Approximation of Geometric Fractional Brownian Motion

8/9/2019 Approximation of Geometric Fractional Brownian Motion

1/19

Approximation of geometric fractional Brownian

motion

Esko Valkeila

TKK, Institute of Mathematics

Mid-term Conference of AMaMeF, Vienna, September 2007


2/19


3/19

Fractional Brownian motion as a model in financeArbitrage with gfBm

Geometric fractional Brownian motion (gfBm) eBH

is not asemimartingale, and hence there are arbitrage opportunities by theFTAP of Delbaen and Schachermayer. The known examples ofarbitrage are based on continuous trading [Dasgupta, Shiryaev,Salopek], or more and more dense discontinuous trading [Rogers,

Cheridito]; but all these arbitrage strategies are apparently difficultto realize in practice.One can show that if one uses Skorohod integrals to defineself-financing strategies, the arbitrage disappears [Hu-ksendal,Elliot-van den Hook], but this kind of integrals are difficult tointerpret economically [Sottinen-Valkeila, Bjork-Hult].In pricing model with transaction costs the arbitrage opportunitieswith gfBm again disappear [Guasoni,Guasoni-Rasonyi-Schachermayer].


4/19

Approximation to geometric fBmIntroduction

Sometimes approximations give additional information about theproperties of the limiting model. We study here the setup wherewe have sequence of positive processes (Sn,FnPn) approximating

(eBH

,F, P), the fractional Black-Scholes model; the approximation

is understood in the sense of weak convergence.

One approximation is based on fractional binomial tree [Sottinen];as a financial model this approximation is complete, but allowsarbitrage.

Another one is based on shot noise process approximation to fBm[Kluppelberg-Kuhn]; as a financial model this is not complete, butdoes not allow arbitrage.


5/19

Approximation to geometric fBmMotivation

Yet another type of approximation is based on mixed Brownian fractional Brownian model; here

St = eWt

1

22t+BHt ;

in this approximation there is a hedging price for European options[Schoenmakers-Kloeden, Bender-Sottinen-Valkeila], and this

approximation allows arbitrage, at least in principle, if H ( 12 , 34 ],and there is no-arbitrage, if H > 34 [Cheridito].

What happens to the hedging price, if 0? For an Europeancall this limit will be

(S0 K)+;

here K is the strike and the interest rate is assumed to be r = 0.Clearly this is difficult to interpret.

We give one more approximation, where the approximating pricingmodel is complete and arbitrage-free.


6/19

One more approximation to gFbmA complete and no-arbitrage approximation

We will the use the teletraffic approximation to fractional

Brownian motion [Gaigalas-Kaj]. This goes as follows: let G be acontinuous distribution function with heavy tails. i.e.

1 G(t) t(1+) (1)

as t with (0, 1).Take i to be the interarrival times of a renewal counting processN. Assume that i G for i 2; for the first interarrival time 1assume that it has the distribution G0 =

1

t

0 (1 Gs)ds, so that

the renewal counting process

Nt =k=1

1{kt}

is stationary, where 1 = 1 and k := 1 + + k.


7/19

One more approximation to gFbmA complete and no-arbitrage approximation, part II

Take now independent copies N(i) of N, numbers am

0,am such that

m

am

; (2)

using the terminology of Gaigalas and Kaj we can speak of fast

connection rate.Define the workload process W(m, t) by

W(m, t) =m

k=1

N(i)t ;

note that the process Nm is a counting process, since theinterarrival distribution is continuous. We have thatEW(m, t) = mt

, since W(m, t) is a stationary process.


8/19

One more approximation to gFbmApproximation as a semimartingale

For the following proposition see Gaigalas and Kaj:

Proposition Assume (1) and (2). Let

Y

m

(t) :=

3

2(1 )(2 )2

W(m, amt) m1amt

m1

2 a12m

.

Then Ym converges weakly [in the Skorohod space D] to afractional Brownian motion BH, where H = 1 2 .

The process Ym is a semimartingale in its own filtration and alsoin the big filtration Fm, where Ft = {N

(i)s : s t, i = 1, . . . , m},

and Fmt = Famt.


9/19

One more approximation to gFbmApproximation as a semimartingale, continuation

To simplify notation put

c(, ) := 3

2

(1 )(2 )

2,

cm := m1

2 a1

2m and A

mt :=

m1amtcm

.Since the process Ym is a semimartingale, it has a semimartingaledecomposition

Ym = Mm + Hm; (3)

here Hm = Bm Am and Bm is the compensator of the normalizedaggregated counting process W. Note that the process Hm is acontinuous process with bounded variation.


10/19

One more approximation to gFbmApproximation as a semimartingale, continuation

Up to a constant we have that the square bracket of themartingale part Mm of the semimartingale Ym is

[Mm

, Mm

]t = C

W(m, amt)

ma2m .

But our assumption imply that [Mm, Mm]tL1(P)

0, as m .

With the Doob inequality we obtain that supst |Mms |

P 0, and

fBm is the limit of a sequence of continuous processes withbounded variation.


11/19

One more approximation to gFbmEnd of act one

It is not difficult to check that the solution to the linear equation

dSmt = SmtdY

ms

converges weakly in the Skorohod space to geometric fractionalBrownian motion.But we know from the previous discussion that here fBm is a limitof processes with bounded variation. But it is well known that inmarket models driven by continuous processes with bounded

variation there are arbitrage opportunities, unless the driver processis equal to the interest rate.


12/19

Further properties of the approximationCompleteness

Next we show that our approximation is complete. We can use theresults of Dzhaparidze he shows using pathwise arguments, thatcounting process models are always compete.Let N be a counting process, > 0 and > 0 are constants, andconsider the pathwise solution S to the following linear equation

dSt = St (dNt dt) with S0 = s;

then the unique solution to this is

St = setst

(1 + Ns) = set (1 + )Nt .

Denote the jump times N by k, k = 1, 2, . . . .


13/19

Further properties of the approximationCompleteness, continuation

The Poisson probabilities pj() are defined by pj() =j

j! e for

> 0. One can include the value = 0 by defining pj(0) = j0,where

j0 =

1, if j = 0

0, if otherwise

is the Kronecker delta. Put also F(j0; ) =

j>j0 pj().

Let WT be a functional of the price process path St, 0 t T. Ifthe price process S has a state sk(T), then the value functionalWT has a state wk(T). Recall that a market model is complete, if

we can find a self-financing strategy such that

WT = VT = v +

T0

sdSs.


14/19

Further properties of the approximationCompleteness, continuation

Dzhaparidze shows that if the define the strategy by

s =j=0

pj ( (T s))(1 + ) wj+k(T) wj+k+1(T)

, (4)

then we obtain self-financing strategy , which replicates the claimW(T); in (4) = .Consider the pricing of an European option (ST K)

+ in thePoisson market model. Then the fair price CE of this option isgiven by

C

E

= S0F(j0; (1 + )T) KF(j0; T), (5)where

j0 = logKS0

+ T

log(1 + ) (6)


15/19

Approximation price

We can now apply the above to the approximating model Sm. Wehave that

= m = c(, )a

2m

m1

2

,

= m = c(, )a

2m

m1

2

and

= m =m

m= .

From (2) we obtain that m 0, m 0, and if K > S0, thenj0 , and if K < S0, then j0 . Put this in (5) and weobtain that the limiting price is (S0 K)

+.

F h f h


16/19

Further properties of the approximationNo-arbitrage property

The basic randomness of the approximating pricing modelsequence Sm comes from the workload process Wm. We can showthat there exists a probability measure Qm such that Wm is aPoisson process with intensity m

.

What happens with the approximation? Recall that Sm

= E(Ym

).But Ym = Mm + Lm, where Lm is a continuous process, and hence[Mm, Lm] = 0. So using Yors formula for stochastic exponents wecan write the approximating sequence as

Smt = S0eLm

t E(Mm)t,

where E(Mm)ucp 1 with respect to the measure Pm.

F h i f h i i


17/19

Further properties of the approximationNo-arbitrage property

We know that the approximation (Sm,Fm, Pm) weakly convergesto the geometric fBm. On the other hand, with respect to themartingale measure Qm the sequence Ym is a martingale

sequence, Ymucp

0 with respect to Qm, and Smucp

S0 withrespect to Qm. So in the price (S0 K)

+ as a limit

(S0 K)+ = lim

mEQm(S

mT K)

+

for the European call.

R f


18/19

References

Bender, C., Sottinen, T., and Valkeila, E. (2006).

Pricing by hedging beyond semimartingales.Preprint.

Dzhaparidze, K.O. (2000).Introduction to option pricing in a securities market.CWI SYLLABUS, Amsterdam.

Gaigalas, R., and Kaj, I. (2003).Convergence of scaled renewal processes and a packet arrivalmodel.Bernoulli, 9, 671703.

Guasoni, P., (2006).No arbitrage under transaction costs, with fractional Brownianmotion and beyond.Math. Finance, 16, (2006), 569582.

R f t


19/19

References, cont

Kluppelberg, C., and Kuhn, C. (2004).Fractional Brownian motion as a weak limit of Poisson shot

noise processes with applications to finance.Stochastic Processes and their Applications, 113, 333251.

Schoenmakers, J., Kloeden, P. (1999)Robust option replication for a Black-Scholes model extended

with nondeterministic trends.JAMSA, 12, 113120.

Sottinen, T. (2001).Fractional Brownian motion, random walks and binary marketmodels.Finance and Stochastics, 5, 343355.

Valkeila, E. (2007).On the approximation of geometric fractional Brownianmotion.

13 pages. In preparation.

Documents

Approximation of Geometric Fractional Brownian Motion