Ornstein-Uhlenbeck Processes - Michael...

Ornstein-UhlenbeckProcesses

Michael Orlitzky

Michael Orlitzky Towson University

Introduction

Goal. To introduce a new financial dervative.

• No fun.

• I’m bad at following directions.

• The derivatives based on Geometric Brownian Motiondon’t model reality anyway.

So we’re going to replace the underlying stochastic processinstead.


Introduction

Claim. GBM doesn’t work in real life.

Proof.Bekaert and Hodrick (1992), Bessembinder and Chan (1992),Breen, Glosten, and Jagannathan (1989), Campbell and Ammer(1993), Campbell and Hamao (1992), Chan (1992), Chen(1991), Chen, Roll, and Ross (1986), Chopra, Lakonishok, andRitter (1992), DeBondt and Thaler (1985), Engle, Lilien, andRobbins (1987), Fama and French (1988a, 1988b, 1990), Ferson(1989, 1990), Ferson, Foerster, and Keim (1993), Ferson andHarvey (1991a, 1991b), Ferson, Kandel, and Stambaugh (1987),Gibbons and Ferson (1985), Harvey (1989), Jegadeesh (1990),Keim and Stambaugh (1986), King (1966), Lehmann (1990), Loand MacKinlay (1988, 1990, 1992), and Poterba and Summers(1988).


Wiener Processes

Definition (Wiener Process). You should know thisalready [1] (Chapter 4).

Recall two important properties:

1. W0 = 0.

2. s < t implies [Wt −Ws] ∼ N[0,√t− s

]Note. As the time period t− s approaches infinity, so does thestandard deviation!


Wiener Processes

0.2 0.4 0.6 0.8 1.0

−1.0

−0.5

0.5

1.0

1.5

WtWtWtWt


Wiener Processes

What’s wrong here?

• If you’re unlucky, the Wiener process can go bonkers.

• When it does, it tends to stay there.

This doesn’t accurately reflect what happens in real life.


Geometric Brownian Motion

The Black-Scholes model that we’ve been using assumes thatthe stock dynamics follow Geometric Brownian Motion:

dXt = αXtdt+ σXtdWt

X0 = x0

with solution,

Xt = x0 exp

{(α− σ

2

2

)t+ σWt

}

Since this contains Wt, it inherits all of its problems.


Geometric Brownian Motion

0.0 0.2 0.4 0.6 0.8 1.0

0.5

1.0

1.5

2.0

2.5 α=1σ=1/5

x0exp{(α−σ2

2)t+σWt

}x0exp

{(α−σ2

2)t}


Ornstein-Uhlenbeck

Definition (Ornstein-Uhlenbeck Process). TheOrnstein-Uhlenbeck process is a stochastic process withdynamics,

dUt = θ (µt − Ut) dt+ σdWtU0 = u0

where Wt is a Wiener process.

• Can be seen as a modification of a Wiener process.• µt is the mean of the process.• θ is the tendency of the process to return to the mean.


Ornstein-Uhlenbeck

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

θ=2σ=1/5µ=1dUt=θ(µ−Ut)dt+σdWt


Ornstein-Uhlenbeck

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

θ=2σ=1/5µ=1/2dUt=θ(µ−Ut)dt+σdWt


Ornstein-Uhlenbeck

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

θ=10σ=1/5µ=1/2dUt=θ(µ−Ut)dt+σdWt


Ornstein-Uhlenbeck

Theorem. When µ is a constant, the solution to theOrnstein-Uhlenbeck SDE is given by,

Ut = µ[1− e−θt

]+ x0e

−θt + σN [0, ξ]

where,

ξ =

√1

2θ[1− e−2θt]


Ornstein-Uhlenbeck

Proof.

• Look up the solution on Wikipedia.• Assume that it’s correct.• Then, the solution is of the form,

f(xt, t) = xteθt

• Apply Itô’s formula.

(or you can apply Proposition 5.3 from our textbook withA = −θ and bt = µθ)


Ornstein-Uhlenbeck

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

1.2

θ=2σ=1/5µ=1Ut


Lo & Wang Process

That’s great, but we want to price some derivatives.

Let Pt be some price process, and pt = ln (Pt). We’re going toassume that pt has the dynamics,

dpt = (−θ (pt − ηt) + η) dt+ σdWtp0 = p0

We see that pt is shifted by ηt – we want to get rid of that.


Lo & Wang Process

We can rewrite that SDE as,

d (pt − ηt) = −θ (pt − ηt) dt+ σdWt

Now if we let ηt = µt, the right-hand side looks like theOrnstein-Uhlenbeck SDE!

d (pt − µt) = −θ (pt − µt) dt+ σdWt


Lo & Wang Process

We’ll rename pt − µt = qt to reduce the amount of ugly. Thuswe have,

dqt = −θqtdt+ σdWt

Compare with Geometric Brownian Motion:

d ln (Xt) =

(α− σ

2

2

)dt+ σdWt

Since we have a function of t in the dt term, our process isactually a slight generalization of GBM. It allows us to modeltrends and predictability.


Lo & Wang Process

Oh, and we can still solve our new process explicitly [3]:

qt = e−θtq0 + σ

t∫0

e−θ(t−s)dWs

Translating back into the original price process...

pt = µt + e−θt (p0 − µ0) + σN [0, ξ]

where again,

ξ =

√1

2θ[1− e−2θt]


Lo & Wang Process

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

1.2

µt=0θ=1σ=1/5

P0exp{µt+e

−θt(p0−µ0) +σN[0,ξ]

}P0exp

{µt+e

−θt(p0−µ0)

}


Lo & Wang Process

0.0 0.2 0.4 0.6 0.8 1.0

0.5

1.0

1.5

2.0

2.5

3.0

µt=tθ=1σ=1/5

P0exp{µt+e

−θt(p0−µ0) +σN[0,ξ]

}P0exp

{µt+e

−θt(p0−µ0)

}


Lo & Wang Process

0.2 0.4 0.6 0.8 1.0

1.0

2.0

3.0

4.0

5.0

µt=0θ=5σ=1/5

P0exp{µt+e

−θt(p0−µ0) +σN[0,ξ]

}P0exp

{µt+e

−θt(p0−µ0)

}


Lo & Wang Process

0.0 0.2 0.4 0.6 0.8 1.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

µt=3tθ=5σ=1P0exp

{µt+e

−θt(p0−µ0) +σN[0,ξ]

}P0exp

{µt+e

−θt(p0−µ0)

}


Lo & Wang Process

0.2 0.4 0.6 0.8 1.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0µt=5tθ=10σ=1/5

P0exp{µt+e

−θt(p0−µ0) +σN[0,ξ]

}P0exp

{µt+e

−θt(p0−µ0)

}


Option Pricing

Theorem. The Black-Scholes formula works for the processwe defined earlier, qt.

This sounds rather amazing at first (see [3] for a citation), butthere is a caveat. While the formula still works, the numericalvalue of σ will differ between GBM and Ornstein-Uhlenbeck.

The short version:

σ2OU = σ2GBM ·

[θτ(

1− e−θτ)−1]

(Here, τ is the length of the time period over which yourobservations are made.)


Option Pricing

Example (IBM, March 2012).

Table: Observed prices for March 2012

Date Closing Price

2012-03-01 whatever

2012-03-02 whatever

2012-03-05 whatever

2012-03-06 whatever

. . . . . .

2012-03-30 whatever


Option Pricing

190.0 195.0 200.0 205.0 210.0 215.0 220.0

5.0

10.0

15.0

20.0

25.0

θ=1FGBM(t=0, s)FOU(t=0, s)


Option Pricing

2.0 4.0 6.0 8.0 10.0θ

5.0

10.0

15.0

t=0s=205[FOU−FGBM]


Option Pricing

As θ increases, the tendency to vary from the mean decreases.Therefore, a large θ corresponds to a “predictable” stock.

Conclusion. Options on a predictable stock are morevaluable!


References

[1] Björk, Tomas. Arbitrage Theory in Continuous Time, 3rdEd. Oxford University Press, Oxford, 2009.

[2] Doob, J. L. The Brownian Movement and StochasticEquations. The Annals of Mathematics, Second Series, Vol.43, No. 2 (Apr., 1942), pp. 351-369.

[3] Lo, A. W.; Wang, J.: Implementing Option Pricing Modelswhen asset returns are predictable. The Journal of Finance50, pp. 87–129, 1995.

[4] Stein, William A. et al. Sage Mathematics Software (Version5.0), The Sage Development Team, 2012,http://www.sagemath.org.

[5] Uhlenbeck, G. E. and Ornstein, L. S. On the Theory ofBrownian Motion. Phys. Rev. 36, pp. 823-841. 1930.


Ornstein-Uhlenbeck Processes - Michael...

Documents

Transcript of Ornstein-Uhlenbeck Processes - Michael...