Brownian Motion as Limit of Random Walk
Claim 1 A (µ, σ) Brownian motion is the limiting case of
Brownian Motion as Limit of Random Walk (continued)
• (continued) – Here
Xi ≡
+1 if the ith move is to the right,
−1 if the ith move is to the left.
– Xi are independent with
Prob[ Xi = 1 ] = p = 1 − Prob[ Xi = −1 ].
• Recall E[ Xi ] = 2p − 1 and Var[ Xi ] = 1 − (2p − 1)2.
Brownian Motion as Limit of Random Walk (continued)
• Therefore,
E[ Y (t) ] = n(∆x)(2p − 1), Var[ Y (t) ] = n(∆x)2 [
1 − (2p − 1)2 ] .
• With ∆x ≡ σ√
∆t and p ≡ [ 1 + (µ/σ)√
∆t ]/2, E[ Y (t) ] = nσ√
∆t (µ/σ)√
∆t = µt, Var[ Y (t) ] = nσ2∆t [
1 − (µ/σ)2∆t]
→ σ2t, as ∆t → 0.
Brownian Motion as Limit of Random Walk (concluded)
• Thus, { Y (t), t ≥ 0 } converges to a (µ, σ) Brownian motion by the central limit theorem.
• Brownian motion with zero drift is the limiting case of symmetric random walk by choosing µ = 0.
• Note that
Var[ Y (t + ∆t) − Y (t) ]
=Var[ ∆x Xn+1 ] = (∆x)2 × Var[ Xn+1 ] → σ2∆t.
• Similarity to the the BOPM: The p is identical to the probability in Eq. (27) on p. 256 and ∆x = ln u.
Geometric Brownian Motion
• Let X ≡ { X(t), t ≥ 0 } be a Brownian motion process.
• The process
{ Y (t) ≡ eX(t), t ≥ 0 }, is called geometric Brownian motion.
• Suppose further that X is a (µ, σ) Brownian motion.
• X(t) ∼ N(µt, σ2t) with moment generating function E
[
esX(t) ]
= E [ Y (t)s ] = eµts+(σ2ts2/2) from Eq. (20) on p 148.
Geometric Brownian Motion (continued)
• In particular,
E[ Y (t) ] = eµt+(σ2t/2), Var[ Y (t) ] = E [
Y (t)2 ]
− E[ Y (t) ]2
= e2µt+σ2t (
eσ2t − 1) .
0.2 0.4 0.6 0.8 1 Time (t) -1
1 2 3 4 5 6 Y(t)
Geometric Brownian Motion (continued)
• It is useful for situations in which percentage changes are independent and identically distributed.
• Let Yn denote the stock price at time n and Y0 = 1.
• Assume relative returns
Xi ≡ Yi Yi−1
are independent and identically distributed.
Geometric Brownian Motion (concluded)
• Then
ln Yn =
∑n i=1
ln Xi
is a sum of independent, identically distributed random variables.
• Thus { ln Yn, n ≥ 0 } is approximately Brownian motion.
– And { Yn, n ≥ 0 } is approximately geometric Brownian motion.
Continuous-Time Financial Mathematics
A proof is that which convinces a reasonable man;
a rigorous proof is that which convinces an unreasonable man.
— Mark Kac (1914–1984) The pursuit of mathematics is a divine madness of the human spirit.
— Alfred North Whitehead (1861–1947), Science and the Modern World
Stochastic Integrals
• Use W ≡ { W (t), t ≥ 0 } to denote the Wiener process.
• The goal is to develop integrals of X from a class of stochastic processes,a
It(X) ≡
∫ t 0
X dW, t ≥ 0.
• It(X) is a random variable called the stochastic integral of X with respect to W .
• The stochastic process { It(X), t ≥ 0 } will be denoted by ∫
X dW .
aKiyoshi Ito (1915–2008).
Stochastic Integrals (concluded)
• Typical requirements for X in financial applications are:
– Prob[∫t
0 X2(s) ds < ∞ ] = 1 for all t ≥ 0 or the stronger ∫ t
0 E[ X2(s) ] ds < ∞.
– The information set at time t includes the history of X and W up to that point in time.
– But it contains nothing about the evolution of X or W after t (nonanticipating, so to speak).
– The future cannot influence the present.
• { X(s), 0 ≤ s ≤ t } is independent of { W (t + u) − W (t), u > 0 }.
Ito Integral
• A theory of stochastic integration.
• As with calculus, it starts with step functions.
• A stochastic process { X(t) } is simple if there exist 0 = t0 < t1 < · · · such that
X(t) = X(tk−1) for t ∈ [ tk−1, tk), k = 1, 2, . . . for any realization (see figure on next page).
t0 t
1 t
2 t
3 t
4 t
5
X t
a f
t
Ito Integral (continued)
• The Ito integral of a simple process is defined as It(X) ≡
n∑−1 k=0
X(tk)[ W (tk+1) − W (tk) ], (50) where tn = t.
– The integrand X is evaluated at tk, not tk+1.
• Define the Ito integral of more general processes as a limiting random variable of the Ito integral of simple stochastic processes.
Ito Integral (continued)
• Let X = { X(t), t ≥ 0 } be a general stochastic process.
• Then there exists a random variable It(X), unique almost certainly, such that It(Xn) converges in probability to It(X) for each sequence of simple
stochastic processes X1, X2, . . . such that Xn converges in probability to X.
• If X is continuous with probability one, then It(Xn) converges in probability to It(X) as
δn ≡ max1≤k≤n(tk − tk−1) goes to zero.
Ito Integral (concluded)
• It is a fundamental fact that ∫
X dW is continuous almost surely.
• The following theorem says the Ito integral is a martingale.
– A corollary is the mean value formula
E
[ ∫ b a
X dW ]
= 0.
Theorem 17 The Ito integral ∫
X dW is a martingale.
Discrete Approximation
• Recall Eq. (50) on p. 489.
• The following simple stochastic process { bX(t) } can be used in place of X to approximate the stochastic
integral ∫t
0 X dW ,
X(s)b ≡ X(tk−1) for s ∈ [ tk−1, tk), k = 1, 2, . . . , n.
• Note the nonanticipating feature of bX.
– The information up to time s,
{ bX(t), W (t), 0 ≤ t ≤ s },
cannot determine the future evolution of X or W .
Discrete Approximation (concluded)
• Suppose we defined the stochastic integral as
n∑−1 k=0
X(tk+1)[ W (tk+1) − W (tk) ].
• Then we would be using the following different simple stochastic process in the approximation,
Y (s)b ≡ X(tk) for s ∈ [ tk−1, tk), k = 1, 2, . . . , n.
• This clearly anticipates the future evolution of X.a
aSee Exercise 14.1.2 of the textbook for an example where it matters.
X
t
X
t
$ Y
(a) (b)
$X
Ito Process
• The stochastic process X = { Xt, t ≥ 0 } that solves Xt = X0 +
∫ t 0
a(Xs, s) ds +
∫ t 0
b(Xs, s) dWs, t ≥ 0 is called an Ito process.
– X0 is a scalar starting point.
– { a(Xt, t) : t ≥ 0 } and { b(Xt, t) : t ≥ 0 } are stochastic processes satisfying certain regularity conditions.
• a(Xt, t): the drift.
• b(Xt, t): the diffusion.
Ito Process (continued)
• A shorthanda is the following stochastic differential equation for the Ito differential dXt,
dXt = a(Xt, t) dt + b(Xt, t) dWt. (51) – Or simply
dXt = at dt + bt dWt.
– This is Brownian motion with an instantaneous drift at and an instantaneous variance b2t.
• X is a martingale if at = 0 (Theorem 17 on p. 491).
aPaul Langevin (1904).
Ito Process (concluded)
• dW is normally distributed with mean zero and variance dt.
• An equivalent form of Eq. (51) is dXt = at dt + bt√
dt ξ, (52)
where ξ ∼ N(0, 1).
Euler Approximation
• The following approximation follows from Eq. (52), X(tb n+1)
= bX(tn) + a( bX(tn), tn) ∆t + b( bX(tn), tn) ∆W (tn),
(53) where tn ≡ n∆t.
• It is called the Euler or Euler-Maruyama method.
• Recall that ∆W (tn) should be interpreted as W (tn+1) − W (tn), not W (tn) − W (tn−1).
• Under mild conditions, bX(tn) converges to X(tn).
More Discrete Approximations
• Under fairly loose regularity conditions, Eq. (53) on p. 498 can be replaced by
X(tb n+1)
= bX(tn) + a( bX(tn), tn) ∆t + b( bX(tn), tn)√
∆t Y (tn).
– Y (t0), Y (t1), . . . are independent and identically distributed with zero mean and unit variance.
More Discrete Approximations (concluded)
• An even simpler discrete approximation scheme:
X(tb n+1)
= bX(tn) + a( bX(tn), tn) ∆t + b( bX(tn), tn)√
∆t ξ.
– Prob[ ξ = 1 ] = Prob[ ξ = −1 ] = 1/2.
– Note that E[ ξ ] = 0 and Var[ ξ ] = 1.
• This is a binomial model.
• As ∆t goes to zero, bX converges to X.
Trading and the Ito Integral
• Consider an Ito process dSt = µt dt + σt dWt. – St is the vector of security prices at time t.
• Let ϕt be a trading strategy denoting the quantity of each type of security held at time t.
– Hence the stochastic process ϕtSt is the value of the portfolio ϕt at time t.
• ϕt dSt ≡ ϕt(µt dt + σt dWt) represents the change in the value from security price changes occurring at time t.
Trading and the Ito Integral (concluded)
• The equivalent Ito integral, GT(ϕ) ≡
∫ T 0
ϕt dSt =
∫ T 0
ϕtµt dt +
∫ T 0
ϕtσt dWt, measures the gains realized by the trading strategy over the period [ 0, T ].
Ito’s Lemma
A smooth function of an Ito process is itself an Ito process.
Theorem 18 Suppose f : R → R is twice continuously differentiable and dX = at dt + bt dW . Then f (X) is the Ito process,
f (Xt)
= f (X0) +
∫ t 0
f′(Xs) as ds +
∫ t 0
f′(Xs) bs dW +1
2
∫ t 0
f′′(Xs) b2s ds for t ≥ 0.
Ito’s Lemma (continued)
• In differential form, Ito’s lemma becomes df (X) = f′(X) a dt + f′(X) b dW + 1
2 f′′(X) b2 dt.
(54)
• Compared with calculus, the interesting part is the third term on the right-hand side.
• A convenient formulation of Ito’s lemma is df (X) = f′(X) dX + 1
2 f′′(X)(dX)2.
Ito’s Lemma (continued)
• We are supposed to multiply out
(dX)2 = (a dt + b dW )2 symbolically according to
× dW dt
dW dt 0
dt 0 0
– The (dW )2 = dt entry is justified by a known result.
• Hence (dX)2 = (a dt + b dW )2 = b2 dt.
• This form is easy to remember because of its similarity to the Taylor expansion.
Ito’s Lemma (continued)
Theorem 19 (Higher-Dimensional Ito’s Lemma) Let W1, W2, . . . , Wn be independent Wiener processes and
X ≡ (X1, X2, . . . , Xm) be a vector process. Suppose
f : Rm → R is twice continuously differentiable and Xi is an Ito process with dXi = ai dt + ∑n
j=1 bij dWj. Then df (X) is an Ito process with the differential,
df (X) =
∑m i=1
fi(X) dXi + 1 2
∑m i=1
∑m k=1
fik(X) dXi dXk, where fi ≡ ∂f/∂Xi and fik ≡ ∂2f /∂Xi∂Xk.
Ito’s Lemma (continued)
• The multiplication table for Theorem 19 is
× dWi dt
dWk δik dt 0
dt 0 0
in which
δik =
1 if i = k, 0 otherwise.
Ito’s Lemma (continued)
• In applying the higher-dimensional Ito’s lemma, usually one of the variables, say X1, is time t and dX1 = dt.
• In this case, b1j = 0 for all j and a1 = 1.
• As an example, let
dXt = at dt + bt dWt.
• Consider the process f(Xt, t).
Ito’s Lemma (continued)
• Then
df = ∂f
∂Xt dXt + ∂f
∂t dt + 1 2
∂2f
∂Xt2 (dXt)2
= ∂f
∂Xt (at dt + bt dWt) + ∂f
∂t dt +1
2
∂2f
∂Xt2 (at dt + bt dWt)2
=
( ∂f
∂Xt at + ∂f
∂t + 1 2
∂2f
∂Xt2 b2t )
dt + ∂f
∂Xt bt dWt. (55)
Ito’s Lemma (continued)
Theorem 20 (Alternative Ito’s Lemma) Let W1, W2, . . . , Wm be Wiener processes and
X ≡ (X1, X2, . . . , Xm) be a vector process. Suppose
f : Rm → R is twice continuously differentiable and Xi is an Ito process with dXi = ai dt + bi dWi. Then df (X) is the following Ito process,
df (X) =
∑m i=1
fi(X) dXi + 1 2
∑m i=1
∑m k=1
fik(X) dXi dXk.
Ito’s Lemma (concluded)
• The multiplication table for Theorem 20 is
× dWi dt
dWk ρik dt 0
dt 0 0
• Above, ρik denotes the correlation between dWi and dWk.
Geometric Brownian Motion
• Consider geometric Brownian motion Y (t) ≡ eX(t) – X(t) is a (µ, σ) Brownian motion.
– Hence dX = µ dt + σ dW by Eq. (49) on p. 473.
• As ∂Y/∂X = Y and ∂2Y /∂X2 = Y , Ito’s formula (54) on p. 504 implies
dY = Y dX + (1/2) Y (dX)2
= Y (µ dt + σ dW ) + (1/2) Y (µ dt + σ dW )2
= Y (µ dt + σ dW ) + (1/2) Y σ2 dt.