ACTLseminar2008-YChoi

8/7/2019 ACTLseminar2008-YChoi

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Fractional Brownian Motion

Yangho Choi

angho Choi (The University of Connecticut) Fractional Brownian Motion Nov. 18, 2008 1 / 19
http://find/http://goback/


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Contents

Contents

Introduction

Original Black-Scholes FormulaFractional Brownian Motion

Applications of Wick-Ito Stochastic Calculus in Finance

Other Developments & Future Works

References



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Introduction

Introduction

History

Bachelier, L (1900): Arithmetic Brownian Motion

dS = dt + dB

Samuelson, P.A. (1964): Geometric Brownian Motion

dS

S= dt + dB

Osborne, M.F.M. (1959): The use of independent and normallydistributed random variable to model the logarithm of stock prices

ln

St+

St

N(0, 2)

Mandelbrot & Taylor (1963, 1967): A fractal behavior in stock price



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Introduction

Introduction

Present

There exist two apparent problems in the Black-Scholes formula.Namely financial processes are not wholly Gaussian and Markovian indistribution.

1 The heavy-tailed distribution of financial assets.2 Long-range dependence of stock returns.

Possible solutions for the above problems1 A series of models has been developed using more general or

heavy-tailed Levy processes. [Boyarchenko& Levendorskii]2 A series of academic studies further purporting the existence of a

non-Markovian market. [Lo& MacKinlay]

New era

Fractional Brownian Motion(fBM) deals with the second problemwhile still assuming a Gaussian process


I d i


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Introduction

Introduction

FBMwithH=0.3

400 500 600FBMwithH=0.5

40500

-5 0- 1 0 0

400 500 60 0FBMwithH=0,7

30 0 400 500 600 700

Figure 1: FBM plot for differentHurst 's exponents


O i i l Bl k S h l F l


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Original Black-Scholes Formula


From the geometric Brownian motion,

dSS

= dt + dB.

Suppose we have an option whose value V(S, t) depends only on S, t.From the Taylor series expansion

dV = VS

dS+ Vt

dt + 12

2

VS2

(dS)2 + .Using Itos lemma, we can write

dV = SV

S

+V

t

+1

2

2S22V

S2 dt + S

V

S

dB.

Now construct a portfolio consisting of one option and number of theunderlying asset S. This number is as yet specified. The value of thisportfolio is

= V S.angho Choi (The University of Connecticut) Fractional Brownian Motion Nov. 18, 2008 6 / 19

Original Black Scholes Formula


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The jump in the value of this portfolio in one time-step is

d

= r

dt = r(V S)dt. (1)Also we find that

follows the random walk

d = dV dS=

S

V

S+

V

t+

1

22S2

2V

S2 SS

dt

+SV

S dB. (2)

Here, we can eliminate the random component in this random walk bychoosing = VS . From (1) and (2),

V

t+

1

22S2

2V

S2+ rS

V

S rV = 0.




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The fractional Brownian Motion is a generalization of the Brownian

Motion, changing the exponent from H = 1/2 to any real number in therange 0 < H < 1, by

BH(t) = BH(0) +1

(H + 1/2)

t

K(t t)dB(t)

where () ia the gamma function and

K(t t) =

(t t)H1/2, if 0 t t(t

t)H1/2

(

t)H1/2, if t

0.

The fractional Brownian Motion with Hurst parameter H is the continuousGaussian process {BH(t)}t0, E[BH(t)] = 0 and whose covariance isgiven by E[BH(t)BH(s)] =

12{|t|2H + |s|2H |t s|2H} for all t, s.




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Theorem

A fBM process BH(t)t0 has the following properties1 H-self similarity, i.e., BH(t) =

HBH(t) for any > 0.

2 Stationary increments, i.e., BH(t + h) BH(h)) d= BH(t) BH(0)for all h.

3 IfH > 12 , then it has long-range dependence, i.e.,n=1 Cov(BH(1), BH(n + 1) BH(n)) =

4 IfH= 1/2, then it is non-Markovian and not a semimartingale.5 IfH = 12 , it has independent increments

6 The covariance between future and past increments is positive if

H > 12 and negative ifH 0, where and = 0 constants

We have the following results.

The return of stock price is

S(t) = s expBH(t) + t 1

2

2t2H , t 0No arbitrage exists in the fBM market (M(t), S(t))




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Theorem

A derivative with bounded payoff f(S(T)) has price given by C(t, S(t))that satisfies,

Ct + H

2t2H1S2 2CS2

+ rSCS rC = 0C(T, S) = f(S).

Compare with the classical Black-Scholes(PDE)

C

t+

1

22S2

2C

S2+ rS

C

S rC = 0.

Note: The Black-Scholes PDEs may be recovered from each other withthe transformation of 2HtH 12 .




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Theorem

The European call price at t [0, T] with strike price K and maturity T isgiven by

C(t, S(t)) = S(t)N(d1) Ker(Tt)N(d2),

where d1 =ln(

s(t)K

)+r(Tt)+22(T2Ht2H)

T2H

t2H

and d2 = d1

T2H

t2H.

The result reveals a surprising yet natural difference between fBS and theclassical Black-Scholes formula. Compare with the classical Black-Scholesformula,

C(t, S(t)) = S(t)N(d1)

Ker(Tt)N(d2)

where d1 =ln(

S(T)K

)+(r+122)(Tt)

Tt and d2 = d1

T t.

Clearly, the difference lies with parameters (T t) and (T2H t2H) of theclassical and fractional versions, respectively.




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Classical and fractional Black-Scholes Greeks

classical Black-Scholes fractional Black-Scholes

= CS = N(d1) N(d1)

= 2

CS2 =N(d

1)

STtN(d

1)

ST2Ht2H = Ct = =

Ct =

SN(d1)2Tt rKer(Tt)N(d2)) Ht2H1

SN(d1)T2Ht2H

rK er(Tt)N(d2))V = C = S

T tN(d1) S

T2H t2HN(d1)

= Cr = K(T t)er(Tt)N(d2) K(T t)er(Tt)N(d2)


Other Developments


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Other Developments

Other Developments

An apparent inconvenience of the Wick-Ito Calculus developed is itsrestraint to H (12 , 1). The idea starts with defining a new operator,MH, such that for f S(R),[Elliott& Hoek]

MHf(x) =

12(H

12) cos(

2(H

12)) R

f(xt)f(x)

|t|32H

dt, 0 < H < 121

2(H 12) cos(

2(H 1

2))

R

f(t)

|tx| 32Hdt, 12 < H < 1

f(x) H = 12

Analogous to the BM case, multidimensional Wick-Ito stochastic

calculus can be constructed.[Biagini&Oksendal]The purely fractional Black-Sholes models become arbitrage-free withWick-self financing strategies. But the notion of a Wick self-financingportfolios seems to be void of a sound economic interpretation, if it isto be interpreted in a real world sense. [Bender, Sottinen& Valkeila]


Future Work& References
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Future Work& References

Future Work

One researcher defines a fractional Brownian Motion indexed by either asphere or a hyperbolic if 0 < H 12 and only one dimensional case. I willtry to get an asymptotic solution for multidimensional fractional BrownianMotion.References

Bender, C., Sottinen, T. and Valkeila, E. (2006). Arbitrage withfractional brownian motion?. Theory of Stochastic Processes, Vol. 12(28), no. 3-4, 2006.

Biagini, F., B. Oksendal, A. Sulem, and N. Wallner (2003). An

introduction to white noise theory and Malliavin calculus for fractionalBrownian motion. Preprint, No. 2, January 2003, University of Oslo.

Boyarchenko, Svetlana I. and Sergei Z. Levendorskii. Non-GaussianMerton-Black-Scholes Theory. World Scientific Publishing Company,2002.


References


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References

Elliott, R.J. and J. van der Hoek (2001). Fractional Brownian motion

and financial modeling. In Trends in Mathematics, Birkhauser VerlagBasel/Switzerland, 2001, p. 140-151.

Hu, Y. and B. Oksendal (2000). Fractional white noise calculus andapplication to Finance. Preprint, University of Oslo.

Karatzas, Ioannis and Steven E. Shreve. Brownian Motion and

Stochastic Calculus:2nd Edition. Springer-Verlag New York, Inc.,1991.

Lo, Andrew W. and Archie Craig MacKinlay. A Non-Random WalkDown Wall Street. Princeton University Press, 1999.

Wick, G. C. (1950). The evaluation of the collinear matrix. PhysicsReview 80, 268-272.


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