Diffusion Processes and Ito’s
Lemma
Question: If we model asset prices as
continuous time stochastic processes,
can we identify trading strategies that
hedge price risk?
1. We want to show that Brownian motion can be
viewed as the limit of a discrete time process.
Define a stochastic process z(t) at date t and let the discrete
change in time be the interval Δt.
How does z(t) change over time?
From (8.3) we can write the first two moments of z(T) – z(0)
Continuous time limit
Now take the limit of z(T) – z(0) as the number of intervals goes
to infinity, assuming that the є are IID. Then, by the Central
Limit Theorem
and z(T) – z(0) has a normal distribution.
The central limit theorem states conditions under which the mean of a
sufficiently large number of IID (identically and independently distributed)
random variables, each with finite mean and variance, will be
approximately normally distributed.
A Wiener (continuous time) process dz(t) exhibits
Brownian motion – a particular type of Markov
stochastic process which has two properties - Δz = є√Δt during a small period of time Δt
Δz for any two Δt are independent (a Markov process)
z(t) exhibits Brownian motion (a random walk)
An Itô integral is
the change in z(t) over a finite interval [0,T]
2. We can use the basic Wiener process (8.7) to
construct a generalized Wiener process (8.11)
arithmetic Brownian motion
If μ and σ can be “time-varying,” we can describe x(t) as a
stochastic differential equation - - an Itô process
We can characterize the stochastic process
F(x(t), t) by using Itô’s lemma
Then x(t) = eF(t) has a lognormal distribution over any discrete interval, and
geometric Brownian motion is lognormally distributed over any time
interval.
3. Geometric Brownian motion is used for modeling
stock prices
We note that a generalized Wiener process with constant drift rate and
constant variance rate would fail to account for the characteristic that the
expected return (μ) required by investors is independent of the stock price
(S). (We assume the stock pays no dividends.)
We need to have it that the expected return (expected drift divided by the
stock price) is a constant.
Then, the expected increase in S = μSΔt
And, if the volatility of stock price = 0, we also have it that ΔS = μSΔt
geometric Brownian motion
In the limit as Δt →0,
or,
dS = μS dt
dS/S = μ dt
Integrating from [0,T] we get ST = S0e μT
=> when the variance rate is zero, the stock price grows at a continuously
compounded rate μ per unit of time.
In reality stock prices exhibit volatility, so we need to modify this approach.
Let’s assume that σ is proportional to S (i.e., investors are just as uncertain
at $10 /share as they are at $50/share)
Then,
or,
dS = μS dt + σ S dz
dS/S = μ dt + σ dz
a geometric Brownian motion process
A discrete time example
Assume a stock that pays no dividends, has volatility σ = .30, and has
expected return μ = .15
If the process for stock price is
dS = 0.15 S dt + 0.30 S dz
dS/S = 0.15 dt + 0.30 dz
(price)
(return)
Rewriting in discrete time intervals,
ΔS/S = 0.15*Δt + 0.30*є√Δt where є ~ N(0,1)
For time interval of 1 week (0.0192 years),
ΔS/S = 0.00288 + 0.0416 є
ΔS = 0.00288 S + 0.0416 S є
Figure 8.2: Sample paths for geometric Brownian
motion, where μ=0.10, σ=0.30, T=2
Problem #2 (Pennacchi)
Let P be a price index, such as the Consumer Price Index. Let M equal the
nominal supply of money in the economy. Assume P and M each follow
geometric Brownian motion processes,
dP
p
dM
M
  p dt   p dz p
  m dt   m dz m
with dzp dzm = ρdt.
Monetary economists define real money balances, m = M/P.
Derive the stochastic process for m (the bivariate case).
Applying Itô’s lemma to the two state variables
m
2
2
2
2

m

m

m

m ( dM )( dP )
1
1
2
dm 
dM 
dP 
( dM ) 
( dP ) 
2
2
P
2 M
2 P
M P
M
 1 dM  M2 dP  0  M3  p2 P 2 dt  12  p P  m M dt
P
P
P
P
 m dM  m dP  m  p2 dt  m  p  m dt
M
P
Therefore,
dm
m
  m d t   m d z m   p d t   p d z p  
 m   p  
2
p
2
p
   p m  d t
   p m  d t   m d z m   p d z p
which can be written as
dm
m
where
  dt   dz
2
   m   p   p   p m ,  dz   m dz m   p dz p ,  2   m2   p2  2  p m .
So, m also follows a geometric Brownian motion process
4. Extending diffusion processes to include jump
processes (Pennacchi, Ch. 11)
Brownian motion processes may not be realistic for modeling economic and
financial time series where the paths of the random variables are not
continuous, rather they exhibit discontinuities or “jumps.” Think of these
as asset price shocks.
We need to augment the diffusion process with another source of uncertainty
to capture that price discontinuity. For this purpose we can use the sum of
a Brownian motion diffusion process and a Poisson jump process.
•
•
•
•
Note: A Poisson process is a continuous-time counting process N( ) with the following
properties:
N(0) = 0,
the numbers of occurrences counted in disjoint intervals are independent from each other,
the probability distribution of the number of occurrences counted in any time interval only
depends on the length of the interval, and
none of the occurrences are simultaneous.
Jumps in continuous time
• Assume the continuous time process for a stock,
where dz is the Wiener process and dq is a Poisson jump process with the
characteristics
During each time interval (dt) there is a λdt probability that q will jump once
during the interval, where λ is assumed to be a constant and (Y-1) is i.i.d.
When a jump occurs dS = (Y-1)S, and S → YS.
Itô’s lemma for jump diffusion processes
• Let c(S,t) be the value of a variable that is a twice-differentiable function
of S(t), and S(t) follows the jump diffusion process
• Extending Itô’s lemma to this case, the value of c(S,t) is
Define μc as the instantaneous expected rate of return on c, i.e., E[dc/c],
conditional on a jump not occurring.
From (11.5) we get
where kc(t) is the time-varying expected jump size of c(S,t), and
Valuing the contingent claim
Now develop a portfolio that includes a contingent claim (call option) with
price c, an underlying asset that follows the jump diffusion process in
(11.1), and a riskless asset that pays return r.
Let the proportions invested in the portfolio be: w1 in the underlying asset, w2
in the contingent claim, w3 = (1 - w1 - w2) in the riskless asset.
The instantaneous rate of return on the portfolio, with substitutions for dS/S
from (11.1) and dc/c from (11.7), is
Imperfect hedge (jump size is random)
Hedge portfolio return with jump - during “quiet times” the call writer earns a profit, but during the
infrequent “active times” the call writer realizes a loss