8 t St i+1 i X 2 r X x) = P X x) if same for all...

Stat 39100/FinMath 34600 Lecture 9

STATIONARY AND NON-GAUSSIAN DISTRIBUTIONS

Xti=

return (log Sti+1− log Sti

), orvolatility (σ2

ti), or

interest rate (rti), etc

Stationary distribution of Xti:

F (x) = P (Xti≤ x) if same for all i

or = limn→∞

1

n

n∑

i=1

I(Xti≤ x) if meaningful

or = limi→∞

P (Xti≤ x) if meaningful

Returns of Geometric Brownian motion:

Xti= µ∆ti + σ∆Wti

∼ N(µ∆t, σ2∆t)

HenceF = N(µ∆t, σ2∆t)

Winter 2005 1 Per A. Mykland


Ifd log St = µdt + σdWt

then (law of large numbers)

log St

t⇒ µ

log St, St has no stationary distribution.



The Vasicek (or Ornstein-Uhlenbeck) Model

drt = K(α − rt)dt + σdWt

Set ut = rt − α, get

dut = −Kutdt + σdWt

Ito:deKtut = eKtdut + KeKtutdt

= eKtσdWt

eKtut − eKsus = σ

∫ t

s

eKudWu

= N

(

0, σ2

∫ t

s

e2Kudu

)

= N

(

0,σ2

2K(e2Kt − e2Ks)

)

or

ut = e−K(t−s)us + N

(

0,σ2

2K(1 − e−2K(t−s))

)

Stationary distribution: t → ∞

ut → N

(

0,σ2

2K

)

and so rt → N

(

α,σ2

2K

)

Also,

1

n

n∑

i=1

I(uti≤ x) ∼ N

(

0,σ2

2K

)

(x)



In practice, to get stationary density f = F ′:

• Collect Xtiover many periods ti → ti+1

• Make histogram of result



APPROACHES TO NON GAUSSIANSTATIONARY DISTRIBUTIONS

• Jumps• State dependent drift, volatility — for example:

dXt = µ(Xt)dt + σ(Xt)dWt

Stationary density

f(x) = C1

σ(x)2exp

(

2

∫ x

x0

µ(y)

σ2(y)dy

)

• Stochastic volatility — for example

dXt = µtdt + σtdWt

dσ2t = K(α − σ2

t )dt + γdBt

perhaps correlation between dW , dB: d[W, B]t =ρdt

Or Heston model



Jump ProcessesThe simplest case: the Poisson process.

Nt is a Poisson process with rate λ if:

• Nt only evolves by jumping (and stays still betweenjumps), and the jump size is always 1

• Nt − Ns is independent of Fu for t ≥ s ≥ u

• N0 = 0• E(Nt) = λt

Some main properties of a Poisson process:

• Nt − λt is a martingale• P (jump in (t, t + δ) | Ft) = λδ + op(δ)• P (more than one jump in (t, t + δ) | Ft) = op(δ)• the distribution of Nt−Ns given Fu is Poisson(λ(t−

s)):

P (Nt −Ns = k | Fu) =((λ(t − s))k

k!exp{−λ(t− s)}

Time varying (possibly random) λt (“counting process”):

• Nt only evolves by jumping (and stays still betweenjumps), and the jump size is always 1

• Nt −∫ t

0λudu is a martingale

• N0 = 0



A frequent stock price model: the compound Poisson pro-

cess:

log St − log S0 =

Nt∑

i=1

Zi

More generally: jump-diffusion

Xt = X0 + µt + σWt +

Nt∑

i=1

Zi

The Z’s (which are iid), N , W are independent



Ito’s formula

df(Xt, t) = f ′x(Xt−, t)dXt + f ′

t(Xt−, t)dt

+ 12f ′′

xx(Xt−, t)d[Xc, Xc]t

+ ∆f(Xt) − f ′x(Xt−)∆Xt

Xct = continuous part of Xt, say,

= µt + σWt

Xt− = lims↑t

Xs

∆Xt = Xt − Xt−

= jump at time t, say,

= Zi if Nt− = i − 1, Nt = i

dXt = dXct + ∆Xt

where Nt is a Poisson process, and the Zi are i.i.d. andindependent of Nt.



Ito’s formula on St scale: St = exp(Xt)

∆ log St ={

Zi if jump0 otherwise

Hence, if jump:

∆St = St − St−

= St− exp(Zi) − St−

= (exp(Zi) − 1)St−

d exp(Xt) = exp(Xt−)dXt + 12 exp(Xt−)d[Xc, Xc]t

+ ∆ exp(Xt) − exp(Xt−)∆Xt

dSt = St−dXt + 12St−σ2dt

+ ∆St − St−∆Xt

= St−dXct + 1

2St−σ2dt + ∆St

dSt

St−

= dXt + 12σ2dt +

∆St

St−

− ∆Xt

= dXct + 1

2σ2dt +∆St

St−

= (µ + 12σ2)dt + σdWt +

∆St

St−



HEDGING A SECURITY Vt = f(St, t)

Suppose r = 0

df(St, t) = f ′x(St−, t)dSt hedgable

+ (f ′t(St−, t) + 1

2f ′′ss(St−, t)σ2)dt will eliminate later

+ ∆f(St, t) − f ′s(St−, t)∆St

last line

= f(St, t) − f(St−, t) − f ′s(St−, t)∆St

= 12f ′′

ss(St−, t)∆S2t + . . . (1)

not hedgable unless f ′′ss(s, t) = 0 or unless

∆St = J = (eZ − 1) = fixed if jump

in which case

(∆St)2 =

{

J2 if jump0 otherwise

= J∆St

and so on, so that:

(1) = 12f ′′

ss(St−, t)J∆St +1

3!f ′′′

sss(St−, t)J2∆St + . . .

= g(St−, t)∆St



In this case

df(St, t) = dSt + dt terms

+ g(St−, t)∆St

If two options fi(St, t), i = 1, 2

df1(St, t) −g1(St−, t)

g2(St−, t)df2(St, t)

= dSt + dt terms.

dt terms must be zero under any risk neutral measure

One option completes market.

In general: p Brownian motions, q jump sizes:

need p + q − 1 options to complete market.



Variation processes for jump processes. For pro-cesses of this type, 〈·, ·〉 is not the same as [·, ·]. Weuse the following definitions: [X, · · · , X] is given by

[X, · · · , X︸︷︷︸

p times

]t = lim∑

(Xti+1− Xti

)p

=

Xt if p = 1lim

∑(Xti+1

− Xti)2 if p = 2

∑∆Xp

u if p > 2

On the other hand, 〈X, · · · , X〉 is the predictable com-ponent in the Doob-Meyer decomposition of [X, · · · , X]:

[X, · · · , X] = 〈X, · · · , X〉︸︷︷︸

predictable

+ martingale .

Important examples of variation processes. For thePoisson process,

[N, · · · , N ]t = Nt

and〈N, · · · , N〉t = λt.

For the compound Poisson:

[log S, · · · , log S]t =

Nt∑

i=1

Zpi



and〈log S, · · · , log S〉t = E(Zp)λt.

From Ito’s formula:

df(Xt) = f ′(Xt−)dXt + 12f ′′(Xt−)d[Xc, Xc]t

+ ∆f(Xt) − f ′x(Xt−)∆Xt

= f ′(Xt−)dXt + 12f ′(Xt−)d[X, X]t

+1

3!f ′′′(Xt−)d[X, X, X]t + . . .

since

∆f(Xt) = f(Xt) − f(Xt−)

= f ′(Xt−)∆Xt + 12f ′′(Xt−)(∆Xt)

2

+1

3!f ′′′(Xt−)(∆Xt)

3 + ...

andd[X, X]t = d[Xc, Xc]t + (∆Xt)

2

This is the Taylor expansion interpretation of Ito’s formula


8 t St i+1 i X 2 r X x) = P X x) if same for all...

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