Xin Guo

IBM T.J.Watson Research Center,P.O.Box218,Yorktown,

NY10598,USA

Abstract

How can one relate stockfluctuations and information-based human ac-tivities?We present a model of an incomplete market by adjoining the Black-Scholes exponential Brownian motion model for stockfluctuations with a hidden Markov process,which represents the state of information in the investors’community.The drift and volatility parameters take differ-ent values depending on the state of this hidden Markov process.Standard option pricing procedure under this model becomes problematic.Yet,with an additional economic assumption,we provide an explicit closed-form for-mula for the arbitrage-free price of the European call option.Our model can be discretized via a Skorohod imbedding technique.We conclude with an example of a simulation of IBM stock,which shows that,not surprisingly, information does affect the market.

AMS classification:60J65;60J10;Keywords:Black-Scholes;Hidden Markov processes;Inside information;Ar-bitrage;Equivalent martingale measure

1Motivation

We begin our discussion with the classic case of Bre-X,a Canadian gold mining company.The mineral company stumbled on what looked like a huge gold cache in Indonesia.Consequently,the stock price sky-rocketed for a while.Then,all of a sudden,the volatility increased by orders of magnitude due to heavy inside tradings.The reason turned out to be that a privileged few were aware of the fraudulent gold assays performed by the company.The honeymoon was over and the stocks crashed when this news became public(cf.Figure1,New York Times, May5,1997).Bre-X perished.

Figure1:The rise and fall of Bre-X.(Source:New York Times,May5,1997.)

One of the morals of the story is:volatility increased when there was incom-

2plete information—some people knew that the assays were fraudulent while the rest did not.In general,it is reasonable to think that the existence of a one-sided group of insiders(in this case short-sellers)will drive the market faster since they will always be ready to sell if a buyer appears.They‘know’something(or believe they know something)that they think is worth money because others do not know what they know(or believe they know).

This story well-exemplifies the role of distribution of information in the in-vestors’community.Information is rarely,if ever shared,simultaneously by ev-eryone.It is exactly this time difference that may create an arbitrage opportunity, which never exists long.It appears,but is removed immediately once everyone gets the‘information’.And this information(or the lack of it)drives‘human activity’in the stock market.

With the view of understanding the stock market better,the obvious question arises:Can and how does one link market movement and human activity?This would tantamount to modeling stockfluctuations with information.This is the theme of our paper.

A guided road map.Section2details the hidden Markov process that models stockfluctuations and information changes.Section3shows the option pricing scheme for our model.Section4outlines one possible discretization of our model. Section5discusses why our model is fundamentally different from other models, such as stochastic volatility models.Finally,Section6provides some preliminary empirical evidence.

32A hidden Markov model with information

We incorporate the existence of inside information by modeling thefluctuations of a single stock price X t using an equation of the form

dX t X tµεt dt X tσεt dW t(1)

whereεt is an additional stochastic process representing the state of information in the investor community.εt is independent of W t,the Weiner process.For each state i,there is a known drift parameterµi and a known volatility parameter σi.µεtσεt take different values whenεt is in different states.

We assume thatεεt is a Markov process which moves among a few(say, 2or3)states.εt0at those times t at which the price change is not abnormal and people believe that they are all well-informed in a seemingly complete mar-ket;in this stateµεtµ0σεtσ0.But the processεt may take other values than zero.εt1when there are wildfluctuations in the stock price and peo-ple suspect that some individuals or groups have extra information which is not circulating among the mass of investors and thus would possibly bring wilderfluc-tuations depending on the reaction of the investors.Here,µεtµ1σεtσ1.µ1may be larger or smaller thanµ0depending on the nature of inside information, therefore this state may divide into two extra states where informed investors be-lieve the company will prosper or decline.Furthermore,some inside groups may actually be misled and the model could include a state which would indicate that there is a group of investors who erroneously believe that the company’s fortunes are going to change for the positive,and another state for the negative.More generally,one can use the state space S012N forεt to model more complex information structures.

4If we assume that theσ’s are distinct then it is no loss of generality to assume thatεt is actually observable,since the local quadratic variation of X t in any small interval to the left of t will yieldσεt exactly.(For details,see McKean, 1969.)Hence,even if X t is not Markovian,X tεt is jointly so.

It is conceivable that sometimes insiders will try to manipulate their buying and selling in such a way that the existence of such information is not detectable from the change of volatilities,namelyσ’s are identical.The problem of de-tecting the state change ofεt whenσ’s remain unchanged appears to be hard

to solve mathematically.It is plausible that change in information distribution, hence predictability,manifests itself in the diffusion coefficient in the form of both stochastic volatility and drift.

A two-state model.For ease of exposition,we focus on the two-state case,in whichεt alternates between0and1such that

0when the market seems complete,and

εt

(2)

1when some people have(or believe they have)inside information; whereσ0σ1.

Suppose,further,that each piece of informationflow is a random process Y i, and Y1Y2Y n being i.i.d processes,then their super-imposed process(under minor technical restrictions)is Poisson.Therefore,letλi denote the rate of leaving state i,τi the time of leaving state i,then

Pτi t eλi t i01(3) The memoryless property of this process is plausible in that,from a practical standpoint,the informationflow be identified more easily otherwise.

53Option pricings and arbitrage

3.1Completing the market—new securities COS

It is easy to see that the model is not“complete”,according to Harrison and Pliska (1981),Harrison and Kreps(1979),because of the additional processεt.In other words,εt is a bounded adapted process with respect to theσ-algebra F t generated by X t(denoted as F X),but is not adapted to theσ-algebra generated by W t(written as F W).

One way(by D.Duffie)to complete the market is as follows:at each time t, there is a market for a security that pays one unit of account(say,a dollar)at the next timeτt inf u tεuεt that the Markov chainεt changes state. That contract then becomes worthless(i.e.,has no future dividends),and a new contract is issued that pays at the next change of state,and so on.Under natural pricing,this will complete the market,and provide unique arbitrage-free prices to the hedge options on the underlying risk asset.(For reference,see Harrison and Pliska,1981).

One can think of this as an insurance contract that compensates its holder for any losses that occur when the next state change occurs.Of course,if one wants to hedge a given deterministic loss C at the next state change,one holds C of the current change-of-state(COS)contracts.

3.2Pricing and no arbitrage

As an assumption analogous to the assumption of the pricing of the underlying risky“stock,”it is natural to propose that the current COS contract trade for a

6

price of

V t E e r kεtτt t F t(4) where k:01ℜis given,and can be thought of as a risk-premium coefficient. More precisely,the current COS contract price is

V t Jεt(5)

λi

J i

(8)

rλQ i

Of course,J J Q,and therefore

rλi

λQ iThe same exercise applied to the underlying risky-asset implies that its price pro-cess S must have the form

dS t r dεt S t dt S tσεt dB Q(10)

where B Q is a standard Brownian motion under Q.

Now the usual techniques from Harrison and Kreps(1979)and Harrison and Pliska(1981)can be applied to get complete market and unique pricing for any derivatives with appropriate square-integrable cashflows.

Theorem1Given Eq.(10),COS,and a riskless interest rate r,the arbitrage free price of a European call option with expiration date T and strike price K is:

V i T K r E Q e rT X T Kε0i(11)

e rT

∞

T

yρln y K m t v t f i t T dtdy(12)

whereρx m t v t is the normal density function with expectation m t and variance v t,and

f0t T eλ1T eλ1λ0t

T t

λ0λ1

t12J12λ0λ1Ttλ0λ1T212

λ1J02λ0λ1Ttλ0λ1T212(14) m t d1d012σ20σ21t r d112σ21T(15) v tσ20σ21tσ21T(16)

8where J a z is the Bessel function such that(cf.Oberhettinger and Badii,1973)

J a z 1

n!Γa n1

(17)

Y a z cotπa J a z cscπa J a z(18)

In particular,whenµ0µ1σ1σ0,we have f i t Tδt T,and therefore above the equations reduce to the classical Black-Scholes formula for European options.

The key idea is to calculate the probability distribution function of a“telegraph process”.This was obtained independently and earlier by Di Masi et.al.(1994). Our Laplace transform based approach is entierly different.The details of our proof are given in Appendix A.

Comparing Eq.(10)with a geometric Brownian motion process with drift r and varianceσ,dεt is of special interest to us.A careful examination reveals that this very extra term dεt differentiates our model from the standard stochastic volatility and Markov volatility models,in that it invalidates the martingale pricing approach.Moreover,it provides us a way to“understand”theflow of the infor-mation.The drift differs from the riskless interest rate r by d1d0when there is some informationflow and hence the arbitrage opportunity emerges.It also sug-gests the difference between the case of“pure noise”(i.e.,σ0σ1d0d1)and the case when there may exist an inside information(i.e.,σ0σ1d1d0).

4A discretization of the CRR type

To facilitate numerical simulations,we present one way of discretizing our con-tinuous market model along the vein of Cox,Ross,and Rubinstein(Cox,Ross, and Rubinstein,1979).It is worth pointing out that this methodology applies also

9to the general case where the the hidden Markov processεt takes more than two states,i.e.,the state space can be S01N,when,for example,more complex information patterns can be imposed.

Suppose the time interval0t is divided into n sub-intervals such that t nh. Let X X k where S k is a price at time kh,and define:

Xεk

k

X kεk X khεkh(19) then the following recurrence is obtained,

X nεnηεnεn1

n X n1εn1(20) whereηi j n are i.i.d random variables taking values u j with probability p jδi1j 1δi1j eλi h and1u j with probability1p jδi1j1δi1j eλi h re-spectively(i j01),where

a i eλi h u i eσi h05σ2i h

h

(21)

By the memoryless property ofτi Xεk k i01is a Markov chain.More pre-cisely,the Markov chain X Xεn n with initial state X0x is a random walk on the set E x xu r rσ0mσ1n m n Z u eof X n and X.Via a Skorohod imbedding technique,we prove that˜X n converges to ˜X with probability one,hence the convergence in distribution of X

n to X.

Proof Sketch:[of Theorem2]The key here is to calculate the characteristic func-tions of Y t ln X t and Yεn n,(cf.Feller,1971)where

Y t Y0

t

µε1

h j01(23)

Without loss of generality,let Yε0

0,let

f j t E E e icY tε0j

E e t0icµεic12c2σ2εdsε0j(24) then starting from time0,between time0and h,eitherεhε0j with prob-ability1eλj h,orεhε0j with probability eλj h and the process starts afresh because of the memoryless property of the exponential function.Therefore we have

f j t eλj h e h icµj ic12c2σ2j f j t hλj h f1j t o h(25) By Taylor’s expansion,we have

f j t1icµj ic 1

2

c2σ2jλj h o h f j t f j t h o h

λj h f1j t(26) thus f0and f1satisfyfirst order system of ODE’s

f0t icµ0ic12c2σ20λ0f0tλ0f1t

f1t icµ1ic12c2σ21λ1f1tλ1f0t

(27)

11with initial conditions f j01f j0icµj12c2σ2j j01.This system is stable and has a unique solution for anyfixed c.

On the other hand,let Yεn n ln Xεn n as defined in Eq.(23),and let

f j k E E e icY kε0j j010k n(28) Then,we have

f0k E E e icYεk kε00

E e icYεk1

k1icσ0h1p

0eλ0hε10

1eλ0h e icYεk1

k1icσ1h1p

1ε11

f0k1e icσ0hλ0h

λ0h f1k1e icσ1h1p1

f0k1e icσ0hλ0hλ0h f1k1

f0k1f0n1hλ012f0n1c2σ20hµ012σ20ich f0k1

λ0h f1k1o h(29) By linear interpolation,it is not hard to see that when h0,there exists f j∞which satisfies

f0∞t icµ01

2

c2σ20λ0f0∞tλ0f1∞t(30)

Similarly,we have

f1∞t icµ11

2

c2σ21λ1f1∞tλ1f0∞t(31)

and initial conditions f j∞01f j∞0icµj12c2σ2j.Therefore,by the uniqueness of the solution to the ODE Eq.(27),f i f i∞.Hence Theorem2 follows immediately.

12

5Our model vs.other models

There has been extensive work on modeling stockfluctuations with stochastic volatility(cf.Anderson,1996;Hull and White,1987;Stein,1991;Wiggins, 1987),Markov volatility(Di Masi et.al.1994),and uncertain volatility(Avel-laneda,Levy,and Par´a s,1995).Many efforts have been made to studyfinancial markets with different information levels among investors(cf.Duffie and Huang, 1986;Ross,19;Anderson,1996;Karatzas and Pikovsky,1996;Guilaume et. al.,1997;Grorud and Pontier,1998;Imkeller and Weisz,1999).Especially,Kyle (1985)considered a dynamic model of inside trading with sequential auctions,in which the informational content of prices and the values of private information to an insider are examined.Lo and Wang(1993)used Ornstein-Uhlenbeck(O-U)processes in their adjustment to the Black-Scholes model(Black and Scholes, 1973;Merton1973)to induce the drift term via option formula.

Our model vs.stochastic volatility.It is worth pointing out that the model we propose here is fundamentally different from various models with stochastic volatility or uncertain volatility.This is because the driftµεt is also driven by the hidden Markov processεt,which,in consequence,changes the option pricing methodology.

Our model vs.Markov volatility.The work of Di Masi et.al.(1994)on Markov volatility emphasizes more on aspects of hedging strategies than of op-tion pricing issues.Moreover,they seemed to have overlooked the fact that the martingale pricing approach would not be applicable for the general case when the drift term is non-zero,therefore their pricing procedure would beflawed.

13Our model vs.O-U processes.O-U processes have the nice property that when the price goes too negative,the drift term will“pull”it back,which makes eco-nomic sense.It is however known to probabilists that O-U process is a rescaled Brownian motion;therefore it does have its limitations to adjusting the Black-Scholes model.Moreover,for tractability reasons,the model proposed by Lo and Wang(1993)assumes afixed volatility.

In our model,the information component is reflected in both the drift and volatility term.Therefore,not surprisingly,the option pricing formula is function-ally dependent on the drift.Moreover,unlike the trending O-U processes which are Markovian,in our model while X t is not Markovian,X tεt is jointly so.It provides a simple and feasible way to connect historical data and current sit-uation.Furthermore,it captures our earlier intuition that the market never exists independently of the information distribution.

Remark.An anticipated criticism comes from the usage of the term“inside in-formation”,perhaps due to its not-so-glorious image in our(hopefully)efficient market.It is,therefore,worth pointing out that“inside information”is merely a convenient way to describe the hidden Markov processεt,which is driven by some market force,and can be interpreted in a broader way to reflect the“noise”exemplified by dεt,that goes beyond the Black-Scholes and other standard mod-els.

For pricings of other types of hedge options such as perpetual lookback op-tions,Russian options,perpetual American options,interested readers are referred to(Guo,1999).

146Empirical results and conclusions

We conclude our paper with an example borrowed from the ongoing empirical study with Ed.Pednault at IBM(Guo and Pednault,2000).Figure6illustrates our notion of“information structure”that is present in the IBM stock price.

Figure2:IBM stock price.

There are many immediate questions that spring to our mind.We recognize that it is not completely realistic to assume thatσ0σ1.We believe,however, the assumption that information structure is closely related tofluctuations in drift and volatilities stands to reason,while the mathematical tractability is retained.It is also far from clear how drift and volatility are intrinsically related with respect to change in information distribution.This question requires further investigation

15and statisticians and experimental economists might be able to provide possible answers.There are also cases in which the emergence of“inside information”will have a certain delay time,such thatσεt andµεt will be generated by different processesεt andεt.It will be interesting to study these models.Furthermore, another direction is to model the actions investors undertake upon receiving new information.

It is worth pointing out that our option pricing approach relies heavily on a rather strong economic assumption:the existence of a COS contract.It is not clear how feasible this assumption is.This brings up a even more basic question:is there a better way to incorporate information distribution into stockfluctuations?

Indeed,a simple model like ours,whose primary goal is to capture the real-ity in stock market without sacrificing tractability,has already gone beyond the boundary of the general framework of martingale approach.Therefore,we feel that our model may serve as little acorn from which great oaks could grow.

Acknowledgements

This model was jointly developed with Larry Shepp.He deserves special thanks for motivating me to pursue it further.Darrell Duffie,Dan Ocone,and Michael Harrison made valuable suggestions to improving both the content and presenta-tion of this paper.Ed.Pednault provided me with Figure6.Part of this work was supported by DIMACS.I thank the hospitality of the University of California at Berkeley.

16References

Anderson,T.G,1996,Return volatility and trading volume:an informationflow interpre-tation of stochastic volatility,Journal of Finance,51,169–204.

Avellaneda,M.,Levy,P.,and Par´a s,A.,1995,Pricing and hedging derivative securities in markets with uncertain volatilities,Applied Mathematical Finance,2,73–88.

Back,K.,1992,Insider trading in continuous time,Review of Financial Studies5,387–409.

Bachelier,L.,1900,Theorie de la Speculation Annales Scientifiques de L’´Ecole Normale Sup´e rieure,3d ser.,17,21–88.

Ball,C.A.,1993,A review of stochastic volatility models with applications to option pricing,Financial Markets,Institutions and Instruments2,55–69.

Billingsley,P.,1968,Convergence of Probability Measures(Wiley,New York). Black,F.and Scholes,M.,1973,The pricing of options and corporate liabilities,Journal of Political Economy81,637–654.

Cox,J.,Ross,S.,and Rubinstein,M.,1979,Option pricing,a simplified approach,Journal of Financial Economics7,229–263.

Delbaen,F.and Schachermayer,W.,1994,A general version of the fundamental theorem of asset pricing,Mathematique Annales,463–520.

Di Masi,G.B.,Yu,M.,Kabanov,and Runggaldier,W.J.,1994,Mean-variance hedging of options on stocks with Markov volatility,Theory of Probability and Its Applications 39,173–181.

Duffie,D.,and Huang,C.F.,1986,Multiperiod security markets with differential infor-mation,Journal of Mathematical Economics15,283–303.

Feller,W.,1971,An Introduction to Probability Theory and Its Applications,V ol.2(John Wiley&Sons).

F¨o llmer,H.,and Schweizer,M.,1990,Hedging of contingent claims under incomplete

17market,in:M.H.A.Davis and R.J.Elliott,eds.,Applied Stochastic Analysis,Stochastic Monographs,V ol.5,(London,Gordon and Breach)3–414.

Grorud,A.,and Pontier,M.,1998,Insider trading in a continuous time market model, International Journal of Theoretical and Applied Finance1,331–347.

Guilaume,D.M.,Dacorogna,M.,Dav´e,R.,Muller,U.,Olsen,R.,and Pictet,P.,1997, From the bird’s eye to the microscope,a survey of stylized facts of the intra-daily foreign exchange market,Finance and Stochastics1,95–129.

Guo,X,1999,Inside Information and Stock Fluctuations.Ph.D.dissertation,Department of Mathematics,Rutgers University.

Guo,X.,and Pednault,E.,2000,Identifying the states of a hidden Markov model of stock pricefluctuations,Manuscript.

Harrison,M.,and Kreps,D.,1979,Martingales and arbitrage in multiperiod securities markets,Journal of Economics Theory20,381–408.

Harrison,M.and Pliska,S.,1981,Martingales and stochastic integrals in the theory of continuous trading,Stochastic Processes and Their Applications11,215–260.

Hull,J.and White,A.,1987,The pricing of options on assets with stochastic volatility, Journal of Finance2,281–300.

Kurtz,T.,1985,Approximation of Population Processes(CBMS–NSF Regional Confer-ence Series in Applied Mathematics36,Society for Industrial and Applied Mathematics (SIAM)).

Jacod,J.and Shiryaev,A.N.,1997,Local martingale and the fundamental asset pricing theorems in the discrete-time case,Labo de probabilities453,2–16.

Karatzas,I.,and Pikovsky,I.,1996,Anticipative portfolio optimization,Advances in Ap-plied Probability28,1095–1122.

Kushner,H.J.and Huang,H.,1984,On the weak convergence of a sequence of general stochastic differential equations to a diffusion,SIAM Journal of Applied Mathematics40, 528–541.

18

Kyle,A.,1985,Continuous auctions and insider trading,Econometrica53,1315–1335. Naik,V.,1993,Option valuation and hedging strategies with jumps in the volatility of asset return,Journal of Finance48(5),1969-1984.

Pag`e s,H.,1987,Optimal Consumption and Portfolio When Markets Are Incomplete, Ph.D.dissertation,Department of Economics,Massachusetts Institute of Technology. Ross,S.A.,19,Information and volatility,the no-arbitrage martingale approach to timing and resolution irrelevancy,Journal of Finance44,1–8.

Samuelson,P.,1973,Mathematics of speculative price(with an appendix on continuous-time speculative processes by Merton,R.C.),SIAM Review15,1–42.

Skorohod,A.V.,1956,Limit theorem for stochastic processes,Theory of Probability and Its Applications1,262–290.

Stein,E.M.,and Stein,C.J.,1991,Stock prices distribution with stochastic volatility,an analytic approach,Review of Financial Studies4,727–752.

Wiggins,J.B.,1987,Option values under stochastic volatility.Theory and empirical evidence,Journal of Financial Economics19,351–372.

A Proof of Theorem1

Since the arbitrage price of the European option is the discounted expected value of X t under the equivalent martingale measure Q,we have

V i T K r E Q e rT X T Kε0i(32) Recalling that

X t X0exp

t

r dεs12σ2εs ds

t

σεs dW s(33)

19

the key point is to calculate the instantaneous distribution of X T .Now let Y t

ln X t ,then

Y t

Y 0

t

r

d εs

1

2σ2εs

ds

t

σεs dW s

(34)

If we consider the probability distribution function f i t T ,where f i t T is the

probability distribution function of T i ,(T i being the total time between 0and T during which εt

0,starting from state i ),then by the well-known property of

conditional expectations,we have

V i T K r

E e

rT

X T

K

ε0

i

e rT E E X T

K

T i ε0

i

e rT E i E X T

K

T i F ε0

i

e

rT

∞0

T

y ρln y

K x m t v t f i t T dydt

(35)

where x

X 0,ρx m t v t

is the normal density function with expectation

m t and variance v t .Clearly,we have:m t

d 1d 0

12σ20

σ21t

r

d 1

12σ21T

(36)v t

σ20

σ21t

σ21T

(37)

and

ρx m t v t

1

2πv t

exp

x

m t

2

ψi r T E e r T0χ0εs dsε0i

∞

e rt

f i t T dt

L r f i T;(40) then we have

ψi r T e rT eλi Tδi0eλi Tδ1i0

T

eλi uλiψ1i T u e ruδi0du(41) i.e.,

ψ0r T e rT eλ0T

T

eλ0uλ0ψ1T u e ru du(42)

ψ1r T eλ1T

T

eλ1uλ1ψ0T u du(43)

Taking Laplace transforms on both sides,and writing

L sψi r L s L r f i T r

∞

e sTψi r T dT

ˆψi r s(44)

then

ˆψ0r s1

r sλ0

ˆψ1r s(45)

ˆψ1r s1

sλ1

ˆψ0r s(46) Solving these equations,we obtain:

ˆψ0r s sλ0

λ1Taking the inverse Laplace transform on Eq.(47)with respect to r yields

L1rˆψ0r s w1λ0

sλ1

w

exp sw sλ0

sλ1exp sw

sλ0

sλ1

exp

s sλ0λ1 sλ1

w

λ0

sλ1

w v

eλ1v eλ1λ0w L1s e wsλ0λ1w

s

e wsλ0λ1wwhere J a z’s are Bessel functions as given in Eqs.(17),and(18). Therefore we have(for w v):

L1s sλ0λ1

sλ1

w v

eλ1v eλ1λ0w

v

δu w

v u

λ0λ1w

12J

12λ0λ1wvλ0λ1w212

λ0J02λ0λ1wvλ0λ1w212(53)

Thus we obtain f0w v,the distribution function of T0,such that

f0w v L1s L1rˆψ0r s w v

eλ1v eλ1λ0w v w

λ0λ1w2

12J12λ0λ1wvλ0λ1w212

λ1J02λ0λ1wvλ0λ1w212(55) Now the theorem is immediate.

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