308	CHAPTER 10. TARGET-ZONE MODELS

all variables except interest rates are in logarithms. The time derivative of a function x(t) is denoted with the ‘dot’ notation, xú (t) = dx(t)/dt. In order to work with these models, you need some background in stochastic calculus.

10.1Fundamentals of Stochastic Calculus

Let x(t) be a continuous-time deterministic process that grows at the constant rate, η such that, dx(t) = ηdt. Let G(x(t), t) be some possibly time-dependent continuous and di erentiable function of x(t). From calculus, you know that the total di erential of G is

dG =	∂G	dx(t) +	∂G	dt.	(10.1)
	∂x		∂t

If x(t) is a continuous-time stochastic process, however, the formula for the total di erential (10.1) doesn’t work and needs to be modiÞed. In particular, we will be working with a continuous-time stochastic process x(t) called a di usion process where the growth rate of x(t) randomly deviates from η,

dx(t) = ηdt + σdz(t).

(10.2)

ηdt is the expected change in x conditional on information available at t, σdz(t) is an error term and σ is a scale factor. z(t) is called a Wiener

process or Brownian motion and it evolves according to,
√
z(t) = u t,			(10.3)

iid

where u N(0, 1). At each instant, z(t) is hit by an independent draw u from the standard normal distribution. InÞnitesimal changes in z(t)

can be thought of as

√

√

√

dz(t) = z(t + dt) − z(t) = ut+dt t + dt − ut

t = u˜

dt, (10.4)

√

√

where ut+dt t + dt N(0, t

+ dt) and u

t

t

N(0, t) deÞne the new

1

random variable u˜ N(0, 1).

The di usion process is the continuous-

time analog of the random walk with drift η. Sampling the di usion

√

√

1

√

√

Since E[ut+dt

t] = 0, and Var[ut+dt

t + dt−ut

t + dt−ut t] = t+dt−t = dt,

√

√

√

iid

ut+dt

t + dt − ut

t deÞnes a new random variable, u˜

dt, where u˜ N(0, 1).

10.1. FUNDAMENTALS OF STOCHASTIC CALCULUS				309
x(t) at discrete points in time yields
x(t + 1) − x(t) =	Zt t+1 dx(s)
= η Zt t+1 ds + σ Zt t+1 dz(s)				(10.5)
=	η + σu˜.	|z(t {z−	}
		+1)	z(t)

If x(t) follows the di usion process (10.2), it turns out that the total di erential of G(x(t), t) is

	∂G		∂G		σ2 ∂2G
dG =		dx(t) +		dt +				dt.	(10.6)
	∂x						2
			∂t		2 ∂x

This result is known as Ito’s lemma. The next section gives a nonrigorous derivation of Ito’s lemma and can be skipped by uninterested readers.

Ito’s Lemma

Consider a random variable X with Þnite mean and variance, and a positive number θ > 0. Chebyshev’s inequality says that the probability that X deviates from its mean by more than θ is bounded by its variance divided by θ2

P{|X − E(X)| ≥ θ} ≤	Var(X)
		.	(10.7)
	θ2

If z(t) follows the Wiener process (10.3), then E[dz(t)] = 0 and Var[dz(t)2] = E[dz(t)2] −[Edz(t)]2 = dt. Apply Chebyshev’s inequality to dz(t)2, to get

P{|[dz(t)]2 − E[dz(t)]2| > θ} ≤ (dtθ2)2 .

Since dt is a fraction, as dt → 0, (dt)2 goes to zero even faster than dt does. Thus the probability that dz(t)2 deviates from its mean dt becomes negligible over inÞnitesimal increments of time. This suggests

310	CHAPTER 10. TARGET-ZONE MODELS

that you can treat the deviation of dz(t)2 from its mean dt as an error term of order O(dt2).2 Write it as

dz(t)2 = dt + O(dt2).

Taking a second-order Taylor expansion of G(x(t), t) gives

∆G =		∂G		∆x(t) +	∂G	∆t
		∂x			∂t			∆t2	+ 2∂x∂t[∆x(t)∆t]#
+ 2 "				∂x2 ∆x(t)2 + ∂t2
	1			∂2G			∂2G			∂2G
+	O(∆t2),								(10.8)

where O(∆t2) are the ‘higher-ordered’ terms involving (∆t)k with k > 2. You can ignore those terms when you send ∆t → 0.

If x(t) evolves according to the di usion process, you√know that ∆x(t) = η∆t + σ∆z(t), with ∆z(t) = u ∆t, and (∆x)2 = η2(∆t)2 + σ2(∆z)2 + 2ησ(∆t)(∆z) = σ2∆t + O(∆t3/2). Substitute these expressions into the square-bracketed term in (10.8) to get,

	∂G			∂G		σ2 ∂2G			(∆t) + O(∆t3/2). (10.9)
∆G =			(∆x(t)) +		(∆t) +
	∂x					2
				∂t		2 ∂x

As ∆t → 0, (10.9) goes to (10.6), because the O(∆t3/2) terms can be ignored. The result is Ito’s lemma.

10.2The Continuous—Time Monetary Model

A deterministic setting. To see how the monetary model works in continuous time, we will start in a deterministic setting. As in chapter 3, all variables except interest rates are in logarithms. The money market equilibrium conditions at home and abroad are

m(t) − p(t)	=	φy(t) − αi(t),	(10.10)
m (t) − p (t)	=	φy (t) − αi (t).	(10.11)

2An O(dt2) term divided by dt2 is constant.

10.2. THE CONTINUOUS—TIME MONETARY MODEL

311

International asset-market equilibrium is given by uncovered interest parity

i(t) − i (t) = sú(t).	(10.12)
The model is completed by invoking PPP
s(t) + p (t) = p(t).	(10.13)
Combining (10.10)-(10.13) you get
s(t) = f(t) + αsú(t),	(10.14)

where f(t) ≡ m(t) − m (t) − φ[y(t) − y (t)] are the monetary-model ‘fundamentals.’ Rewrite (10.14) as the Þrst-order di erential equation

sú(t)	−		s(t)		=	−f(t)	.	(10.15)
			α
						α
The solution to (10.15) is3
s(t) = α Zt ∞ e(t−x)/αf(x)dx
	1
=	1	et/α Zt			∞ e−x/αf(x)dx.			(10.16)
	α

A stochastic setting. The stochastic continuous-time monetary model is

m(t) − p(t)

=

φy(t) − αi(t),

(10.17)

m (t) − p (t)

=

φy (t) − αi (t),

(10.18)

i(t) − i (t)

= Et[sú(t)],

(10.19)

s(t) + p (t)

=

p(t).

(10.20)

3To verify that (10.16) is a solution, take its time derivative

α−2et/α

sú(t) = α et/α ·dt

Zt

e−x/αf(x)dx¸ + ·Zt

e−x/αf(x)dx¸

1

d

∞

∞

1

1

et/α Zt

∞

= −

f(t) +

e−x/αf(x)dx

α

α2

= −

1

f(t) +

1

s(t)

α

α

Therefore, (10.16) solves (10.15).

	312	CHAPTER 10. TARGET-ZONE MODELS
	Combine (10.17)-(10.20) to get
	E	[sú(t)]	−	s(t)		=	−f(t)	,	(10.21)
				α			α
	t

which is a Þrst-order stochastic di erential equation. To solve (10.21), mimic the steps used to solve the deterministic model to get the continuoustime version of the present-value formula

	1	∞ e(t−x)/αEt[f(x)]dx.
s(t) =	α Zt		(10.22)

To evaluate the expectations in (10.22) you must specify the stochastic process governing the fundamentals. For this purpose, we assume that the fundamentals process follow the di usion process

df(t) = ηdt + σdz(t),		(10.23)
√

where η and σ are constants, and dz(t) = u dt is the standard Wiener process. It follows that

f(x) − f(t) =	Zt x df(r)dr
=	Z x ηdr + Z x σdz(r)
	t	t	q
=	η(x − t) + σu			(x − t).		(10.24)

Take expectations of (10.24) conditional on time t information to get the prediction rule

Et[f(x)] = f(t) + η(x − t),

(10.25)

and substitute (10.25) into (10.22) to obtain

s(t) = α Zt ∞ e

α

[f(t) + η(x − t)]dx

1

(t−x)

= α

e (f − ηt) Zt

e−

dx +ηe

Zt

xe−

dx

1

t/α

∞ x/α

t/α

∞

x/α

a

b

|

{z

}

|

{z

}

= αη + f(t),

(10.26)