is given in Proposition 4.
The dynamical equation satisfied by the price process is analysed in Section XVII, where
we demonstrate in Proposition 5 that the driving process is a Brownian motion, just as in
the constant parameter case. In Section XVIII we derive the price of a European-style call
option in the case for which the information rate is time dependent.
Our approach is based on the idea that first one models the cash flows, then the in-
formation processes, then the filtration, and finally the prices. In Section XIX, we solve
the corresponding “inverse” problem. The result is stated in Proposition 6. Starting from
the dynamics of the conditional distribution of the impending payoff, which is driven by a
Brownian motion adapted to the market filtration, we construct (a) the random variable
that represents the relevant market factor, and (b) an independent Brownian bridge repre-
senting irrelevant information. These two combine to generate the market filtration. We
conclude in Section XX with a general multi-factor extension of the time-dependent setup,
for which the dynamics of the resulting price processes are given in Propositions 7 and 8.
II. THE MODELLING FRAMEWORK
In asset pricing we require three ingredients: (a) the cash flows, (b) the investor prefer-
ences, and (c) the flow of information to market participants. Translated into a mathematical
language, these ingredients amount to the following: (a′) cash flows are modelled as random
variables; (b′) investor preferences are modelled with the determination of a pricing kernel;
and (c′) the market information flow is modelled with the specification of a filtration. As we
have indicated above, asset pricing theory conventionally attaches more weight to (a) and
(b) than to (c). In this paper we emphasise the importance of ingredient (c).
Our theory will be based on modelling the flow of information accessible to market par-
ticipants concerning the future cash flows associated with a position in a financial asset.
We start by introducing the notation and assumptions employed in this paper. We model
the financial markets with the specification of a probability space (Ω,F ,Q) on which a fil-
tration {Ft}0≤t<∞ will be constructed. The probability measure Q is understood to be the
risk-neutral measure, and the filtration {Ft} is understood to be the market filtration. All
asset-price processes and other information-providing processes accessible to market partic-
ipants will be adapted to {Ft}.
Several simplifying assumptions will be made that will enable us to concentrate our
efforts on the problems associated with the flow of market information. The first of these
assumptions is the use of the risk-neutral measure. The “real” probability measure does
not enter into the present investigation. Expectation in the measure Q will be denoted by
E[−]. Our second assumption is that we take the default-free system of interest rates to
be deterministic. The absence of arbitrage then implies that the corresponding system of
discount functions {PtT }0≤t≤T<∞ can be written in the form PtT = P0T/P0t for t ≤ T , where
{P0t}0≤t<∞ is the initial discount function, which we take to be part of the initial data of
the model. The function {P0t}0≤t<∞ is assumed to be differentiable and strictly decreasing,
and to satisfy 0 < P0t ≤ 1 and limt→∞ P0t = 0.
We also assume, for simplicity, that cash flows occur at pre-determined dates. Now clearly
for some purposes we would like to allow for cash flows occurring effectively at random
times—in particular, at stopping times associated with the market filtration. But in the
present exposition we want to avoid the idea of a “prespecified” filtration with respect to
which stopping times are defined. We take the view that the market filtration is a “derived”
notion, generated by information about upcoming cash flows, and by the values of cash
flows when they occur. We shall therefore regard a “randomly-timed” cash flow as being a
set of random cash flows occurring at various times, and with a joint distribution function
that ensures only one of these flows is non-zero. Hence in our view the ontological status of
a cash flow is that its timing is definite, only the amount is random—and that cash flows
occurring at different times are, by their nature, different cash flows.
III. MODELLING THE CASH FLOWS
First we consider the case of an asset that provides a single isolated cash flow occurring
at time T , represented by a random variable DT . We assume that DT ≥ 0. The value St of
the cash flow at any earlier time t in the interval 0 ≤ t < T is then given by the discounted
conditional expectation of DT :
St = PtTE [DT |Ft] . (1)
In this way we model the price process {St}0≤t<T of a limited-liability asset that pays a
single dividend DT at time T . The construction of the price process here is carried out in
such a way as to guarantee an arbitrage-free market if other assets are priced by the same
method (see Davis [7] for a closely related point of view). We shall use the terms “cash flow”
and “dividend” more or less interchangeably. If a more specific use of one of these terms
is needed, then this will be evident from the context. We adopt the convention that when
the dividend is paid the asset price goes “ex-dividend” immediately. Hence in the example
above we have limt→T St = DT and ST = 0.
In the case where the asset pays a sequence of dividends DTk (k = 1, 2, . . . , n) on the
dates Tk the price (for t < T1) is given by
PtTkE [DTk |Ft] . (2)
More generally, for all t ≥ 0, and taking into account the ex-dividend behaviour, we have
1{t<Tk}PtTkE [DTk |Ft] . (3)
It will be useful if we adopt the convention that a discount bond also goes ex-dividend
on its maturity date. Thus in the case of a discount bond we assume that the price of the
bond is given, for dates earlier than the maturity date, by the product of the principal and
the relevant discount factor. But at maturity (when the principal is paid) the value of the
bond drops to zero. In the case of a coupon bond, there is likewise a downward jump in
the price of the bond at the time a coupon is paid (the value lost may be captured back in
the form of an “accrued interest” payment). In this way we obtain a consistent treatment
of the “ex-dividend” behaviour of all of the asset price processes under consideration. With
this convention it follows that price processes are right continuous with left limits.
IV. MODELLING THE INFORMATION FLOW
Now we present a simple model for the flow of market information, following Brody et
al. [3]. We consider first the case of a single distribution, occurring at time T , and assume
that market participants have only partial information about the upcoming cash flow DT .
The information available in the market about the cash flow is assumed to be contained in
a process {ξt}0≤t≤T defined by:
ξt = σtDT + βtT . (4)
We call {ξt} the market information process. This process is composed of two parts. The
term σtDT contains the “true information” about the dividend, and grows in magnitude as
t increases. The process {βtT}0≤t≤T is a standard Brownian bridge over the time interval
[0, T ]. Thus {βtT } is Gaussian, β0T = 0, βTT = 0, the random variable βtT has mean zero,
and the covariance of βsT and βtT for s ≤ t is s(T − t)/T . We assume that DT and {βtT }
are independent. Thus the information contained in the bridge process is “pure noise”.
We assume that the market filtration {Ft} is generated by the market information process:
Ft = σ({ξs}0≤s≤t). The dividend DT is therefore FT -measurable, but is not Ft-measurable
for t < T . Thus the value of DT becomes “known” at time T , but not earlier. The
bridge process {βtT } is not adapted to {Ft} and thus is not directly accessible to market
participants. This reflects the fact that until the dividend is paid the market participants
cannot distinguish the “true information” from the “noise” in the market. The introduction
of the Brownian bridge models the fact that market perceptions, whether valid or not, play
a role in determining asset prices. Initially, all available information is used to determine
the a priori probability distribution for DT . The parameter σ represents the rate at which
information about the true value of DT is revealed as time progresses. If σ is low, the value
of DT is effectively hidden until very near the time of the dividend payment; whereas if σ is
high, then the value of the cash flow is for all practical purposes revealed quickly.
In the example under consideration we have made some simplifying assumptions concern-
ing the information structure. For instance, we assume that σ is constant. In Section XIII,
however, we consider a time-dependent information flow rate. We have also assumed that
the random dividend DT enters directly into the structure of the information process, and
enters linearly. As we shall indicate later, a more general and natural setup is to let the
information process depend on a random variable XT which we call a “market factor”; then
the dividend is regarded as a function of the market factor. This arrangement has the ad-
vantage that it easily generalises to the situation where a cash flow might depend on several
independent market factors, or indeed where cash flows associated with different financial
instruments have one or more factors in common.
Given the market information structure described above for a single cash flow, we proceed
to construct the associated price dynamics. The price process {St} for a share in the firm
paying the specified dividend is given by formula (1). It is assumed that the a priori
probability distribution of the dividend DT is known. This distribution is regarded as part
of the initial data of the problem, which in some cases can be calibrated from knowledge
of the initial price of the asset along with other price data. The general problem of how
the a priori distribution is obtained is an important one—any asset pricing model has to
confront this issue—which we defer for later consideration. The initial distribution is not to
be understood as being “absolutely” determined, but rather represents the “best estimate”
for the distribution given the data available at that time, in accordance with what one
might call a Bayesian point of view. We note the fact that the information process {ξt} is
Markovian (see Brody et al. [3], and Rutkowski and Yu [22]). Making use of this property
together with the fact that DT is FT -measurable we deduce that
St = 1{t<T}PtTE [DT |ξt] . (5)
If the random variable DT that represents the payoff has a continuous distribution, then the
conditional expectation in (5) can be expressed in the form
E [DT |ξt] =
xπt(x) dx. (6)
Here πt(x) is the conditional probability density for the random variable DT :
πt(x) =
Q(DT ≤ x|ξt). (7)
We implicitly assume appropriate technical conditions on the distribution of the dividend
that will suffice to ensure the existence of the expressions under consideration. Also, for con-
venience we use a notation appropriate for continuous distributions, though corresponding
results can be inferred for discrete distributions, or more general distributions, by slightly
modifying the stated assumptions and conclusions.
Bearing in mind these points, we note that the conditional probability density process
for the dividend can be worked out by use of a form of the Bayes formula:
πt(x) =
p(x)ρ(ξt|DT = x)
p(x)ρ(ξt|DT = x)dx
. (8)
Here p(x) denotes the a priori probability density for DT , which we assume is known as an
initial condition, and ρ(ξt|DT = x) denotes the conditional density for the random variable
ξt given that DT = x. Since βtT is a Gaussian random variable with mean zero and variance
t(T − t)/T , we deduce that the conditional probability density for ξt is
ρ(ξt|DT = x) =
2πt(T − t)
−(ξt − σtx)
2t(T − t)
. (9)
Inserting this expression into the Bayes formula we get
πt(x) =
p(x) exp
T−t(σxξt −
σ2x2t)
p(x) exp
T−t(σxξt −
σ2x2t)
. (10)
We thus obtain the following result for the asset price:
Proposition 1. The information-based price process {St}0≤t≤T of a limited-liability asset
that pays a single dividend DT at time T with a priori distribution
Q(DT ≤ y) =
p(x) dx (11)
is given by
St = 1{t<T}PtT
xp(x) exp
T−t(σxξt −
σ2x2t)
p(x) exp
T−t(σxξt −
σ2x2t)
, (12)
where ξt = σtDT + βtT is the market information.
V. ASSET PRICE DYNAMICS IN THE CASE OF A SINGLE CASH FLOW
In order to analyse the properties of the price process deduced above, and to be able
to compare it with other models, we need to work out the dynamics of {St}. One of the
advantages of the model under consideration is that we have an explicit expression for
the price at our disposal. Thus in obtaining the dynamics we need to find the stochastic
differential equation of which {St} is the solution. This turns out to be an interesting
exercise because it offers some insights into what we mean by the assertion that market
price dynamics should be regarded as an “emergent phenomenon”. To obtain the dynamics
associated with the price process {St} of a single-dividend paying asset let us write
DtT = E[DT |ξt]. (13)
Evidently, DtT can be expressed in the form DtT = D(ξt, t), where D(ξ, t) is defined by
D(ξ, t) =
xp(x) exp
T−t(σxξ −
σ2x2t)
p(x) exp
T−t(σxξ −
σ2x2t)
. (14)
A straightforward calculation making use of the Ito rules shows that the dynamical equation
for {DtT} is given by
dDtT =
T − t
T − t
ξt − σTDtT
dt+ dξt
. (15)
Here Vt is the conditional variance of the dividend:
Vt = Et
(DT − Et[DT ])2
x2πt(x) dx−
xπt(x) dx
. (16)
Therefore, if we define a new process {Wt}0≤t<T by setting
Wt = ξt −
T − s
σTDtT − ξs
ds, (17)
we find, after some rearrangement, that
dDtT =
T − t
VtdWt. (18)
For the dynamics of the asset price we thus have
dSt = rtStdt + ΓtTdWt, (19)
where rt = −d lnP0t/dt is the short rate, and the absolute price volatility ΓtT is
ΓtT = PtT
T − t
Vt. (20)
A slightly different way of arriving at this result is as follows. We start with the condi-
tional probability πt(x). Then, using the notation above, for its dynamics we obtain
dπt(x) =
T − t
(x−DtT )πt(x) dWt. (21)
Since according to (5) the asset price is given by
St = 1{t<T}PtT
xπt(x) dx, (22)
we can infer the dynamics of {St} from the dynamics of the conditional probability {πt(x)},
once we take into account formula (16) for the conditional variance.
As we shall demonstrate later, the process {Wt} defined in (17) is an {Ft}-Brownian
motion. Hence from the point of view of the market it is the process {Wt} that drives
the asset price dynamics. In this way our framework resolves the paradoxical point of view
usually adopted in financial modelling in which {Wt} is regarded on the one hand as “noise”,
and yet on the other hand also generates the market information flow. And thus, instead
of hypothesising the existence of a driving process for the dynamics of the markets, we are
able from the information-based perspective to deduce the existence of such a process.
The information-flow parameter σ determines the overall magnitude of the volatility.
In fact, σ plays a role analogous to the similarly-labelled parameter in the Black-Scholes-
Merton theory. Thus, we can say that the rate at which information is revealed in the
market determines the magnitude of the volatility. Everything else being the same, if we
increase the information-flow rate, then the market volatility will increase as well. According
to this point of view, those mechanisms that one might have thought were destined to
make markets more efficient—e.g., globalisation of the financial markets, reduction of trade
barriers, improved communications, a robust regulatory environment, and so on—can have
the effect of increasing market volatility, and hence market risk, rather than reducing it.
VI. EUROPEAN-STYLE CALL OPTIONS
Before we turn to the consideration of more general cash flows and market information
structures, let us consider the pricing of a derivative on an asset for which the price process
is governed by (19). Specifically, we consider the valuation of a European call option on
such an asset, with strike price K, and exercisable at a fixed date t. The option is written
on an asset that pays a single dividend DT at time T > t. The initial value of the option is
C0 = P0tE
(St −K)+
. (23)
Inserting the expression for St derived in the previous section into this formula, we obtain
C0 = P0t E
x πt(x)dx−K
. (24)
For convenience we write the conditional probability πt(x) in the form
πt(x) =
pt(x)
pt(x)dx
, (25)
where the “unnormalised” density pt(x) is defined by
pt(x) = p(x) exp
T − t
σxξt − 12σ
. (26)
Substituting (26) into (24) we find that the value of the option is
C0 = P0tE
(PtTx−K) pt(x)dx
, (27)
where
pt(x)dx. (28)
The random variable 1/Φt can be used to introduce a measure B on (Ω,Ft), which we call
the “bridge measure”. The option price can then be written:
C0 = P0tE
(PtTx−K) pt(x)dx
. (29)
The special feature of the bridge measure, as we establish in Section XIV in a more general
context, is that the random variable ξt is Gaussian under B. In particular, under B we find
that {ξt} has mean 0 and variance t(T − t)/T . Since pt(x) can be expressed as a function
of ξt, when we calculate the expectation in (29) we obtain a tractable formula for C0.
To determine the value of the option we define a constant ξ∗ (the critical value) by the
following condition:
(PtTx−K) p(x) exp
T − t
σxξ∗ − 1
σ2x2t
dx = 0. (30)
Then the expectation in (29) can be performed and we find that the option price is
C0 = P0T
x p(x)N
− z∗ + σx
dx− P0tK
p(x)N
− z∗ + σx
dx, (31)
where N(x) is the standard normal distribution function, and
T − t
, z∗ = ξ∗
t(T − t)
. (32)
We see that a tractable expression of the Black-Scholes-Merton-Merton type is obtained. The
option pricing problem, even for general p(x), reduces to an elementary numerical problem.
It is interesting to note that although the probability distribution for the price St is not of
a “standard” type, nevertheless the option valuation problem remains a solvable one.
VII. EXAMPLES OF SPECIFIC DIVIDEND STRUCTURES
In this section we consider the dynamics of assets with various dividend structures. First
we look at a simple asset for which the cash flow is exponentially distributed. The a priori
probability density for DT is thus of the form
p(x) =
exp (−x/δ) , (33)
where δ is a constant. We can regard the idea of an exponentially distributed payout as
a model for the situation where little is known about the probability distribution of the
dividend, apart from its mean. Then from formula (12) we find that the asset price is:
St = 1{t<T}PtT
x exp(−x/δ) exp
T−t(σxξt −
σ2x2t)
exp(−x/δ) exp
T−t(σxξt −
σ2x2t)
. (34)
We note that S0 = P0T δ, so we can calibrate δ by use of the initial price. The integrals
in the numerator and denominator in the expression above can be worked out explicitly.
Hence, we obtain a closed-form expression for the price in the case of a simple asset with an
exponentially-distributed cash flow:
St = 1{t<T}PtT
B2t /At
2πAt N(Bt/
, (35)
where At = σ
2tT/(T − t) and Bt = σTξt/(T − t)− δ−1.
Next we consider the case of an asset for which the single dividend paid at T is gamma-
distributed. More specifically, we assume the probability density is of the form
p(x) =
(n− 1)!
xn−1 exp(−δx), (36)
where δ is a positive real number and n is a positive integer. This choice for the probability
density also leads to a closed-form expression for the share price. We find that
St = 1{t<T}PtT
t Fk(−Bt/
k−n+1
n−k−1
t Fk(−Bt/
, (37)
where At and Bt are as above, and
Fk(x) =
zk exp
dz. (38)
A recursion formula can be worked out for the function Fk(x). This is given by
(k + 1)Fk(x) = Fk+2(x)− xk+1 exp
, (39)
from which it follows that F0(x) =
2πN(−x), F1(x) = e−
x2 , F2(x) = xe
2πN(−x),
F3(x) = (x
2 + 2)e−
x2 , and so on. In general, the polynomial parts of {Fk(x)}k=0,1,2,... are
related to the Legendre polynomials.
VIII. MARKET FACTORS AND MULTIPLE CASH FLOWS
In this section we proceed to consider the more general situation where the asset pays
multiple dividends. This will allow us to consider a wider range of financial instruments.
Let us write DTk (k = 1, . . . , n) for a set of random dividends paid at the pre-designated
dates Tk (k = 1, . . . , n). Possession of the asset at time t entitles the bearer to the cash
flows occurring at times Tk > t. For simplicity we assume n is finite. For each value of
k we introduce a set of independent random variables XαTk (α = 1, . . . , mk), which we call
market factors or X-factors. For each value of α we assume that the market factor XαTk is
FTk-measurable, where {Ft} is the market filtration.
For each value of k, the market factors {XαTj}j≤k represent the independent elements that
determine the cash flow occurring at time Tk. Thus for each value of k the cash flow DTk is
assumed to have the following structure:
DTk = ∆Tk(X
, XαT2 , ..., X
), (40)
where ∆Tk(X
, XαT2 , ..., X
) is a function of
j=1mj variables. For each cash flow it is, so
to speak, the job of the financial analyst (or actuary) to determine the relevant independent
market factors, and the form of the cash-flow function ∆Tk for each cash flow. With each
market factor XαTk we associate an information process {ξ
}0≤t≤Tk of the form
ξαtTk = σ
XαTkt+ β
. (41)
Here σαTk is a parameter, and {β
} is a standard Brownian bridge over the interval [0, Tk].
We assume that the X-factors and the Brownian bridge processes are all independent. The
parameter σαTk determines the rate at which the market factorX
is revealed. The Brownian
bridge represents the associated noise. We assume that the market filtration {Ft} is gener-
ated by the totality of the independent information processes {ξαtTk}0≤t≤Tk for k = 1, 2, . . . , n
and α = 1, 2, . . . , mk. Hence, the price of the asset is given by
1{t<Tk}PtTkEt [DTk ] . (42)
IX. GEOMETRIC BROWNIAN MOTION MODEL
The simplest application of the X-factor technique arises in the case of geometric Brow-
nian motion models. We consider a limited-liability company that makes a single cash
distribution ST at time T . We assume that ST has a log-normal distribution under Q, and
can be written in the form
ST = S0 exp
rT + ν
TXT − 12ν
, (43)
where the market factor XT is normally distributed with mean zero and variance one, and
r > 0 and ν > 0 are constants. The information process {ξt} is taken to be of the form
ξt = σtXT + βtT , (44)
where the Brownian bridge {βtT} is independent of XT , and where the information flow rate
is of the special form
. (45)
By use of the Bayes formula we find that the conditional probability density is of the Gaussian
form:
πt(x) =
2π(T − t)
2(T − t)
Tx− ξt
, (46)
and has the following dynamics
dπt(x) =
T − t
Tx− ξt
πt(x)dξt. (47)
A short calculation then shows that the value of the asset at time t < T is given by
St = e
−r(T−t)
Et[ST ]
= e−r(T−t)
Tx− 1
ν2Tπt(x)dx
= S0 exp
rt+ νξt − 12ν
. (48)
The surprising fact in this example is that {ξt} itself turns out to be the innovation process.
Indeed, it is not too difficult to verify that {ξt} is an {Ft}-Brownian motion. Hence, setting
Wt = ξt for 0 ≤ t < T we obtain the standard geometric Brownian motion model:
St = S0 exp
rt+ νWt − 12ν
. (49)
We see therefore that starting with an information process of the form (44) we are able to
recover the familiar asset price dynamics given by (49).
An important point to note here is that the Brownian bridge process {βtT } appears quite
naturally in this context. In fact, if we start with (49) then we can make use of the following
orthogonal decomposition of the Brownian motion (see, e.g., Yor [24]):
. (50)
The second term in the right, independent of the first term on the right, is a standard
representation for a Brownian bridge process:
βtT = Wt −
WT . (51)
Thus by writing XT = WT/
T and σ = 1/
T we find that the right side of (50) is indeed
the market information. In other words, formulated in the information-based framework, the
standard Black-Scholes-Merton theory can be expressed in terms of a normally distributed
X-factor and an independent Brownian bridge noise process.
X. DIVIDEND GROWTH
As an elementary example of a multi-dividend structure, we shall look at a simple growth
model for dividends in the equity markets. We consider an asset that pays a sequence of
dividends DTk , where each dividend date has an associated X-factor. Let {XTk}k=1,...,n be a
set of independent, identically-distributed X-factors, each with mean 1 + g. The dividend
structure is assumed to be of the form
DTk = D0
XTj , (52)
where D0 is a constant. The parameter g can be interpreted as the dividend growth factor,
and D0 can be understood as representing the most recent dividend before time zero. For
the price of the asset we have:
St = D0
1{t<Tk}PtTkEt
. (53)
Since the X-factors are independent, the conditional expectation of the product appearing
in this expression factorises into a product of conditional expectations, and each such con-
ditional expectation can be written in the form of an expression of the type we have already
considered. As a consequence we are led to a tractable family of dividend growth models.
XI. ASSETS WITH COMMON FACTORS
The multiple-dividend asset pricing model introduced in Section VIII can be extended in
a very natural way to the situation where two or more assets are being priced. In this case
we consider a collection of N assets with price processes {S(i)t }i=1,2,...,N . With asset number
(i) we associate the cash flows {D(i)Tk} paid at the dates {Tk}k=1,2,...,n. We note that the dates
{Tk}k=1,2,...,n are not tied to any specific asset, but rather represent the totality of possible
cash-flow dates of any of the given assets. If a particular asset has no cash flow on one of the
dates, then it is assigned a zero cash-flow for that date. From this point, the theory proceeds
exactly as in the single asset case. That is to say, with each value of k we associate a set of
X-factors XαTk (α = 1, 2, . . . , mk), and a system of market information processes {ξ
}. The
X-factors and the information processes are not tied to any particular asset. The cash flow
occurring at time Tk for asset number (i) is given by a cash flow function of the form
(XαT1, X
, ..., XαTk). (54)
In other words, for each asset each cash flow can depend on all of the X-factors that have
been “activated” at that point. Thus for the general multi-asset model we have the following
price process system:
1{t<Tk}PtTkEt
. (55)
It is possible in general for two or more assets to “share” an X-factor in association with
one or more of the cash flows of each of the assets. This in turn implies that the various
assets will have at least one Brownian motion in common in the dynamics of their price
processes. We thus obtain a natural model for the correlation structures in the prices of
these assets. The intuition is that as new information comes in (whether “true” or “bogus”)
there will be several different assets all affected by the news, and as a consequence there will
be a correlated movement in their prices.
XII. ORIGIN OF UNHEDGEABLE STOCHASTIC VOLATILITY
Based on the general model introduced in the previous sections, we are now in a position
to make an observation concerning the nature of stochastic volatility in the equity markets.
In particular, we shall show how a natural framework for stochastic volatility arises in the
information-based framework. This is achieved without the need for any ad hoc assumptions
concerning the dynamics of the stochastic volatility. In fact, a very specific dynamical model
for stochastic volatility is obtained—thus leading to a possible means by which the theory
proposed here might be tested.
We shall work out the volatility associated with the dynamics of the asset price process
{St} given by (42). The result is given in Proposition 2 below. First, as an example, we
consider the dynamics of an asset that pays a single dividend DT at T . We assume that the
dividend depends on the market factors {XαT }α=1,...,m. For t < T we then have:
St = PtTE
X1T , . . . , X
∣ ξ1tT , . . . , ξ
= PtT
· · ·
∆T (x
1, . . . , xm) π1tT (x1) · · ·πmtT (xm) dx1 · · ·dxm. (56)
Here the various conditional probability density functions παtT (x) for α = 1, . . . , m are
παtT (x) =
pα(x) exp
σα x ξαtT − 12(σ
α)2 x2t
pα(x) exp
σα x ξαtT − 12(σα)2 x2t
, (57)
where pα(x) denotes the a priori probability density function for the factor XαT . The drift of
{St}0≤t<T is given by the short rate. This is because Q is the risk-neutral measure, and no
dividend is paid before T . Thus, we are left with the problem of determining the volatility
of {St}. We find that for t < T the dynamical equation of {St} assumes the form:
dSt = rtStdt +
ΓαtTdW
t . (58)
Here the volatility term associated with factor number α is given by
ΓαtT = σ
T − t
PtT Cov
X1T , . . . , X
, XαT
, (59)
and {W αt } denotes the Brownian motion associated with the information process {ξαt }, as
defined in (17). The absolute volatility of {St} is of the form
(ΓαtT )
. (60)
For the dynamics of a multi-factor single-dividend paying asset we can thus write dSt =
rtStdt+ΓtdZt, where the {Ft}-Brownian motion {Zt} that drives the asset-price process is
ΓαsT dW
s . (61)
The point to note here is that in the case of a multi-factor model we obtain an unhedgeable
stochastic volatility. That is to say, although the asset price is in effect driven by a single
Brownian motion, its volatility in general depends on a multiplicity of Brownian motions.
This means that in general an option position cannot be hedged with a position in the
underlying asset. The components of the volatility vector are given by the covariances of
the cash flow and the independent market factors. Unhedgeable stochastic volatility thus
emerges from the multiplicity of uncertain elements in the market that affect the value of
the future cash flow. As a consequence we see that in this framework we obtain a natural
explanation for the origin of stochastic volatility.
This result can be contrasted with, say, the Heston model [12], which despite its popularity
suffers from the fact that it is ad hoc in nature. Much the same can be said for the various
generalisations of the Heston model used in commercial applications. The approach to
stochastic volatility proposed in the present paper is thus of a new character. Expression
(58) generalises to the case for which the asset pays a set of dividends DTk (k = 1, . . . , n),
and for each k the dividend depends on the X-factors {{XαTj}
α=1,...,mj
j=1,...,k }. The result can be
summarised as follows.
Proposition 2. The price process of a multi-dividend asset has the following dynamics:
dSt = rt St dt +
1{t<Tk}
σαkTk
Tk − t
PtTk Cov
DTk , X
dW αkt
DTkd1{t<Tk}, (62)
where DTk = ∆Tk(X
, XαT2 , · · · , X
) is the dividend at time Tk (k = 1, 2, . . . , n).
XIII. TIME-DEPENDENT INFORMATION FLOW
We consider now a generalisation of the foregoing material to the situation in which the
information-flow rate varies in time. The time-dependent problem is of relevance to many
circumstances. For example, there will typically be more activity in a market during the day
than at night—such a consideration is important for short-term investments. Alternatively,
it may be that the annual report of a firm is going to be published on a specified day—in
this case much more information concerning the future of the firm may be made available
on that day than normal.
We begin our analysis of the time-dependent case by considering the situation where
there is a single cash flow DT occurring at T , and the associated market factor is the cash
flow itself. In this way we can focus our attention on mathematical issues arising from the
time dependence of the information flow rate. Once these issues have been dealt with, we
shall consider more complicated cash-flow structures. For the market information process
we propose an expression of the form
ξt = DT
σsds+ βtT , (63)
where the function {σs}o≤s≤T is taken to be deterministic and nonnegative. We assume that
σ2sds < ∞. The price process {St} of the asset is then given by
St = 1{t<T}PtTE [DT |Ft ] . (64)
where the market filtration is, as in the previous sections, assumed to be generated by the
information process.
Our first task is to work out the conditional expectation in (64). This can be achieved
by use of a change-of-measure technique, which will be outlined in Section XIV. It will
be useful, however, to state the result first. We define the conditional probability density
process {πt(x)} by setting
πt(x) =
Q (DT ≤ x| Ft) . (65)
The following result is obtained:
Proposition 3. Let the information process {ξt} be given by (63). Then the conditional
probability density process {πt(x)} for the random variable DT is given by
πt(x) =
p(x) e
x( 1T−t ξt
σsds+
σsdξs)− 1
σsds)
σ2sds
p(x) e
x( 1T−t ξt
σsds+
σsdξs)− 1
σsds)
σ2sds
. (66)
We deduce at once from Proposition 3 that the conditional expectation of the random
variable DT is
DtT =
xp(x) e
x( 1T−t ξt
σsds+
σsdξs)− 1
σsds)
σ2sds
p(x) e
x( 1T−t ξt
σsds+
σsdξs)− 1
σsds)
σ2sds
. (67)
The associated price process {St} is then given by St = 1{t<T}PtTDtT .
XIV. CHANGES OF MEASURE FOR BROWNIAN BRIDGES
Since the information process is a Brownian bridge with a random drift, we shall require
formulae relating a Brownian bridge with drift in one measure to a standard Brownian bridge
in another measure to establish Proposition 3 . We proceed as follows. First we recall a well-
known integral representation for the Brownian bridge. Let the probability space (Ω,F ,Q)
be given, with a filtration {Gt}0≤t<∞, and let {Bt} be a standard {Gt}-Brownian motion.
Then the process {βtT}, defined by
βtT = (T − t)
T − s
dBs, (68)
for 0 ≤ t < T , and by βtT = 0 for t = T , is a standard Brownian bridge over the interval
[0, T ]. Expression (68) converges to zero as t → T ; see, e.g., Karatzas and Shreve [16],
Protter [21]). The filtration {Gt} is larger than the market filtration {Ft}. In particular,
since {βtT} is adapted to {Gt} we can think of {Gt} as the filtration describing the information
available to an “insider” who can distinguish between what is noise and what is not.
Let DT be a random variable on (Ω,F ,Q). We assume thatDT is G0-measurable and that
DT is independent of {βtT}. Thus the value of DT is known “all along” to the insider, but
not to the typical market participant. For simplicity in what follows we assume that DT is
bounded; this condition can be relaxed with the introduction of an appropriate Novikov-type
condition; but we will not pursue the more general situation here. Define the deterministic
nonnegative process {νt}0≤t≤T by
νt = σt +
T − t
σsds, (69)
and let {ξt} be defined as in (63). We define the process {Λt}0≤t<T by the relation
= exp
νsdBs − 12D
ν2sds
. (70)
With these elements in hand, we fix a time horizon U ∈ (0, T ) and introduce a probability
measure B on GU by the relation
dB = Λ−1U dQ. (71)
Then we have the following facts: (i) The process {W ∗t }0≤t<U defined by
W ∗t = DT
νsds+Bt (72)
is a B-Brownian motion. (ii) The process {ξt} defined by (63) is a B-Brownian bridge and
is independent of DT . (iii) The random variable DT has the same probability law with
respect to B and Q. (iv) The conditional expectation for any integrable function f(DT ) of
the random variable DT can be expressed in the form
Q[f(DT )|F ξt ] =
f(DT )Λt
∣F ξt
∣F ξt
] . (73)
We note that the measure B is independent of the specific choice of the time horizon U in
the sense that if B is defined on GU ′ for some U ′ > U , then the restriction of that measure
to GU agrees with the measure B as already defined.
When we say that {ξt} is a B-Brownian bridge what we mean, more precisely, is that
ξ0 = 0, that {ξt} is B-Gaussian, that EB[ξt] = 0, and that
B[ξsξt] =
s(T − t)
for 0 ≤ s ≤ t ≤ U . Thus with respect to the measure B the process {ξt}0≤t≤U has the
properties of a standard [0, T ]-Brownian bridge that has been truncated at time U . The fact
that {ξt} is a B-Brownian bridge can be verified as follows. By (63), (68), and (72) we have
ξt = DT
σsds+ (T − t)
T − s
σsds+ (T − t)
T − s
(dW ∗s −DTνsds)
σsds− (T − t)
T − s
+ (T − t)
T − s
dW ∗s
= (T − t)
T − s
dW ∗s , (75)
where in the final step we used the relation
T − s
νsds =
T − t
σsds. (76)
This formula can be verified explicitly by differentiation, which then gives us (69). In (75)
we see that {ξt} has been given the standard integral representation of a Brownian bridge.
We remark, incidentally, that (73) can be thought of a variation of the Kallianpur-Striebel
formula appearing in the literature of nonlinear filtering (see, for example, Bucy and Joseph
[4], Davis and Marcus [7], Fujisaki et al. [9], Kallianpur and Striebel [15], Krishnan [17],
and Liptser and Shiryaev [18]).
XV. DERIVATION OF THE CONDITIONAL DENSITY
We have introduced the idea of measure changes associated with Brownian bridges in
order to introduce formula (73), which involves the density process {Λt}, which in (70) is
defined in terms of the Q-Brownian motion {Bt}. On the other hand, the expectations in
(73) are conditional with respect to the information generated by {ξt}. Therefore, it will be
convenient to express {Λt} in terms of {ξt}. To do this we substitute (72) in (70) to obtain
Λt = exp
s − 12D
ν2sds
. (77)
We then observe, by differentiating (75), that
dξt = −
T − t
dt+ dW ∗t . (78)
Substituting this relation in (77) we obtain
Λt = exp
νsdξs +
T − s
νsξsds
ν2sds
. (79)
In principle at this point all we need to do is to substitute (69) into (79) to obtain the
result for {Λt}. In practice, further simplification can be achieved. To this end, we note
that by taking the differential of the coefficient of DT in the exponent of (79) we get
νsdξs +
T − s
νsξsds
dξt +
T − t
T − t
dξt +
T − t
T − t
σsds+
σsdξs
. (80)
Then integrating both sides of (80) we obtain:
νsdξs +
T − s
νsξsds =
T − t
σsds+
σsdξs. (81)
Similarly, by taking the differential of the coefficient of −1
D2T in the exponent of (79) and
making use of (69), we find
ν2t dt =
σ2t + 2
T − t
σsds+
(T − t)2
T − t
σ2sds
. (82)
Therefore, by integrating both sides of (82) we obtain an identity for the coefficient of −1
D2T .
It follows by virtue of the two identities just obtained that {Λt} can be expressed in terms
of {ξt}. More explicitly, we have
Λt = exp
T−t ξt
σsds+
σsdξs
σ2sds
. (83)
Note that by transforming (79) into (83) we have eliminated a term having {ξt} in the
integrand, thus achieving a considerable simplification. Proposition 3 can then be deduced
if we use equation (76) and the basic relation
Q (DT ≤ x| Ft) = EQ
1{DT≤x}
. (84)
In particular, since DT and {ξt} are independent under the bridge measure, by virtue of
(73), (83), and (84) we obtain
Q (DT ≤ x| Ft) =
p(y) e
y( 1T−t ξt
σsds+
σsdξs)− 1
σsds)
σ2sds
p(y) e
y( 1T−t ξt
σsds+
σsdξs)− 1
σsds)
σ2sds
, (85)
from which we immediately infer Proposition 3 by differentiation with respect to x.
We conclude this section by noting that an alternative expression for {πt(x)}, written in
terms of {W ∗t }, is given by
πt(x) =
p(x) exp
u − 12x
ν2udu
p(x) exp
νudW ∗u − 12x2
ν2udu
. (86)
XVI. CONSISTENCY RELATIONS
Before we proceed to analyse in detail the dynamics of the price process {St}, first we
shall establish a useful dynamical consistency relation satisfied by prices obtained in the
information-based framework. By “consistency” we have in mind the following. Suppose
that we re-initialise the information process at an intermediate time s ∈ (0, T ) by specifying
the value ξs of the information at that time. For the framework to be dynamically consistent,
we require that the remainder of the period [s, T ] admits a representation in terms of a
suitably “renormalised” information process. Specifically, we have:
Proposition 4. Let 0 ≤ s ≤ t ≤ T . The conditional probability πt(x) can be written in
terms of the intermediate conditional probability πs(x) in the form
πt(x) =
πs(x) e
x( 1T−t ηt
σ̃udu+
σ̃udηu)− 1
σ̃udu)
σ̃2udu
πs(x) e
x( 1T−t ηt
σ̃udu+
σ̃udηu)− 1
σ̃udu)
σ̃2udu
, (87)
where
σ̃u = σu +
T − s
σvdv (88)
is the re-initialised market information-flow rate, and
ηt = ξt −
T − t
T − s
ξs (89)
is the re-initialised information process.
The fact that {ηt}s≤t≤T represents the updated information process bridging the interval
[s, T ] can be seen as follows. First we note that ηs = 0 and that ηT = ξT . Substituting (63)
in (89) we find that
ηt = DT
σ̃udu+ γtT , (90)
where σ̃u is as defined in (88), and
γtT = βtT −
T − t
T − s
βsT . (91)
A calculation making use of the covariance of the Brownian bridge {βtT} shows that the
Gaussian process {γtT}s≤t≤T is a standard Brownian bridge over the interval [s, T ]. It thus
follows that {ηt} is the information bridge interpolating the interval [s, T ].
To verify (87) we note that (86) can be written in the form
πt(x) =
πs(x) exp
u − 12x
ν2udu
πs(x) exp
νudW ∗u − 12x2
ν2udu
. (92)
The identity (80) then implies that
T − t
σudu+
σudξu +
T − t
T − s
T − t
σ̃udu+
σ̃udηu, (93)
where we have made use of (88) and (89). Similarly, (81) implies that
ν2udu =
T − t
T − s
σ2udu
T − t
σ̃udu
σ̃2udu. (94)
Substitution of (93) and (94) into (92) establishes (87). In particular, the form of (87) is
identical to the original formula (66), modulo the indicated renormalisation of the informa-
tion process and the associated information flow rate.
XVII. EXPECTED DIVIDEND
The goal of sections XIII, XIV, and XV was to obtain an expression for the conditional
expectation (13) in the case of a single-dividend asset in the case of a time-dependent
information-flow rate. In the analysis of the associated price process it will therefore be
useful to work out the dynamics of the conditional expectation of the dividend. In particular,
an application of Ito’s rule to (67), after some rearrangement, shows that
dDtT = νtVt
T − t
ξt − νtDtT
dt+ νtVtdξt, (95)
where {Vt} is the conditional variance of the random variable DT , given by (16). Let us
define a new process {Wt} by
Wt = ξt +
T − s
ξsds−
νsDsTds. (96)
We refer to {Wt} as the “innovation process”. It follows from the definition of {Wt} that
dDtT = νtVt dWt. (97)
Since {DtT } is an {Ft}-martingale we are thus led to conjecture that {Wt} must also be an
{Ft}-martingale. In fact, we have the following result:
Proposition 5. The process {Wt} defined by (96) is an {Ft}-Brownian motion under Q.
Proof. We need to establish that (i) {Wt} is an {Ft}-martingale, and that (ii) (dWt)2 =
dt. Writing Et[−] = EQ[−|Ft] and letting t ≤ u we have
Et [Wu] = Et [ξu] + Et
T − s
νsDsTds
. (98)
Splitting the second two terms on the right into integrals between 0 and t, and between t
and u, we obtain
Et [Wu] = Et[ξu] +
T − s
ξsds−
νsDsTds
T − s
Et[ξs]ds−
νsEt[DsT ]ds. (99)
The martingale property of the conditional expectation implies that Et[DsT ] = DtT for t ≤ s,
which allows us to simplify the last term. To simplify the expression for the expectation
Et[ξs] for t ≤ s we use the tower property:
Et[βsT ] = Et[E[βsT |HT , βtT ]] = Et[E[βsT |βtT ]]. (100)
To calculate the inner expectation E[βsT |βtT ] here we use the fact that the random variable
βsT/(T − s)− βtT/(T − t) is independent of βtT and deduce that
E[βsT |βtT ] =
T − s
T − t
βtT , (101)
from which it follows that
Et [βsT ] =
T − s
T − t
Et[βtT ]. (102)
As a result we obtain
Et[ξs] = DtT
σvdv +
T − s
T − t
Et[βtT ]. (103)
We also recall the definition of {Wt} given by (96), which implies that
T − s
ξsds−
νsDsTds = Wt − ξt. (104)
Therefore, substituting (103) and (104) into (99) we obtain
Et [Wu] = DtT
σsds+Wt − ξt +DtT
T − s
ds−DtT
+Et[βtT ]. (105)
Next we split the first term into an integral from 0 to t and an integral from t to u, and we
insert the definition (69) of {νt} into the fifth term. The result is:
Et [Wu] = Wt +DtT
σsds+ Et[βtT ]− ξt. (106)
Finally, if we make use of the fact that ξt = Et[ξt], and hence that
ξt = DtT
σsds+ Et[βtT ], (107)
we see that {Wt} satisfies the martingale condition. On the other hand, by virtue of (96)
we have (dWt)
2 = dt. We conclude that {Wt} is an {Ft}-Brownian motion under Q. �
XVIII. ASSET PRICES AND DERIVATIVE PRICES
We are now in a position to consider in more detail the dynamics of the price process of
an asset paying a single dividend DT in the case of a time-dependent information flow. For
{St} we have St = 1{t<T}PtTDtT , or equivalently
St = 1{t<T}PtT
xp(x) e
x( 1T−t ξt
σsds+
σsdξs)− 1
σsds)
σ2sds
p(x) e
x( 1T−t ξt
σsds+
σsdξs)− 1
σsds)
σ2sds
. (108)
A calculation making use of (97) shows that for the dynamics of the price we have
dSt = rtStdt + ΓtTdWt, (109)
where the asset price volatility is given by ΓtT = νtPtTVt, where Vt is the conditional variance
of the dividend, given by (16). It should be evident by virtue of its definition that {Vt} is a
supermartingale. More specifically, for the dynamics of {Vt} we obtain
dVt = −ν2t V 2t dt+ νtκtdWt, (110)
where κt denotes the third conditional moment of DT , given by κt = Et [(DT −DtT )3].
Although we have derived (108) by assuming that the price process is induced by the
market information {ξt}, the result to be shown in Section XIX demonstrates that we can
regard the dynamical equation (109) for the price process as given, and then deduce the
structure of the underlying information. The information-based interpretation of the mod-
elling framework, however, is more appealing. According to this interpretation there is a
flow of market information, which is available to all market participants and is represented
by the filtration generated by {ξt}. Given this information, each participant will “act”, in
our interpretation, so as to minimise the risk adjusted future P&L variance associated with
the cash flow under consideration. The future P&L is determined by the value of DT , and
the estimate of DT that minimises its variance is indeed given by the conditional expectation
(13). By discounting this expectation with PtT we recover the price process {St}.
We note that {ΓtT } is “infinitely stochastic” in the sense that all of the higher-order
volatilities (the volatility of the volatility, and so on) are stochastic. These higher-order
volatilities have a natural interpretation: the volatility of the asset price is determined by
the variance of the random cash flow; the volatility of the volatility is determined by the
skewness of DT ; its volatility is determined by the kurtosis of DT ; and so on.
The fact that the asset price in the bridge measure is given by a function of a Gaussian
random variable means that the pricing of derivatives is numerically straightforward. We
have seen this in the case of a constant information-flow rate, but the result holds in the
time-dependent case as well. For example, consider a European-style call option with strike
K and maturity t, where t ≤ T , for which the value is (23). If we express the asset price St
on the option maturity date in terms of the B-Brownian motion {W ∗t } we find
C0 = P0tE
(PtTx−K)p(x) exp
s − 12x
ν2sds
, (111)
where
p(x) exp
s − 12x
ν2sds
dx. (112)
To proceed we shall use the factor 1/Φt in (111) to make a change of measure on (Ω,Ft).
The idea is as follows. We fix a time horizon u at or beyond the option expiration but before
the bond maturity, so t ≤ u < T , and define a process {Φt}0≤t≤u by use of the expression
(112), where now we let t vary in the range [0, u]. By an application of Ito calculus on (112)
we see that dΦt = νtDtTΦt dW
t . On the other hand, it follows from (78) and (96) that the
B-Brownian motion {W ∗t } and the Q-Brownian motion {Wt} are related by
dW ∗t = dWt + νtDtTdt. (113)
Therefore, in terms of {Wt} we have
dΦt = ν
tTΦtdt + νtDtTΦt dWt, (114)
from which it follows that
dΦ−1t = −νtDtTΦ−1t dWt. (115)
Upon integration we deduce
Φ−1t = exp
νsDsT dWs − 12
sT ds
. (116)
Since {νsDsT} is bounded, and s ≤ u < T , we see that {Φ−1s }0≤s≤u is a Q-martingale with
EQ[Φ−1t ] = 1, where t is the option maturity date. Therefore, the factor 1/Φt in (111) can be
used to effect a change of measure Q → B on (Ω,Ft). We note that while on the space (Ω,Gt)
it is the process {Λt} introduced in (70) that defines the measure change from B and Q, on
(Ω,Ft) it is Φt = EQ[Λt|Ft] that defines the relevant measure change. As a consequence, by
changing the measure in (111) we obtain
C0 = P0tE
(PtTx−K)p(x) exp
s − 12x
ν2sds
. (117)
This result should be compared with (27). We note that in the bridge measure the expression
s is a Gaussian random variable with mean zero and variance
ω2t =
ν2sds =
T − t
σ2sds. (118)
Here we have used the relation (82). Therefore, if we set
Y = ω−1t
s , (119)
it follows that Y is B-Gaussian. For the call price we thus have
C0 = P0tE
(PtTx−K)p(x)eωtxY−
ω2t x
, (120)
and hence
C0 = P0t
(PtTx−K)p(x)eωtxy−
ω2t x
dy. (121)
We observe that there exists a critical value y = y∗ such that the argument of the “plus”
function vanishes in the expression above. Thus y∗ is given by
(PtTx−K)p(x) eωtxy
ω2t x
dx = 0. (122)
As a consequence the call price can be written
C0 = P0t
(PtTx−K)p(x)eωtxy−
ω2t x
dy. (123)
The integration in the y variable can be performed, and we deduce the following represen-
tation for the call price:
C0 = P0t
(PtTx−K)p(x)N(ωtx− y∗)dx. (124)
When the cash flow is represented by a discrete random variable and the information-flow
rate is constant, this result reduces to an expression equivalent to the option pricing formula
derived in Brody et al. [3]. If the cash flow is a continuous random variable and the
information flow rate is constant then we recover the expression (31) given in section IV (see
also Rutkowski and Yu [22]).
We conclude this section with the remark that the simulation of {St} is straightforward.
First, we generate a Brownian trajectory, and form the associated Brownian bridge {βtT (ω)}.
We then select a value for DT by a method consistent with the a priori probability density
p(x), and substitute these in the formula ξt(ω) = DT (ω)
σsds + βtT (ω) for some choice
of {σt}. Finally, substitution of {ξt(ω)} in (108) gives us a simulated path {St(ω)}. The
statistics of the process {St} are obtained by repeating this procedure, the results of which
can be used to price derivatives, or to calibrate the information-flow rate {σt}.
XIX. EXISTENCE OF THE INFORMATION PROCESS
We consider now what might be called the “inverse problem” for information-based asset
pricing. The idea is to begin with the conditional density process {πt(x)} and to construct
from it the independent degrees of freedom represented by the X-factor DT and the noise
{βtT}. The setup is as follows. On the probability space (Ω,F ,Q) let {Wt} be a Brownian
motion and let {Ft} be the filtration generated by {Wt}. Let DT be FT -measurable, and
let {πt(x)} denote the associated conditional probability density process. We assume that
{πt(x)} satisfies the stochastic differential equation
dπt(x) = νt(x−DtT )πt(x) dWt, (125)
with the initial condition π0(x) = p(x), where {νt} is given by (69), and where
DtT =
xπt(x) dx. (126)
We define the process {ξt} as follows:
ξt = (T − t)
T − s
dWs + νsDsTds
. (127)
Then we have the following result:
Proposition 6. The random variables DT and βtT = ξt − DT
σsds are Q-independent
for all t ∈ [0, T ]. Furthermore, the process {βtT} is a Q-Brownian bridge.
Proof. To establish the independence of DT and βtT it suffices to verify that
Q[exβtT+yDT ] = EQ[exβtT ]EQ[eyDT ] (128)
for arbitrary x, y. Using the tower property we have
Q[exβtT+yDT ] = EQ
exξt E
e(y−x
σsds)DT
, (129)
where we have inserted the definition of βtT given in the statement of the Proposition. We
consider the inner expectation first. From equation (73) for the conditional expectation of
a function of DT we deduce that
e(y−x
σsds)DT
= Φ−1t
p(z) e(y−x
σsds)z ez
u− 12z
ν2ududz, (130)
where the process {Φt} is defined by (112). We now change the probability measure from
Q to B, so that the term Φ−1t appearing in (130) drops out to give us
exξt E
e(y−x
σsds)DT
p(z) e(y−x
σsds)z ez
u− 12 z
ν2ududz
p(z)EB
ex(T−t)
dW ∗s +(y−x
σsds)z+z
s − 12 z
ν2sds
p(z) e(y−x
σsds)z− 1
ν2sds+
α2sds E
s − 12
α2sds
dz, (131)
where αs = x(T − t)/(T − s) + zνs, and therefore
Q[exβtT+yDT ] =
p(z) e(y−x
σsds)z− 1
ν2sds+
α2sdsdz (132)
Furthermore, making use of relation (76) we have
σsds− 12z
ν2sds +
α2sds
= exp
t(T − t)
. (133)
As a consequence, it follows from (132) that
exβtT+yDT
p(z) eyzdz
t(T − t)
. (134)
This establishes the independence of {βtT} and DT .
The factorisation (134) also shows that the process {βtT } is Q-Gaussian, with mean zero
and variance t(T − t)/T . To establish that {βtT} is a Brownian bridge, we must show that
for s ≤ t the covariance of βsT and βtT is given by s(T − t)/T . Alternatively, it suffices to
analyse the moment generating function E[exβsT+yβtT ]. We proceed as follows. First, using
the tower property we have
exβsT+yβtT
exξs+yξt−(x
σudu+y
σudu)DT
exξs+yξtE
σudu+y
σudu)DT
. (135)
Next, by use of formula (73), the inner expectation can be carried out to give
exβs+yβtT
exξs+yξtΦ−1t
p(z) e−(x
σudu+y
σudu)z ez
u− 12z
ν2ududz
.(136)
If we change the probability measure to B the random variable Φt in the denominator drops
out, and we have
exβs+yβtT
p(z) e−(x
σudu+y
σudu)z− 1
ν2udu E
exξs+yξt+z
dz. (137)
Let us consider the inner expectation first. By defining au = x(T − s)/(T − u) and bu =
y(T − t)/(T − u) + zνu we can write
exξs+yξt+z
. (138)
However, since {W ∗t } is a B-Brownian motion, using the properties of Gaussian random
variable we find that
= exp
a2udu+
b2udu+ 2
aubudu
. (139)
Substituting the definitions of {au} and {bu} into the right side of (139) and combining the
result with the remaining terms in the exponent of the right side of (137) we find that the
terms involving the integration variable z drop out, and we are left with the integral of the
density function p(z), which is unity. Gathering the remaining terms we obtain
exβsT+yβtT
= exp
s(T − s)
t(T − t)
+ 2xy
s(T − t)
. (140)
It follows that the covariance of βsT and βtT for s ≤ t is given by
exβsT+yβtT
x=y=0
s(T − t)
. (141)
This establishes the assertion that {βtT} is a Q-Brownian bridge. �
The result above shows that, for the class of price processes we are considering, even if
at the outset we take the “usual” point of view in financial modelling, and regard the price
process of the asset as being adapted to some “prespecified” filtration, nevertheless it is
possible to deduce the structure of the underlying information-based model.
XX. MULTI-FACTOR MODELS WITH A TIME-DEPENDENT INFORMATION
FLOW RATE
Let us now turn to consider the case of a single cash flowDT that depends on amultiplicity
of market factors {XαTk}
α=1,...,mk
k=1,...,n , where we have the n pre-designated information dates
{Tk}k=1,2,...,n, and where for each value of k we have a set ofmk market factors. For simplicity
we set T = Tn. Each market factor X
is associated with an information process
ξαtTk = X
σαsTkds+ β
, (142)
where XαTk and {β
} are independent. It should be evident that although the random
variable DT representing the cash flow is FT -measurable, the values of some of the X-factors
upon which it depends may be revealed at earlier times. That is to say, the uncertainties
arising from some of the economic elements affecting the value of the cash flow at time T
may be resolved before that time.
Since the X-factors are independent, it follows that for each market factor the associated
conditional density process παtTk(x) takes the form given in (66), and the corresponding
dynamical equation is given by
dπαtTk = ν
xαk − EQ
XαTk |Ft
παtTk dW
t . (143)
The function ναtTk appearing here is given by an expression of the form (69):
ναtTk = σ
Tk − t
σαsTkds. (144)
The innovation process {W αkt } is defined in terms of {ξαtTk} via a relation of the form
W αkt = ξ
Tk − s
ξαsTkds−
ναsTkX
ds. (145)
The conditional expectation EQ[DT |Ft] is thus given by the multi-dimensional integral
DtT =
· · ·
∆T (x
1, . . . , x
1 , . . . , x
n, . . . , x
×πt1(x11) · · ·πt1(x
1 ) · · ·πtn(x1n) · · ·πtn(xmnn ) dx11 · · ·dx
1 · · ·dx1n · · ·dxmnn , (146)
and the price of the asset for t < T is St = PtTDtT . A straightforward application of Ito’s
rule then establishes the following result:
Proposition 7. The price process {St}0≤t<T of an asset that pays a single dividend DT
at time T (= Tn) depending on the market factors {XαTk}
α=1,2,...,mk
k=1,2,...n , satisfies the dynamical
equation
dSt = rtStdt+
ναtTkCovt[DT , X
] dW αkt , (147)
where
DT = ∆T
XαT1 , . . . , X
. (148)
Here Covt[DT , X
] denotes the covariance between the cash-flow DT and the market factor
XαTk , conditional on the information Ft generated by the information processes {ξ
}0≤s≤t.
In the more general case of an asset that pays multiple dividends (see Section VIII) the
price is given by
1{t<Tk}PtTkE
{XαTj}
α=1,2,...,mj
j=1,...,k
. (149)
Proposition 8. The price process {St} of an asset that pays the random dividends DTk on
the dates Tk (k = 1, . . . , n) satisfies the dynamical equation
dSt = rtStdt+
1{t<Tk}ν
Covt[DTk , X
] dW αkt +∆Tkd1{t<T}, (150)
where
DTk = ∆Tk
{XαTj}
α=1,...,mj
j=1,...,k
. (151)
Here Covt[DTk , X
] denotes the covariance between the dividend DTk and the market factor
XαTk , conditional on the market information Ft.
We conclude that the multi-factor, multi-dividend situation is fully tractable when the
information-flow rates associated with the various market factors are time dependent. A
straightforward extension of Proposition 8 allows us to formulate the joint price dynamics of
a system of assets, the associated dividend flows of which may depend on common market
factors. As a consequence, a specific model for stochastic volatility and correlation emerges
for such a system of assets, and it is one of the main conclusions of this paper that such
a model can be formulated. The information-based “X-factor” approach presented here
thus offers a new insights into the nature of volatility and correlation, and as such may find
applications in a number of different areas of financial risk analysis. We have in mind, in
particular, applications to equity portfolios, credit portfolios, and insurance, all of which
exhibit intertemporal market correlation effects. We also have in mind the problem of firm-
wide risk management and optimal capital allocation for banking institutions.
Acknowledgements. The authors thank T. Bielecki, T. Björk, I. Buckley, H. Bühlmann,
S. Carter, I. Constantinou, M. Davis, J. Dear, A. Elizalde, B. Flesaker, H. Geman, V. Hen-
derson, D. Hobson, T. Hurd, M. Jeanblanc, A. Lokka, J. Mao, B. Meister, M. Monoyios,
M. Pistorius, M. Rutkowski, D. Taylor, and M. Zervos for stimulating discussions. The
authors are also grateful for helpful comments made by seminar participants at various
meetings where parts of this work have been presented, including: the Developments in
Quantitative Finance conference, July 2005, Isaac Newton Institute, Cambridge; the Math-
ematics in Finance conference, August 2005, Kruger National Park, RSA; the School of
Computational and Applied Mathematics, University of the Witwatersrand, RSA, August
2005; CEMFI (Centro de Estudios Monetarios y Financieros), Madrid, October 2005; the
Department of Actuarial Mathematics and Statistics, Heriot-Watt University, December
2005; the Department of Mathematics, King’s College London, December 2005; the Bank
of Japan, Tokyo, December 2005; and Nomura Securities, Tokyo, December 2005. DCB
acknowledges support from The Royal Society; LPH and AM acknowledge support from
EPSRC (grant number GR/S22998/01); AM thanks the Public Education Authority of the
Canton of Bern, Switzerland, and the UK Universities ORS scheme, for support.
References.
[1] K. Back, “Insider trading in continuous time”, Rev. Fin. Studies 5, 387-407 (1992).
[2] K. Back and S. Baruch, “Information in securities markets: Kyle meets Glosten and
Milgrom”, Econometrica 72, 433-465 (2004).
[3] D. C. Brody, L. P. Hughston, and A. Macrina, “Beyond hazard rates: a new framework
for credit-risk modelling”, in Advances in Mathematical Finance: Festschrift Volume
in Honour of Dilip Madan (Basel: Birkhäuser, 2007).
[4] R. S. Bucy and P. D. Joseph, Filtering for stochastic processes with applications to
guidance (New York: Interscience Publishers, 1968).
[5] U. Cetin, R. Jarrow, P. Protter, and Y. Yildrim, “Modelling Credit Risk with Partial
Information”, Ann. Appl. Prob. 14, 1167-1172 (2004).
[6] M. H. A. Davis, “Complete-market models of stochastic volatility” Proc. Roy. Soc.
Lond. A460, 11-26 (2004)
[7] M. H. A. Davis and S. I. Marcus, “An introduction to nonlinear filtering” in
Stochastic systems: The mathematics of filtering and identification and application,
M. Hazewinkel and J. C. Willems, eds. (Dordrecht: D. Reidel, 1981).
[8] D. Duffie and D. Lando, “Term structure of credit spreads with incomplete accounting
information” Econometrica 69, 633-664 (2001).
[9] M. Fujisaki, G. Kallianpur, and H. Kunita “Stochastic differential equations for the
non linear filtering problem” Osaka J. Math. 9, 19 (1972).
[10] K. Giesecke, “Correlated default with incomplete information” J. Banking and Finance
28, 1521-1545 (2994).
[11] K. Giesecke and L. R. Goldberg, “Sequential default and incomplete information” J.
Risk 7, 1-26 (2004).
[12] S. L. Heston, “A closed-form solution for options with stochastic volatility with appli-
cations to bond and currency options” Rev. Financial Studies 6, 327-343 (1993).
[13] X. Guo, R. A. Jarrow and Y. Zeng, “Information reduction in credit risk modelling”
working paper (2005).
[14] R. A. Jarrow and P. Protter, “Structural versus reduced form models: a new informa-
tion based perspective” J. Investment Management 2, 34-43 (2004).
[15] G. Kallianpur and C. Striebel, “Estimation of stochastic systems: Arbitrary system
process with additive white noise observation errors” Ann. Math. Statist. 39, 785
(1968).
[16] I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus (Berlin:
Springer, 1997).
[17] V. Krishnan, Nonlinear Filtering and Smoothing (New York: Dover, 2005).
[18] R. S. Liptser and A. N. Shiryaev, Statistics of Random Processes Vols. I and II, 2nd
ed. (Berlin: Springer, 2000).
[19] A. Macrina, “An information-based framework for asset pricing: X-factor theory and
its applications” PhD thesis, King’s College London (2006).
[20] M. O’Hara, Market Microstructure Theory (Cambridge, Massachusetts: Blackwell,
1995).
[21] P. Protter Stochastic Integration and Differential Equations: A New Approach, 2nd
ed. (Berlin: Springer, 2003).
[22] M. Rutkowski and N. Yu, “On the Brody-Hughston-Macrina approach to modelling
of defaultable term structure”, working paper, School of Mathematics, University of
New South Wales, Sydney, downloadable at www.defaultrisk.com (2005).
[23] W. M. Wonham, “Some applications of stochastic differential equations to optimal
nonlinear filtering” J. SIAM A2, 347 (1965).
[24] M. Yor, Some Aspects of Brownian Motion, Part I: Some Special Functionals (Basel:
Birkhäuser, 1992).
[25] M. Yor, Some Aspects of Brownian Motion, Part II: Some Recent Martingale Problems
(Basel: Birkhäuser, 1996).
	Introduction
	The Modelling framework
	Modelling the cash flows
	Modelling the information flow
	Asset price dynamics in the case of a single cash flow
	European-style call options
	Examples of specific dividend structures
	Market factors and multiple cash flows
	Geometric Brownian motion model
	Dividend growth
	Assets with common factors
	Origin of unhedgeable stochastic volatility
	Time-dependent information flow
	Changes of measure for Brownian bridges
	Derivation of the conditional density
	Consistency relations
	Expected dividend
	Asset prices and derivative prices
	Existence of the information process
	Multi-factor models with a time-dependent information flow rate
