The stochastic process followed by stock prices.

Moty Katzman

September 19, 2014

The stochastic process followed by stock prices

The price of a certain stock at a future time t is unknown at thepresent.We think of it as being a random variable St .

What sort of random variable are St for t ≥ 0?

Brownian motion

A Brownian motion is a family of random variables

Bt | t ≥ 0

on some probability space (Ω,F ,P) such that:(1) B0 = 0,(2) for 0 ≤ s < t the increment Bt − Bs is normally distributedwith mean 0 and variance t − s,(3) for any 0 ≤ t1 < t2 < · · · < tn the increments

Bt1 − B0,Bt2 − Bt1 , . . . ,Btn − Btn−1

are independent random variables, and(4) For any ω ∈ Ω the function t 7→ Bt(ω) is continuous.

Three instances of Brownian Motion

corresponding to ω1, ω2, ω3 ∈ Ω.

t

Bt(ω1)

t

Bt(ω2)

t

Bt(ω3)

Does Brownian motion exist?Yes– by Kolmogorov’s Existence Theorem.

Brownian motion has surprising properties; for example,(1) The function t 7→ Bt(ω) is nowhere differentiable withprobability 1,(2) if Bt = x for some t then for any ǫ > 0 the setτ : |τ − t| < ǫ and Bτ = x is infinite with probability one.

Brownian motion is useful for describing the jiggling of prices:buying and selling jiggle prices.

The Ito integral

Brownian motion is an example of a stochastic process i.e., afamily of random variables indexed by time t ≥ 0.We now construct a more general kind of stochastic processeswhose definition is based on Brownian motion.

We want to construct a stochastic process Xt | t ≥ 0 on thesame probability space (Ω,F ,P) on which the Brownia motionBt | t ≥ 0 is definedwith the property that the change of X over an infinitesimal periodof time dt is given by

dX = a(ω, t)dt + b(ω, t)dB

where a and b are themselves stochastic processes on (Ω,F ,P)with continuous paths and where dB is the change in the Brownianmotion over the infinitesimal period of time dt.

Fix any ω ∈ Ω; for any partition P of [0, t] into small intervals[s0, s1], [s1, s2], . . . , [sn−1, sn] where s0 = 0 and sn = t, we computethe sum

ΣP =n−1∑

i=0

b(ω, si )(

B(ω)si+1− B(ω)si

)

.

Define the norm of the partition P to be

‖P‖ = max s1 − s0, s2 − s1, . . . , sn − sn−1 ;

If b satisfies some technical conditions, the limit as ‖P‖ → 0 ofΣP exists.This limit is known as an Ito integral and we denote it with

∫

t

0b(ω, s) dB(ω)s .

We now define the process

Xt(ω) = X0(ω) +

∫

t

0a(ω, s) ds +

∫

t

0b(ω, s) dB(ω)s

for all t ≥ 0.

Example:∫ t

0 s dBs

We compute the integral from definition(1) Take any partition P of [0, t] into small intervals[s0, s1], [s1, s2], . . . , [sn−1, sn] where s0 = 0 and sn = t.(2) The sum

ΣP =n−1∑

i=0

si(

Bsi+1− Bsi

)

is a sum of independent normally distributed random variables withmean 0 and variance s20 (s1 − s0), s

21 (s2 − s1), . . . , s

2n−1(sn − sn−1).

(3) ΣP is normally distributed with mean 0 and variances20 (s1 − s0) + s21 (s2 − s1) + · · ·+ s2

n−1(sn − sn−1).

(4) As ‖P‖ → 0 this variance converges to∫

t

0 s2ds = t3/3.

(5) We conclude that∫

t

0 sdBs is a normally distributed randomvariable with mean 0 and variance t3/3.

Henceforth we write

dX = a(X , t) dt + b(X , t) dB

to denote that fact that X is a stochastic process defined by

Xt(ω) = X0(ω) +

∫

t

0a(Xs(ω), s) ds +

∫

t

0b(Xs(ω), s) dB(ω)s .

We shall refer to stochastic processes of this form Ito processes.

Back to stock prices

As a first approximation we model the proportional increase instock prices as a Brownian motion.We could then derive the following discrete time version

dS

S= σdB

where dS is the change in the stock price over a short time from t

to t + dt, dB = Bt+dt − Bdt and B is a Brownian motion.(In particular proportional increases in S are independent, e.g.,today’s increase in a stock price is independent of tomorrow’sincrease.)

model implies that the values of stocks vary without any long termtrend, i.e., E (dS

S) = 0.

A quick glance at historical data shows that this is not veryplausible:

t

DJIA

Despite the randomness of the value of the DJIA, its long termexponential growth is quite visible.

To take into account this upward trend in stock prices weintroduce a drift term

dS

S= σdB+µdt.

We refer to a process S defined above as a geometric Brownian

motion.

Now

E (dS

S) = µdt.

We shall refer to µ as the expected return of the stockand to σ as the volatility of the stock.

Instances of the process dS = 0.1Sdt + σSdB

t

St

σ = 0

t

St

σ = 0.05

t

St

σ = 0.1

t

St

σ = 0.5

One can raise several objections to this model. Consider thefollowing list of all trades in Vodafone shares between 16:22:08 and16:23:08 on August 16th, 2002:

Trade time Trade price Bid Ask Volume Block price Buy/Sell16:23:08 100.75p 100.75p 101p 180,614 £181,969 SELL16:23:08 100.75p 100.75p 101p 1,185 £1,194 SELL16:23:08 100.75p 100.5p 100.75p 250,000 £251,875 BUY16:23:01 100.5p 100.25p 100.75p 2,500 £2,51216:22:55 100.75p 100.5p 100.75p 28,383 £28,596 BUY16:22:55 100.75p 100.5p 100.75p 3,000 £3,022 BUY16:22:55 100.75p 100.5p 100.75p 248,815 £250,681 BUY16:22:55 100.75p 100.5p 100.75p 179,802 £181,151 BUY16:22:55 100.75p 100.5p 100.75p 40,000 £40,300 BUY16:22:42 100.5p 100.5p 100.75p 1,500 £1,508 SELL16:22:41 100.75p 100.5p 100.75p 23,117 £23,290 BUY16:22:39 100.527p 100.5p 100.75p 10,000 £10,053 SELL16:22:38 100.75p 100.5p 100.75p 48,500 £48,864 BUY16:22:38 100.75p 100.5p 100.75p 150,000 £151,125 BUY16:22:38 100.75p 100.5p 100.75p 25,000 £25,188 BUY16:22:38 100.75p 100.5p 100.75p 26,500 £26,699 BUY16:22:29 100.688p 100.5p 100.75p 25,000 £25,172 BUY16:22:18 100.75p 100.5p 100.75p 25,000 £25,188 BUY16:22:08 100.75p 100.5p 100.75p 11,000 £11,082 BUY

(1) The model allows any real number to be a value for S , but inreal life there is a smallest unit.(2) We assume that prices are changing continuously but tradesoccur at discrete times.(3) Shares are often traded through market makers.They buy at the bid price and sell at the ask price. So there aretwo prices! Sometimes it is crucial to model both.(4) Sometimes, e.g., during market crashes, changes seem to havea “memory”.We should regard our model as an approximation to real life pricesand trades, not as an accurate description.

Ito’s Lemma.

Consider a stochastic process Xt whose change over a smallinterval of time from t to t + dt is given by

dX = a(X , t)dt + b(X , t)dB

where a(x , t) and b(x , t) are functions of x and t.Consider a new stochastic process Y = G (X , t) where G (x , t) is afunction of x and t.What sort of process is Y ?Theorem: (Ito’s Lemma)Assume that G (x , t) is twice continuously differentiable withrespect to x and continuously differentiable with respect to t.The process Y is also an Ito process.In fact,

dY =

(

∂G

∂xa +

∂G

∂t+

1

2

∂2G

∂x2b2)

dt +∂G

∂xb dB .

”dB2 = dt”

Write dB = Bt+∆t − Bt as√∆tZ for some standard normal Z .

Note: 1 = Var(Z ) = E(Z 2)− E(Z )2 = E(Z 2).So, E(dB2) = E(∆tZ 2) = ∆t E(Z 2) = ∆t, andVar(dB2) = ∆t2 Var(Z 2) is of order ∆t2.As ∆t → 0, Var(dB2) ≪ ∆t = E(dB2), so dB2 “converges to”∆t.

(Very) informal proof of Ito’s Lemma:

Using the Taylor series for G we can write

dY =∂G

∂x∆X +

∂G

∂t∆t +

1

2

∂2G

∂x2∆X 2 +

∂2G

∂x∂t∆X∆t+

1

2

∂2G

∂t2∆t2 + higher order terms.

Now

(∆X )2 = (a∆t + bdB)2 = b2∆t + higher terms in ∆t

So for small ∆t we approximate

dY ≈

∂G

∂x∆X +

∂G

∂t∆t +

1

2

∂2G

∂x2b2∆t+

higher degree terms in ∆X ,∆t =

∂G

∂x(a∆t + b∆B) +

∂G

∂t∆t +

1

2

∂2G

∂x2b2∆t+

higher degree terms in ∆B ,∆t.

An Example:Use Ito’s Lemma to show that

d(

B2t

)

= dt + 2Bt(d Bt).

Deduce that∫

t

0BsdBs =

1

2B2t − t

2.

Let Xt be the process with dX = 0× dt + 1× dBt and considerthe process Yt = G (Xt , t) where G (x , t) = x2. We have

∂G

∂x= 2x ,

∂2G

∂x2= 2,

∂G

∂t= 0

so Ito’s Lemma implies that d(B2t ) = dY =

(

2X × 0 + 0 +1

22× 12

)

dt + 2X × 1 dB = dt + 2BtdB .

Integrating both sides of the equation above gives∫

t

0d(

B2s

)

= t + 2

∫

t

0BsdBs ⇒

∫

t

0BsdBs =

1

2B2t − t

2.

The stochastic process followed by forward stock pricesConsider a forward contract on stock paying no dividends maturingat time T ; let F (t) be its forward price at time t ≥ 0:

F (t) = S(t)er(T−t),

where S(t) is the spot price of the stock at time t.Regard F as a function of s and t, i.e., F = F (s, t) = ser(T−t):

∂F

∂s= er(T−t),

∂2F

∂s2= 0 and

∂F

∂t= −rser(T−t).

Our model assumes that

dS = µSdt + σSdB

so Ito’s Lemma implies that

dF =(

er(T−t)µS − rSer(T−t))

dt + er(T−t)σSdB =

(µ− r)Fdt + σFdB ,

i.e., F follows a geometric Brownian motion with drift µ− r .

The stochastic process followed by the logarithm of stock

prices

Let S be the spot price of a certain stock at time t and letG = G (s, t) = log s. Since

∂G

∂s=

1

s,

∂2G

∂s2=

−1

s2and

∂G

∂t= 0

and since dS = µSdt + σSdB Ito’s Lemma implies that

dG =

(

1

SµS − σ2S2

2S2

)

dt +σS

SdB =

(

µ− σ2

2

)

dt + σdB .

Consider a fixed time T in the future and the spot price ST at thattime: we see that the logarithm of the proportional price change ofthe stock,

logST

S= log ST − log S ,

is normally distributed

with expected value

(

µ− σ2

2

)

T

and variance σ2T .

Example:Consider the price S of a stock with current price S0 = $10,expected annual return of µ = 15% and annual price volatility (i.e.,standard deviation) of σ = 20%.The stock price follows a geometric Brownian motion

dS = µS dt + σS dB = 0.15S dt + 0.2S dB

and for any time T in the future the price of the stock at time T

satisfies log ST ∼ N(

log S0 +(

µ− σ2

2

)

T , σ2T)

=

N(

log 10 + 0.13T , 0.22T)

. So for example, one can say that with90% confidence in 6 months

log 10 + 0.065 − 0.2√2× 1.645 < log S1/2 <

log 10 + 0.065 +0.2√2× 1.645

i.e., with 90% confidence the logarithm of stock price in 6 monthswill be between approximately 2.135 and 2.6 and so the stock priceitself will be between approximately $8.46 and $13.47 with 90%confidence.

The End