SEEM 5670 – Advanced Models in Financial Engineering
Sept. 19, 2016
Professor: Nan Chen
Lecture 3: Ito’s Formula and the Black-Scholes Option Pricing
Theory
1
Part I: Ito’s Formula
1.1
Ito Integral and Ito Processes
• Let Π = {t0 , t1 , · · · , tn } be a partition of [0, T ]. Assume that a stochastic process ∆t is a
simple process, i.e., ∆t is constant in t on each subinterval [tj , tj+1 ). Consider the Ito integral
of ∆t with respect to W :
Z
t
∆s dWs .
It =
0
It is natural to define it as follows: for any tk ≤ t ≤ tk+1 , then
It =
k−1
X
∆tj [Wtj+1 − Wtj ] + ∆tk [Wt − Wtk ].
j=0
Theorem 1 The Ito integral {It , t ≥ 0} is a martingale and
EIt2 = E
Z t
0
∆2s ds .
Theorem 2 The quadratic variation accumulated up to t by the Ito integral I is
Z t
[I, I]t =
0
∆2s ds.
• For general integrand ∆t , we approximate it by a sequence of simple processes such that
Z T
lim E
n→+∞
For each ∆n , the Ito integral
Rt
0
0
|∆nt − ∆t |2 dt = 0.
∆ns dWs has already been defined. Define
Z t
It =
Z t
∆s dWs = lim
n→+∞ 0
0
∆ns dWs
• Properties of the Ito integral.
Theorem 3 (i) The sample paths of It are continuous;
(ii) For eachR t, It is Ft -measurable;
R
R
(iii) If It = 0t ∆s dWs and
J = 0t Γs dWs , then It ± Jt = 0t (∆s ± Γs )dWs ; furthermore, for
Rt t
every constant c, cIt = 0 c∆s dWs ;
(iv) It is a martingale;
R
(v) EIt2 = E[R 0t ∆2s ds];
(vi) [I, I]t = 0t ∆2s ds.
1
SEEM 5670 – Advanced Models in Financial Engineering
Sept. 19, 2016
Professor: Nan Chen
Example 1 Compute
Z t
0
1
1
Ws dWs = Wt2 − t.
2
2
• Let Wt be a BM, and let Ft be an associated filtration. An Ito process is a stochastic process
of the form
Z
Z
t
t
Θs ds,
∆s dWs +
Xt = X0 +
0
0
where X0 is a nonrandom initial point and ∆ and Θ are adapted stochastic processes with
respect to Ft . People usually remember the Ito process by writing it in a differential form:
dXt = ∆t dWt + Θt dt.
Example 2 (Geometric BM; Black-Scholes Model)
Theorem 4 The quadratic variation of the Ito process is
Z t
[X, X]t =
0
1.2
∆2s ds.
Ito-Doeblin Formula
• Let Xt be an Ito process such that
Z T
Z T
∆t dWt
Θt dt +
XT = X0 +
t
t
and let f (t, x) be a function for which the partial derivatives ft (t, x), fx (t, x) and fxx (t, x) are
defined and continuous. Then, for every T ≥ 0,
Z T
Z T
0
0
Z T
+
0
fx (t, Xt )∆t dWt
ft (t, Xt )dt +
f (T, XT ) = f (0, X0 ) +
1
fx (t, Xt )Θt dt +
2
Z T
0
fxx (t, Xt )∆2t dt.
Example 3 (Vasicek interest rate model)
1.3
Multivariate Stochastic Calculus
• A d-dimensional Brownian motion is a process Wt = (Wt1 , · · · , Wtd ) with the following properties:
(i) Each Wti is one-dimensional Brownian motion;
(ii) If i 6= j, then the processes Wti and Wtj are independent.
• The cross variation of W i and W j on [0, T ] is defined as
[W i , W j ]T = lim
kΠk→0
n−1
X
[Wtik+1 − Wtik ][Wtjk+1 − Wtjk ]
k=0
where Π = {t0 , t1 , · · · , tn } be a partition of [0, T ].
2
SEEM 5670 – Advanced Models in Financial Engineering
Sept. 19, 2016
Professor: Nan Chen
Theorem 5 [W i , W j ]T = 0 for all T almost surely.
• Ito formula for multiple processes:
Let f (t, x, y) be a function whose partial derivatives ft , fx , fy , fxx , fxy , and fyy are defined
and continuous. Let Xt and Yt be Ito processes defined as
dXt = θt1 dt + σt11 dWt1 + σt12 dWt2
dYt = θt2 dt + σt21 dWt1 + σt22 dWt2 .
Then, for every T ≥ 0,
df (t, Xt , Yt ) = ft (t, Xt , Yt )dt + fx (t, Xt , Yt )dXt + fy (t, Xt , Yt )dYt
1
1
+ fxx (t, Xt , Yt )d[X, X]t + fxy (t, Xt , Yt )d[X, Y ]t + fyy (t, Xt , Yt )d[Y, Y ]t .
2
2
1.4
Recognizing Brownian motion
• Theorem 6 (Levy) Let Mt , t ≥ 0 be a martingale relative to a filtration {Ft , t ≥ 0}. Assume
that M0 = 0 and it has continuous sample paths, [M, M ]t = t for all t ≥ 0. Then Mt is a
Brownian motion.
2
Part II: Black-Scholes Option Pricing Theory
2.1
Black-Schole-Merton Formula
• Option and its payoffs
• Consider a market having a money market account and a stock. The money account pays a
constant rate of interest r and the stock is modeled by a geometric Brownian motion
dSt = αSt dt + σSt dWt .
Suppose that there is a trader who at each time t has a portfolio valued at Xt . He buys ∆t
shares of the stock and invests Xt − ∆t St in the money market account. Then, Xt follows
dXt = rXt dt + ∆t (α − r)St dt + ∆t σSt dWt .
• Consider a European call option that pays (ST − K)+ at time T . We want to calculate its
price at time t < T . Denote it by c(t, x), which is the price of the option at time t if the stock
price then is St = x. By Ito formula,
d(e
−rt
1
c(t, St )) = e
−rc(t, St ) + ct (t, St ) + αSt cx (t, St ) + σ 2 St2 cxx (t, St ) dt
2
+e−rt σSt cx (t, St )dWt .
−rt
On the other hand, a short option hedging portfolio starts with some initial capital X0 and
invest in the stock and money market account so that the portfolio value Xt at each time
matches with c(t, St ). In other words, X0 = c(0, S0 ) and
d(e−rt c(t, St )) = d(e−rt Xt ).
3
SEEM 5670 – Advanced Models in Financial Engineering
Sept. 19, 2016
Professor: Nan Chen
Applying Ito formula on the right hand side, we have
d(e−rt Xt ) = (α − r)∆t e−rt St dt + σ∆t e−rt St dWt .
This leads to the Black-Scholes-Merton partial differential equation
1
rc(t, x) = ct (t, x) + rxcx (t, x) + σ 2 x2 cxx (t, St )
2
and c(T, x) = (x − K)+ .
Theorem 7 (Black-Scholes-Merton Formula) The solutions to the above PDE is given
by
c(t, x) = xN (d+ (T − t, x)) − Ke−r(T −t) N (d− (T − t, x))
where
"
1
x
σ2
d± = √ log
+ (r ± )τ
σ τ
K
2
#
and N is the cumulative standard normal distribution function.
4
SEEM 5670 – Advanced Models in Financial Engineering
Sept. 19, 2016
Professor: Nan Chen
Homework Set 3 (Due on Sept. 29)
1. Let St satisfy the geometric Brownian motion:
dSt = αSt dt + σSt dWt .
(i). Compute d(log St ).
(ii). Integrate the formula you obtained in (i), and then exponentiate the answer to get that
1
St = S0 exp (α − σ 2 )t + σWt .
2
2. The Vasicek interest rate model is given by
dRt = (α − βRt )dt + σdWt .
(i) Compute d(eβt Rt ).
(ii) Integrate the formula you obtained in (i), and then solve for Rt .
(iii) What is the distribution of Rt ?
3. Let Wt be a Brownian motion, and define
Z t
sign(Ws )dWs ,
Bt =
0
where
(
sign(x) =
1, x ≥ 0,
−1, x < 0.
(i) Show that Bt is a Brownian motion;
(ii) Use Ito formula to compute d[Bt Wt ]. Integrate both sides of the resulting equation and
take expectations. Show that E[Bt Wt ] = 0.
(iii) Verify that
dWt2 = 2Wt dWt + dt.
(iv) Use Ito formula to compute d[Bt Wt2 ].
4. Verify Theorem 3.
5. Let W = {Wt = (Wt1 , Wt2 , Wt3 ) : t ≥ 0} be a 3-dimensional Brownian motion on some
probability space (Ω, F, P ). Consider its distance from the origin
Rt =
q
(Wt1 )2 + (Wt2 )2 + (Wt3 )2 .
Rt is known as the Bessel process with dimension 3. Show that Rt should satisfy the following
integral equation
Z t
1
Rt =
ds + Bt
R
0
s
where B = {Bt : t ≥ 0} is the standard 1-dimensional Brownian motion.
5
SEEM 5670 – Advanced Models in Financial Engineering
Sept. 19, 2016
Professor: Nan Chen
6. Let
dXt = bt dt + dWt ,
where W· is a 1-dim standard Brownian motion and b· is a deterministic function of t. Let
Lt = exp −
Z t
bs dWs −
0
1
2
Z t
0
b2s ds .
Show that L is a martingale.
7. Let σt be a given deterministic function of time and define the process X by
Z t
σs dWs .
Xt =
0
Show that the moment generating function of Xt (for a fixed t) is given by
(
E[exp(uXt )] = exp
u2
2
Z t
0
)
σs2 ds .
What is the probability distribution of Xt from this moment generating function? (Hint:
You may try to apply Ito’s formula on exp(uXt ) and calculate its expectation.)
6