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)
• Recall
E[ Xi ] = 2p − 1,
Var[ Xi ] = 1 − (2p − 1)2.
• Assume n = t/Δt is an integer.Δ
• Its position at time t is
Y (t) = Δx (XΔ 1 + X2 + · · · + Xn) .
Brownian Motion as Limit of Random Walk (continued)
• Therefore,
E[ Y (t) ] = n(Δx)(2p − 1), Var[ Y (t) ] = n(Δx)2
1 − (2p − 1)2 .
• With Δx = σΔ √
Δt and p = [ 1 + (μ/σ)Δ √
Δt ]/2,a E[ Y (t) ] = nσ√
Δt (μ/σ)√
Δt = μt, Var[ Y (t) ] = nσ2Δt
1 − (μ/σ)2Δt
→ σ2t, as Δt → 0.
aIdentical to Eq. (42) on p. 292!
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.
• Similarity to the the BOPM: The p is identical to the probability in Eq. (42) on p. 292 and Δx = ln u.
• Note that
Var[ Y (t + Δt) − Y (t) ]
=Var[ Δx Xn+1 ] = (Δx)2 × Var[ Xn+1 ] → σ2Δt.
Geometric Brownian Motion
• Let X =Δ { X(t), t ≥ 0 } be a Brownian motion process.
• The process
{ Y (t) = eΔ X(t), t ≥ 0 }, is called geometric Brownian motion.
• Suppose further that X is a (μ, σ) Brownian motion.
• By assumption, Y (0) = e0 = 1.
Geometric Brownian Motion (concluded)
• X(t) ∼ N(μt, σ2t) with moment generating function E
esX(t)
= E [ Y (t)s ] = eμts+(σ2ts2/2) from Eq. (27) on p 171.
• In particular,a
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)
A Case for Long-Term Investment
a• Suppose the stock follows the geometric Brownian motion
S(t) = S(0) eN (μt,σ2t) = S(0) etN (μ,σ2/t ), t ≥ 0, where μ > 0.
• The annual rate of return has a normal distribution:
N
μ, σ2 t
.
• The larger the t, the likelier the return is positive.
• The smaller the t, the likelier the return is negative.
aContributed by Dr. King, Gow-Hsing on April 9, 2015. See
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 =Δ { 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) =Δ
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.
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).a
aIt is right-continuous.
J J J J! J" J# : J
= B
J
Ito Integral (continued)
• The Ito integral of a simple process is defined as It(X) =Δ
n−1
k=0
X(tk)[ W (tk+1) − W (tk) ], (75) 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 = maxΔ
1≤k≤n(tk − tk−1)
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
Theorem 17 The Ito integral
X dW is a martingale.
• A corollary is the mean value formula E
b
a
X dW
= 0.
aSee Exercise 14.1.1 for simple stochastic processes.
Discrete Approximation
• Recall Eq. (75) on p. 588.
• The following simple stochastic process { X(t) } can be used in place of X to approximate t
0 X dW ,
X(s) = X(tΔ k−1) for s ∈ [ tk−1, tk), k = 1, 2, . . . , n.
• Note the nonanticipating feature of X.
– The information up to time s,
{ X(t), W (t), 0 ≤ t ≤ s },
cannot determine the future evolution of X or W .
Discrete Approximation (concluded)
• Suppose, unlike Eq. (75) on p. 588, we defined the stochastic integral from
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) = X(tΔ k) 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.
:
J
:
J
;
= >
:
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)
• Typical regularity conditions are:a
– For all T > 0, x ∈ Rn, and 0 ≤ t ≤ T ,
| a(x, t) | + | b(x, t) | ≤ C(1 + | x |) for some constant C.b
– (Lipschitz continuity) For all T > 0, x ∈ Rn, and 0 ≤ t ≤ T ,
| a(x, t) − a(y, t) | + | b(x, t) − b(y, t) | ≤ D | x − y | for some constant D.
aØksendal (2007).
bThis condition is not needed in time-homogeneous cases, where a
Ito Process (continued)
• A shorthanda is the following stochastic differential equationb (SDE) for the Ito differential dXt,
dXt = a(Xt, t) dt + b(Xt, t) dWt. (76) – 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.c
aPaul Langevin (1872–1946) in 1904.
bLike any equation, an SDE contains an unknown, the process Xt.
cRecall Theorem 17 (p. 590).
Ito Process (concluded)
• From calculus, we would expect t
0 W dW = W (t)2/2.
• But W (t)2/2 is not a martingale, hence wrong!
• The correct answer is [ W (t)2 − t ]/2.
• A popular representation of Eq. (76) is dXt = at dt + bt√
dt ξ, (77)
where ξ ∼ N (0, 1).
Euler Approximation
• Define tn = nΔt.Δ
• The following approximation follows from Eq. (77),
X(t n+1)
= X(tn) +a( X(tn), tn) Δt + b( X(tn), tn) ΔW (tn). (78)
• It is called the Euler or Euler-Maruyama method.
• Recall that ΔW (tn) should be interpreted as W (tn+1) − W (tn),
not W (tn) − W (tn−1)!a
aRecall Eq. (75) on p. 588.
Euler Approximation (concluded)
• With the Euler method, one can obtain a sample path X(t 1), X(t2), X(t3), . . .
from a sample path
W (t0), W (t1), W (t2), . . . .
• Under mild conditions, X(tn) converges to X(tn).
More Discrete Approximations
• Under fairly loose regularity conditions, Eq. (78) on p. 598 can be replaced by
X(t n+1)
= X(tn) + a( X(tn), tn) Δt + b( X(tn), tn)√
Δt 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(t n+1)
= X(tn) + a( X(tn), tn) Δt + b( X(tn), tn)√
Δt ξ.
– Prob[ ξ = 1 ] = Prob[ ξ = −1 ] = 1/2.
– Note that E[ ξ ] = 0 and Var[ ξ ] = 1.
• This is a binomial model.
• As Δt goes to zero, X converges to X.a
aHe (1990).
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 =Δ 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) =Δ
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
aA 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.
aIto (1944).
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 (79)
=
f(X) a + 1
2 f(X) b2
dt + f(X) b dW.
• Compared with calculus, the interesting part is the third term on the right-hand side of Eq. (79).
• A convenient formulation of Ito’s lemma is df (X) = f(X) dX + 1
f(X)(dX)2. (80)
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 in Eq. (80).
• 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 = (XΔ 1, 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 /∂XΔ i and fik = ∂Δ 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. (81)
Ito’s Lemma (continued)
Theorem 20 (Alternative Ito’s Lemma) Let W1, W2, . . . , Wm be Wiener processes and
X = (XΔ 1, 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) = eΔ X(t). – X(t) is a (μ, σ) Brownian motion.
– By Eq. (74) on p. 573,
dX = μ dt + σ dW.
• Note that
∂Y
∂X = Y,
∂2Y
= Y.
Geometric Brownian Motion (continued)
• Ito’s formula (79) on p. 605 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.
• Hence
dY
Y =
μ + σ2/2
dt + σ dW. (82)
• The annualized instantaneous rate of return is μ + σ2/2 (not μ).a
aConsistent with Lemma 9 (p. 297).
Geometric Brownian Motion (continued)
• Alternatively, from Eq. (74) on p. 573, Xt = X0 + μt + σ Wt, an explicit (strong) solution.
• Hence
Yt = Y0 eμt+σ Wt,
a strong solution to the SDE (82) where Y0 = eX0.
Geometric Brownian Motion (concluded)
• On the other hand, suppose dY
Y = μ dt + σ dW.
• Then X(t) = ln Y (t) followsΔ dX =
μ − σ2/2
dt + σ dW.