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STOCHASTIC CALCULUS AND AN APPLICATION TO THE
BLACK-SCHOLES MODEL
BRIAN CHEN
Abstract. The following paper develops the basics behind stochastic calculus, which extends the theory of integration to stochastic (random) processes.
The paper starts off with developing Brownian motion and its properties, which
are used to develop the theory behind Itô integration. Several forms of the Itô
integral are presented. A brief overview of the Radon-Nikodym and Girsanov
Theorems are presented. Lastly, we present an application of stochastic calculus with the Black-Scholes Model, a model for pricing options or other claims
based on the arbitrage-free price of a hedging portfolio.
Contents
1. Probability
2. Brownian Motion
3. Stochastic Integration
4. The Black-Scholes Model
5. The Girsanov Theorem
6. The Black-Scholes Model for Pricing Other Claims
Acknowledgments
References
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1. Probability
In this section, we introduce some definitions and concepts in probability that
will be important in this paper.
Definition 1.1. A σ-algebra Σ over a set Ω is a non-empty collection of subsets
such that:
(1) If A ∈ Σ, then Ac ∈ Σ.
(2) If Ai is a countable collection of subsets in Σ, then ∪i Ai ∈ Σ.
Definition 1.2. If A is a collection of subsets of a set Ω, the σ-algebra generated
by A is the smallest σ-algebra containing A.
Definition 1.3. We denote a probability space (Ω, F), where Ω is the set of all
outcomes and F is the collection of events, or subsets of Ω, which forms a σ-algebra.
Definition 1.4. A measure is a non-negative countably additive set function from
a σ-algebra to R\R− . We say that µ is a probability measure if µ is a non-negative
countably additive set function F → [0, 1] and µ(Ω) = 1.
1
2
BRIAN CHEN
Definition 1.5. A discrete random variable X is a variable with values in a countable
P subset S of R and has a probability of taking on any value
P s in S. Furthermore,
P(X
=
s)
=
1,
P(X
=
s)
≥
0,
and
P(X
∈
U
)
=
s∈S
s∈U ∩S P(X = s). Note
that P is a probability measure: 2Ω → [0, 1].
Definition 1.6. A continuous random variable X is a variable with values in an
uncountable subset S of R with a probability measure P such that:
• There exists Ra non-negative, Lebesgue integrable function f such that
P(X ∈ A) = R A f (x)dx.
• P(X ∈ S) = S f (x)dx = 1.
Note that P is a probability measure: 2Ω → [0, 1]. We call the function f the
probability density function of X.
Definition 1.7. For a discrete random variable X, with values in a countable
subset
P
S of R, the expectation of X, denoted as E(X), is given by E(X) = s∈S sP(X =
s). For a continuous random variable Y with values in an uncountable
subset S̄ of
R
R, the expectation of Y , denoted as E(Y ), is given by E(Y ) = S̄ xf (x)dx, where
f is the probability density function of Y .
Definition 1.8. The variance of a discrete
or
continuous random variable X, denoted by Var X, is given by Var X = E X 2 − (E[X])2 .
Definition 1.9. A random variable X is said to be normally distributed with mean
µ and variance σ 2 if its probability density function is given by
(x−µ)2
1
f (x) = √ e− 2σ2 .
σ 2π
The function f is known as the normal (or Gaussian) distribution. The standard
normal distribution is the normal distribution with mean µ = 0 and variance σ 2 = 1.
For a random variable X, we write X ∼ N (µ, σ 2 ) if X is normally distributed with
mean µ and variance σ 2 .
Definition 1.10. We say a random variable X is an integrable random
if
variable
E [|X|] < ∞. We say that X is a square integrable random variable if E |X 2 | < ∞.
Definition 1.11. A filtration Ft is a nested sequence of σ-algebras in F, i.e., for
s < t, Fs ⊂ Ft .
Definition 1.12. A Borel set is a set that can be obtained through a taking a
countable union, taking a countable intersection, or taking the complement, or a
countable composition of the above operations, starting with open sets.
Definition 1.13. Given a Borel set B ∈ R and a random variable X, X −1 (B) is
the pre-image of B, which is an element of Ft .
By default, given random variable Xt , Ft is the smallest σ-algebra that contains
Xt−1 (B) for all Borel sets B ∈ R, i.e., Ft is generated by Xt−1 (B). Intuitively, one
can think of Ft as the “information” from the events up to time t.
Definition 1.14. A random variable X is said to be Ft -measurable if for all Borel
sets B in R, X −1 (B) ∈ Ft .
Definition 1.15. A collection of random variables Xt is said to be adapted to a
filtration Ft if for each t, Xt is Ft -measurable.
STOCHASTIC CALCULUS AND AN APPLICATION TO THE BLACK-SCHOLES MODEL
3
Example 1.16. For any integer t, let Xt be a random variable taking on a value
of either 1 or −1, with probability 12 for each. Then, let S0 = 0 and for integers
t > 0, let St = X1 + · · · + Xt . Then St is adapted to Ft .
Definition 1.17. A collection of random variables Xt adapted to a filtration Ft is
a martingale if:
(1) For any t, Xt is integrable, i.e., E[|Xt |] < ∞.
(2) For any s < t, E(Xt |Fs ) = Xs .
Example 1.18. The random variable St from example 1.16 is a martingale, as
E[St |Fr ] = Sr for r < t and E[|St |] < ∞ for all t.
Definition 1.19. A process Xt adapted to a filtration Ft is a submartingale (supermartingale) if it satisfies condition (1) in the definition of a martingale and the
second condition is replaced with:
For any s < t, E(Xt |Fs ) ≥ Xs (or E(Xt |Fs ) ≤ Xs ), respectively.
Definition 1.20. Two events A and B are independent if P (A ∩ B) = P (A)P (B).
Definition 1.21. We say two random variables X and Y are independent if for
any Borel sets B1 , B2 of R, {X ∈ B1 } and {Y ∈ B2 } are independent events.
Lastly, we will introduce some basic properties regarding expectations. Proofs
can be found in, for example, [1].
Proposition 1.22. Let X, Y be random variables. The following properties are
true:
• If X and Y are independent random variables, then E(XY ) = E(X)E(Y ).
• If a, b are constants, E(aX + bY ) = aE(X) + bE(Y ).
• If X is a Fn -measurable random variable, then E[X|Fn ] = X.
• If X is a Fn -measurable random variable, then E[XY |Fn ] = XE[Y |Fn ].
• (Tower Property) For s < t, E[X|Fs ] = E[E[X|Ft ]|Fs ].
Remark 1.23. The Tower Property implies that E[X] = E[E[X|Ft ]], which can be
seen by letting s = 0.
2. Brownian Motion
A stochastic process is a collection of random variables Xt that depend on time
t. The times can be either discrete or continuous. For the purposes of this paper,
we will mostly be dealing with continuous time. In particular, we will define a
specific type of stochastic process called Brownian motion.
One way to think of Brownian motion is as a continuous analog to a simple
random walk. A simple random walk on a space Rn is a process in which at t = 0,
we have a particle starting at a point on the integer lattice, or Zn . Then, at each
time step, the particle moves a distance 1 in one of the 2n directions, with each
direction being of equal probability. In other words, if the particle at time t is at a
point a = (a1 , · · · , an ) with respect to the standard basis, at time t + 1, the particle
1
for each direction.
moves to a point a ± ei , with probability 2n
Brownian motion is the limit of a simple random walk with both time intervals
and distance traveled
√ per time interval going to 0, with the distance intervals going
to 0 at a rate of t. A more rigorous formulation can be found in [1].
Although we will define Brownian motion in a different, more direct way, the
above is a useful way to conceptualize what Brownian motion is.
4
BRIAN CHEN
Definition 2.1. We say a stochastic process Bt is a Brownian motion with drift
m and variance σ 2 if it satisfies the following properties:
• Independent Increments: The random variable Bt − Bs , for s < t, is independent of the events occurring before time s (i.e., Br for r ≤ s).
• Stationary Increments: For s < t, the distribution of Bt − Bs is the same
as that of Bt−s − B0 .
• Continuity: The function t → Bt is continuous, with probability one.
In the one-dimensional Brownian motion, Bt − Bs is a normal random variable
with mean m(t − s) and variance σ 2 (t − s). The proof of this fact can be found in
[1].
Definition 2.2. We say that Bt is a standard Brownian motion if Bt is a Brownian
motion with m = 0 and σ 2 = 1.
In this paper, we assume that Brownian motion exists. A construction of Brownian motion can be found in [1] or other texts on Brownian motion.
Let us consider several properties of Brownian motion.
Theorem 2.3. With probability one, the function f : t → Bt is nowhere differentiable.
Proof. Let us first show that the function is nowhere differentiable on the interval
[0, 1]. Suppose the function were differentiable at a point t. Then, there would exist
−Bs
= r. This implies that for δ small enough, |Bt − Bs | <
a r such that lims→t Btt−s
2|r(t − s)| for all s ∈ B(t, δ).
For fixed n, consider the function
fn (k) = max{|B(k+2)/n − B(k+1)/n |, |B(k+1)/n − Bk/n |, |Bk/n − B(k−1)/n |}.
If there were a point in [0, 1] such that the function t → Bt were differentiable, then
for large enough n, for some k, fn (k) < M
n for some positive integer M .
For fixed n, k, note that
3
M
M
P fn (k) <
= P |B1/n | <
n
n
#3
"Z M r
n
n − x2 n
=
e 2 dx
2π
−M
n
r 3
x2 n 2M
n
≤
because e− 2 ≤ 1 for all x
n
2π
M3
≤ 3/2 .
n
3
M
M
Let Xn = mink fn (k). Then, P {Xn < M
n } ≤ nP {fn (k) < n } ≤ n1/2 . This goes to
0 as we let n → ∞.
For a fixed M , if we denote AM to be the event that there exists k such that
fn (k) < M
n , P (AM ) goes to 0 as n → ∞. If we take the countable union over
all positive integers M , we are done. Earlier on, we showed that if the function
t → Bt were differentiable, then there would exist k and M such that fn (k) < M
n .
This proof can be extended to show that the function t → Bt has probability zero
of being differentiable on the interval [s, s + 1] for positive integers s. Therefore,
STOCHASTIC CALCULUS AND AN APPLICATION TO THE BLACK-SCHOLES MODEL
5
taking a countable union of these intervals, we see that the function t → Bt has
probability zero of being differentiable on [0, ∞).
Theorem 2.4. Brownian motion Bt with drift m = 0 is a martingale.
Recall that a random process Zt adapted to filtration Ft is a martingale if for
t > s,
E(Zt |Fs ) = Zs .
Proof. This theorem follows from our definition of a Brownian motion, as
E(Bt |Fs ) = E(Bs |Fs ) + E(Bt − Bs |Fs ) = Bs .
Theorem 2.5. If Bt is a standard Brownian motion, then for positive a,
Bat
Yt = √
a
is also a standard Brownian motion.
Proof. This theorem can be proven by checking the conditions of Brownian motion.
Therefore, if we take a Brownian motion and scale time by a factor of a and scale
distance by a factor of √1a , the new process is still a standard Brownian motion.
A useful property of Brownian motion is the Strong Markov Property. However,
before we get to that, we need to define a stopping time.
Definition 2.6. We say that a random variable T that takes on values in [0, ∞]
is a stopping time with respect to {Ft } if the event {T ≤ t} is Ft -measurable. In
other words, at time t, we know whether or not the process has stopped based on
information from time 0 to t.
As an example, let T be the first time a Brownian motion reaches 1. Then, T is
a stopping time. We state the following theorem regarding stopping times.
Theorem 2.7. Brownian motion has the Strong Markov Property: If T is a
stopping time that is finite with probability one, then the random variable Yt =
BT +t − BT is a Brownian motion, and Yt is independent of Bt for t ≤ T .
A proof of this theorem can be found in [3].
Theorem 2.7 is useful in allowing us to prove the following theorem, which can
be used to calculate the probability that a Brownian motion exceeds a certain value
within a given time.
Theorem 2.8. Let Mt = max0≤s≤t Bs , where Bs is a standard Brownian motion
with B0 = 0. Then, for a > 0,
√
P (Mt > a) = 2P (Bt > a) = 2[1 − Φ(a/ t)]
where Φ is the probability density function for the standard Gaussian.
Proof. If we let Ta be inf{t|Bt > a}, then we see that the event Mt > a is equivalent
to the event Ta < t.
P (Bt > a) = P (Ta < t, Bt > a) = P (Mt > a)P (Bt − BTa > 0|Ta < t)
6
BRIAN CHEN
Using the Strong Markov Property, Bt − BTa is a standard Brownian motion if
Ta < t. Therefore,
P (Bt − BTa > 0|Ta < T ) = 1/2.
Using Theorem 2.5 and the fact that the distribution of Brownian motion is normal,
we see that
a
a
P (Bt > a) = P B1 > √ = 1 − Φ √
t
t
√
because Bt = B1 t.
We now define the quadratic variation of a stochastic process, as it will be
important later on.
Definition 2.9. The quadratic variation of a process Xt is given by:
X
hXit = lim
[Xj/n − X(j−1)/n ].
n→∞
j≤tn
Let us use hXis,t to denote hXit − hXis , for s < t. In other words,
X
[Xj/n − X(j−1)/n ].
hXis,t = lim
n→∞
sn≤j≤tn
Let us calculate the quadratic variation of a standard Brownian motion Bt . The
quadratic variation of Bt is given by
X
hBit = lim
[Bj/n − B(j−1)/n ]2 .
n→∞
j≤tn
Note that Bj/n −B(j−1)/n is a normal random variable with mean 0 and variance n1 .
Furthermore, note that the random variables Bj/n − B(j−1)/n and Bi/n − B(i−1)/n
for i 6= j are independent. For fixed n,
1
3
E[(B1/n )2 ] = and E[(B1/n )4 ] = 2 .
n
n
The second result comes from the fact that if X is a normal random variable
with mean 0 and variance σ 2 , then E[X 4 ] = 3σ 4 . This can be checked by using
integration by parts. We can calculate the variance of the random variable (B1/n )2 :
2
Var [(B1/n )2 ] = E[(B1/n )4 ] − (E[(B1/n )2 ])2 = 2
n
P
Now consider the random variable Qn = j≤tn [Bj/n − B(j−1)/n ]2 for fixed n.
Its expectation is given by


X
E[Qn ] = E 
[Bj/n − B(j−1)/n ]2  = tnE[(B1/n )2 ] = t
j≤tn
and its variance is given by


X
2t
Var [Qn ] = Var 
[Bj/n − B(j−1)/n ]2  = tnVar [(B1/n )2 ] = .
n
j≤tn
We note that when we take the limit as n → ∞, the variance goes to 0 while the
expectation remains t. Therefore, hBit = t.
STOCHASTIC CALCULUS AND AN APPLICATION TO THE BLACK-SCHOLES MODEL
7
Definition 2.10. We define the covariation of two processes Xt and Yt , denoted
as hX, Y it , as
X
hX, Y it = lim
[Xj/n − X(j−1)/n ][Yj/n − Y(j−1)/n ]
n→∞
j≤tn
Note that if we let Xt = Yt , the covariation hXt , Xt i is just the quadratic variation of Xt .
3. Stochastic Integration
We begin to develop the theory of Itô integration. Our goal will be to define
integrals for stochastic processes. As examples of stochastic integrals, consider
Z t
Z t
As dBs and
Cs ds,
0
0
where As and Cs are stochastic processes and Bs is a standard Brownian motion.
Although Cs is a random variable, the time integral is defined in its usual sense.
So, we will try to define the first
R tterm.
We will define the integral 0 As dBs in a similar method as that of defining
Rb
Riemann integrals. To define the integral a f dx, we split an interval [a, b] ∈ R into
a partition a = t0 < t1 < · · · < tn = b. Then, we define a step function g so that
for x ∈ [ti , ti + 1), g(x) = f (si ), where si is any number in the interval [ti , ti+1 ).
Pn−1
We define the Riemann sum R =
i=0 g(si )(ti+1 − ti ). Lastly, we take a
sequence of partitions and functions gn where the maximum distance ti+1 − ti
approaches 0. We define ||∆ti || = max0≤i≤n−1 |ti+1 − ti | and say that
Z b
f dx = lim R,
||∆ti ||→0
a
if the limit exists. Stochastic integration will be defined similarly - we start off with
defining the integral over a simple process (analogous to a step function) and then
approximate the integral over other processes as a limit of integrals over simple
processes.
Definition 3.1. We call At a simple process if there exist times 0 = t0 < · · · <
tn = t and random variables Y0 , · · · , Yn such that Yj is Ftj -measurable and At = Yj
for tj ≤ t < tj+1 .
Then, we define the integral of a simple process At as follows:
Z t
j−1
X
Zt =
As dBs =
Yj [Btj+1 − Btj ]
0
i=0
Furthermore, we let:
Z
t
Z
Z
As dBs −
As dBs =
r
t
0
r
As dBs
0
Proposition 3.2. Let At be a bounded, Ft -measurable process with continuous
paths. There exists C such that with probability one, |At | ≤ C for all t. Then, there
(n)
exists a sequence of simple processes At such that for all t,
Z t
2
lim
E[|As − A(n)
s | ]ds = 0
n→∞
0
8
BRIAN CHEN
(n)
and |At | < C for all n, t.
(n)
Proof. We construct a sequence of simple processes At in the following way. Let
(
A0
for t ∈ [0, n1 )
(n)
R (j+1)/n
At =
n j/n
As ds for t ∈ [ nj , j+1
n ) where j ∈ Z, j ≥ 1
(n)
Because |At | ≤ C, it is easy to see that |At | ≤ C for all n and t. Furthermore,
(n)
because t → At is continuous with probability one, At → At . By the Dominated
R t (n)
R t (n)
Convergence Theorem, limn→∞ 0 (As − As )2 ds = 0. Then, E[limn→∞ 0 (As −
Rt
R t (n)
(n)
As )2 ds] = limn→∞ 0 E[(As − As )2 ]ds = 0 by the fact that | 0 (As − As )2 ds| is
(n)
bounded uniformly by (2C)2 t and that (As
− As )2 < (2C)2 .
With this, we can now define the integral of bounded continuous processes. If At
(n)
is a bounded continuous process, there exists a sequence of simple processes At
Rt
(n) 2
such that limn→∞ 0 E[|As − As | ]ds = 0. Then, we define
Z t
Z t
As ds = lim
A(n)
s ds.
n→∞
0
0
Rt
(n)
For fixed t, this limit exists because L2 is complete and limn→∞ 0 E[|As −As |2 ]ds =
0. However, for t in general, t can take on an uncountable number of values. The
proof of the existence of the above limit is more complicated and will not be presented here (see [1] for the proof).
The Itô Integral satisfies several important properties, which are given here.
Theorem 3.3. Let At , Ct be bounded, adapted processes with continuous paths
and let
R t Bt be a standard Brownian motion with respect to a filtration Ft . Let
Zt = 0 As dBs . Then, the following properties hold:
• Linearity: If a, b are constants, then
Z t
Z t
Z t
(aAs + bCs )dBs = a
As dBs + b
Cs dBs
0
0
0
If r < t, then
Z
t
Z
As dBs =
0
r
Z
As dBs +
0
t
As dBs
r
• Variance Rule: Zt is square-integrable and satisfies
Z t
2
V ar[Zt ] = E[Zt ] =
E[A2s ]ds.
0
• Continuity: With probability one, the function t → Zt is continuous.
• Martingale Property: Zt is a martingale with respect to the filtration Ft .
Proof. Linearity of integrals of simple processes follows from definition, and linearity of integrals of a general process At follows from taking a sequence of simple
processes that approximate At . Continuity follows from the continuity of Brownian
motion.
We first prove the martingale property for a simple process At . We want to show
that for r < s, E[Zs |Fr ] = Zr . There exist times 0 = t0 < t1 < · · · and random
STOCHASTIC CALCULUS AND AN APPLICATION TO THE BLACK-SCHOLES MODEL
9
variables Y0 , Y1 , · · · such that At = Yj for tj ≤ t < tj+1 . Let m be the smallest i
such that r ≤ ti and let n be the largest i such that s ≥ ti . Then,
" n−1
#
X
E [Zs |Fr ] = E
Yi (Bti+1 − Bti ) + Ym−1 (Btm − Br ) + Yn (Bs − Btn )|Fr + Zr
i=m
=
n−1
X
E[Yi (Bti+1 − Bti )|Fr ] + E[Ym−1 (Btm − Br ) + Yn (Bs − Btn )|Fr ] + Zr
i=m
Note that each Yi is Fti measurable and Bti+1 − Bti is independent of Fti . Then,
using the tower property of conditional expectation [Prop 1.22],
E[Yi (Bti+1 − Bti )|Fr ] = E[E[Yi (Bti+1 − Bti )|Fti ]|Fr ]
= E[Yi E[(Bti+1 − Bti )]|Fr ] = 0.
Similarly, E[Ym−1 (Btm − Br ) + Yn (Bs − Btn )|Fr ] = 0. Therefore, the martingale
property holds when At is a simple function.
Now, we will prove that the variance rule holds for a simple process At . Let n
be the largest integer such that tn ≤ s. Then,
Zs2 =
n−1
X n−1
X
Yi Yj (Bti+1 − Bti )(Btj+1 − Btj ) +
i=1 j=1
+
n−1
X
n−1
X
Yn Yi (Bs − Btn )(Bti+1 − Bti )
i=1
Yn Yj (Bs − Btn )(Btj+1 − Btj ) + Yn2 (Bs − Btn )2 .
j=1
Now consider i < j. Then,
E(Yi Yj [Bti+1 − Bti ][Btj+1 − Btj ]) = E[E(Yi Yj (Bti+1 − Bti )(Btj+1 − Btj )|Ftj ]
= E[Yi Yj (Bti+1 − Bti )E(Btj+1 − Btj |Ftj )] = 0.
Pn−1
The same holds for j < i and a similar argument shows that E[ i=1 Yn Yi (Bs −
Pn−1
Btn )(Bti+1 − Bti )] = E[ j=1 Yn Yj (Bs − Btn )(Btj+1 − Btj )] = 0.
Therefore,
E[Zt2 ] =
n−1
X
E[Yi2 (Bti+1 − Bti )2 ] + E[Yn2 (Bs − Btn )2 ].
i=1
As before,
E[Yi2 (Bti+1 − Bti )2 ] = E[E(Yi2 (Bti+1 − Bti )2 |Ft )]
= E(Yi2 E((Bti+1 − Bti )2 )|Ft ))
= E[Yi2 ](ti+1 − ti ),
and similarly, E[Yn2 (Bs − Btn )2 ] = E[Yn2 ](s − tn ).
Therefore,
E[Zt ] =
n−1
X
i=1
E[Yi2 ](ti+1 − ti ) + E[Yn2 ](s − tn ) =
Z
t
E[A2s ]ds.
0
For general At , the martingale property and variance rule follow from taking a
sequence of simple functions.
10
BRIAN CHEN
Rt
(n)
To prove the martingale property, let 0 As dBs = Zt = limn→∞ Zt , where
R
t
(n)
(n)
(n)
Zt = 0 As ds and As is a sequence of simple functions approximating At , as
(n)
described above. We know that Zt = limn→∞ Zt . Then, we take the conditional
expectation with respect to Fr , where r < t:
(n)
E[Zt |Fr ] = E[ lim Zt |Fr ]
n→∞
(n)
Because Zt
Therefore,
is bounded, we can apply the Dominated Convergence Theorem.
(n)
E[Zt |Fr ] = lim E[Zt |Fr ] = lim Zr(n) = Zr .
n→∞
n→∞
For the variance rule, we take the limit as n → ∞:
Z t
(n)
2
E[(A(n)
lim E[Zt ] = lim
s ) ]ds
n→∞
(n)
As
n→∞
0
(n)
Zt
Because
and
are bounded, the Dominated Convergence Theorem can be
applied to obtain our desired result:
Z t
E[Zt ] =
E[A2s ]ds.
0
The definition of an Itô integral can be extended to non-bounded processes. Let
(n)
Tn = inf{t : |At | = n}. Define As = As∧Tn , where a ∧ b is defined as min{a, b}.
(n)
Then, As is a bounded process, and we define
Z t
Z t
As dBs = lim
A(n)
s dBs .
n→∞
0
0
(n)
To prove that this limit exists, let Kt = max0≤s≤t |As |. For n > Kt , At = At and
Rt
R t (n)
therefore 0 As dBs = limn→∞ 0 As dBs .
When the Itô integral is extended to non-bounded processes, continuity and
linearity still hold. However, the martingale property does not necessarily hold,
and a slight change must be made to the variance rule.
Theorem 3.4. Let At , Ct be adapted processes with continuous paths and
R tlet Bt be
a standard Brownian motion with respect to a filtration {Ft }. Let Zt = 0 As dBs .
• Linearity: If a, b are constants, then
Z t
Z t
Z t
(aAs + bCs )dBs = a
As dBs + b
Cs dBs
0
0
0
If r < t, then
t
Z
Z
As dBs =
0
r
Z
As dBs +
0
t
As dBs
r
• Variance Rule: Zt satisfies
V ar[Zt ] =
E[Zt2 ]
Z
=
t
E[A2s ]ds,
0
with the possibility that both sides are equal to infinity. If Var Zt < ∞ for
all t, then Zt is a square-integrable martingale.
STOCHASTIC CALCULUS AND AN APPLICATION TO THE BLACK-SCHOLES MODEL 11
• Continuity: With probability one, the function t → Zt is continuous.
Definition 3.5. A process Mt is a local martingale on [0, T ) if there exists a
sequence of stopping times τ1 ≤ τ2 ≤ · · · such that limn→∞ τn = T and Mt∧τj is a
martingale for all j.
Rt
Although Zt = 0 As dBs is not necessarily a martingale if As is not bounded,
Rt
it is a local martingale. We let τj = inf{t : hZit = 0 A2s ds = j}. Then Zt∧τj is a
martingale. Therefore, Zt is a local martingale on [0, T ), where T = limn→∞ τj .
We write the following stochastic differential equation
dXt = At dBt + Ct dt
to indicate that the random process Xt satisfies
Z t
Z t
Cs ds,
As dBs +
Xt = X0 +
0
0
where As and Cs are adapted continuous processes. We will often consider differential equations of the form
dXt = f (t, Xt )dBt + g(t, Xt )dt.
We now present Itô’s Formula, which is analogous to the fundamental theorem
of calculus. The fundamental theorem of calculus states that if f is continuous on
Rb
an interval [a, b] and F is the antiderivative of f , then a f (x)dx = F (b) − F (a).
Itô’s formula is similar to the fundamental theorem of calculus, but is applied to
functions of Brownian motion and time, as long as the function is C 1 with respect
to time and C 2 with respect to the Brownian motion.
Theorem 3.6. (Itô’s Formula) Suppose f (t, x) is C 1 in t and C 2 in x. If Bt is a
standard Brownian motion, then
Z t
Z t
1
f (t, Bt ) = f (0, B0 ) +
δx f (s, Bs )dBs +
[δs f (s, Bs ) + δxx f (s, Bs )]ds
2
0
0
In differential form, this is
1
df (t, Bt ) = δx f (t, Bt )dBt + [δt f (t, Bt ) + δxx f (t, Bt )]dt.
2
Proof. Consider the following partition of [0, t] : {0, t/n, · · · , t}. Then,
(3.7)
f (t, Bt ) = f (0, 0) +
n−1
X
f ((i + 1)t/n, B(i+1)t/n ) − f (it/n, Bit/n ).
i=0
To simplify notation, we let ∆t denote (i + 1)t/n − it/n = t/n and ∆B denote
B(i+1)t/n − Bit/n By Taylor’s Theorem,
f (t + ∆t, Bt + ∆B) =f (t, Bt ) + δt f (t, Bt )∆t + δB f (t, Bt )∆B
1
+ δBB f (t, Bt )(∆B)2 + o1 (∆t) + o2 ((∆B)2 ),
2
12
BRIAN CHEN
2
(∆t)
)
where lim∆t→0 o1∆t
= 0 and lim∆B→0 o2 ((∆B)
= 0. So, we apply Taylor’s
∆B
theorem to (3.7) and take the limit as n → ∞. We see that
f (t, Bt ) − f (0, 0) = lim
n−1
X
n→∞
δt f (it/n, Bit/n )∆t + lim
n→∞
i=0
+ lim
n−1
X
n→∞
+ lim
i=0
n−1
X
n→∞
n−1
X
δB f (it/n, Bit/n )∆B
i=0
n−1
X
1
δBB f (it/n, Bit/n )(∆B)2 + lim
o1 (∆t)
n→∞
2
i=0
o2 ((∆B)2 ).
i=0
We note that the fourth term goes to 0. Similarly, since (∆B)2 ≈ nt (given the
variance of Brownian motion), the fifth term also goes to 0. We see that the
Rt
first and second terms are simple process approximations for 0 δt f (t, Bt )dt and
Rt
δ f (t, Bt )dBt , respectively.
0 B
Pn−1
So, let us consider the third term: limn→∞ i=0 12 δBB f (it/n, Bit/n )(∆B)2 . We
R
t
will prove that it is equal to 21 0 δBB f (t, Bt )dt.
00
Let g(t) = f (Bt ). By continuity, for all > 0, there exists a simple function
g (t) such that for all t, |g(t) − g (t)| < . For any fixed > 0, we want to show
that
Z t
n−1
X
g [B(i+1)t/n − Bi/n ] =
lim
g dt.
n→∞
0
i=0
Note that for every interval [a, b] such that g is constant,
Z b
X
g (∆Bt )2 = g hBia,b = g
lim
ds,
n→∞
a
using the quadratic variation of Brownian motion. Therefore,
Z t
n−1
X
g [B(i+1)t/n − Bi/n ]2 =
lim
g dt.
n→∞
Because |
get that
Pn−1
i=0
0
i=0
[g − g][B(i+1)t/n − Bi/n ]| < lim
n→∞
n−1
X
Pn−1
i=0
[B(i+1)t/n − Bi/n ] = Bn , we
Z
g[B(i+1)t/n − Bi/n ] =
t
gdt.
0
i=0
There is a more general form of Itô’s Formula, which will be stated here. It
extends Itô’s Formula to functions f (Xt , Yt ), where Xt and Yt are Itô processes,
which will be defined below.
Definition 3.8. An Itô process is a process Xt that satisfies
dXt = At dt + Ct dBt ,
where At and Ct are adapted processes with continuous paths.
Note that At and Ct can depend on Xt , Bt , or even on the past path of Bt .
STOCHASTIC CALCULUS AND AN APPLICATION TO THE BLACK-SCHOLES MODEL 13
Theorem 3.9. Let Xt , Yt be Itô processes, where dXt = At dt + Ct dBt and dYt =
Rt dt + St dBt . Suppose f (x, y) is C 2 . Then,
1
df (Xt , Yt ) =δx f (Xt , Yt )dXt + δy (Xt , Yt )dYt + δxx f (Xt , Yt )Ct2 dt
2
1
+ δyy f (Xt , Yt )St2 dt + δxy f (Xt , Yt )Ct St dt.
2
We will not prove this theorem in this paper, as the general ideas are similar to
that of the proof of Theorem 3.6.
An important case of the above theorem is when f is a function of t and Xt ,
where Xt is an Itô process. As such, we will present it as a theorem itself.
Theorem 3.10. Let Xt be an Itô process such that dXt = At dt + Ct dBt . Let
f (t, Xt ) be C 1 in t and C 2 in x. Then,
1
df (t, Xt ) = δx f (t, Xt )dXt + δt f (t, Xt )dt + Ct2 δxx f (t, Xt )dt
2
1 2
= [δx f (t, Xt )At + δt f (t, Xt ) + Ct δxx f (t, Xt )]dt + δx f (t, Xt )Ct dBt .
2
Proof. We let Yt = t, which satisfies dYt = dt + 0dBt , and apply Theorem 3.9. The
second line follows from the fact that dXt = At dt + Ct dBt .
In section 2, we defined the covariation of two stochastic processes. Let us prove
the following theorem regarding the covariation of Itô processes.
Rt
Rt
Rt
Theorem 3.11. Let Xt = X0 + 0 As ds + 0 Cs dBs and Yt = Y0 + 0 A0s ds +
Rt 0
C dBs , where Bt is a standard Brownian motion and At , Ct , A0t , and Ct0 are
0 s
adapted processes with continuous paths. Then, the covariation hX, Y it is given by:
Z t
hX, Y it =
Cs Cs0 ds.
0
Before we prove this theorem, we will first need to prove the following:
Theorem 3.12. Suppose Xt is continuous and Yt has finite variation. Then, the
covariation hX, Y it = 0.
Proof.
| hf, git | =| lim
n→∞
X
[Xj/n − X(j−1)/n ][Yj/n − Y(j−1)/n ]|
j≤tn
≤ lim max |Xi/n − X(i−1)/n ||
n→∞ i≤tn
X
|Yj/n − Y(j−1)/n |
j≤tn
≤ lim max |Xi/n − X(i−1)/n ||Yt − Y0 |.
n→∞ i≤tn
We know that by continuity of Xt (and therefore uniform continuity over [0, t]) that
limn→∞ maxi≤tn |Xi/n − X(i−1)/n | = 0, so we are done.
Returning to Theorem 3.11:
14
BRIAN CHEN
Proof of Theorem 3.11. Using the definition of covariation, we see that
X
hX, Y it = lim
[Xj/n − X(j−1)/n ][Yj/n − Y(j−1)/n ]
n→∞
j≤tn
= lim
n→∞
Z
j/n
X Z
j≤tn
Z
Z
j/n
A0s ds +
(j−1)/n
t
As ds,
=
Z
Z
0
t
j/n
Z
j/n
Cs0 dBs
(j−1)/n
(j−1)/n
0
t
Z t
Z t
Cs dBs,
Cs0 ds
A0s ds +
0
Rt
Cs0 dBs
(j−1)/n
Cs dBs
0
t
t
Cs dBs,
+
Z
j/n
As ds
+
Z t
Z t
0
0
Cs dBs
As ds,
As ds +
0
0
Z
(j−1)/n
(j−1)/n
t
Z
j/n
Z
(j−1)/n
(j−1)/n
Cs dBs
Z
A0s ds
As ds
j/n
+
j/n
0
t
0
t
Rt
Rt
Rt
Because 0 As ds and
have finite variation and 0 Cs dBs and 0 Cs0 dBs
are continuous, by Theorem 3.12, the first
of the
DR three terms
E above sum are equal
Rt 0
t
to 0. Thus, we only have to deal with 0 Cs dBs , 0 Cs dBs .
A0s ds
0
t
0
For a fixed > 0, there exist simple functions C and C such that for all t,
0
|Ct − Ct | < and |Ct − Ct0 | < . We proceed in a similar way as in part of the
proof of Theorem 3.6.
0
For fixed , let us consider intervals [a, b] where C and C are constant. Then,
Z j/n
X Z j/n
X
0
0
lim
Cs dBs
(Bj/n − B(j−1)/n )2
Cs dBs = Cj/n
Cj/n
lim
n→∞
(j−1)/n
n→∞
(j−1)/n
0
(b − a).
= Cj/n
Cj/n
Thus,
lim
n→∞
XZ
j/n
Cs dBs
(j−1)/n
j≤tn
Z
j/n
0
Cs dBs
(j−1)/n
Z
=
t
0
Cs Cs dt.
0
0
So, for simple functions Ct and Ct , the covariation
Z t
Z t
Z t
0
0
Cs dBs ,
Cs dBs =
Cs Cs dt.
0
0
t
0
DR
t
E
Rt
Now we want to show that this approximates 0 Cs dBs , 0 Cs0 dBs . Note that
t
X Z j/n
Z j/n
Z
Z
j/n
X j/n
0
0
Cs dBs
Cs dBs −
Cs dBs
Cs dBs (j−1)/n
(j−1)/n
j≤tn (j−1)/n
j≤tn (j−1)/n
X Z j/n
Z j/n
0
≤ [
Cs dBs +
Cs dBs ]
(j−1)/n
(j≤tn (j−1)/n
Z t
Z t
≤ Cs0 dBs +
Cs dBs .
0
0
So, by making arbitrarily small, we are done.
STOCHASTIC CALCULUS AND AN APPLICATION TO THE BLACK-SCHOLES MODEL 15
Rt
Rt
Theorem 3.13. Let Xt = X0 + 0 As ds+ 0 Cs dBs , where Bt is a standard Brownian motion and At is an adapted process with continuous paths. Then, the quadratic
variation of Xt is given by
Z t
hXit =
Cs2 ds.
0
Similarly, for r < t,
Z
hXir,t =
t
Cs2 ds.
r
Proof. As mentioned earlier, the covariation hX, Xit is equal to the quadratic variation hXit . So, this theorem follows from the more general case, Theorem 3.11. Theorem 3.14. (Stochastic Product Rule) Suppose Xt , Yt satisfy dXt = At dt +
Ct dBt and dYt = Rt dt + St dBt . Then,
d(Xt Yt ) = Xt dYt + Yt dXt + d hX, Y it .
Proof. Let f (x, y) = xy and apply Theorem 3.9.
4. The Black-Scholes Model
One application of stochastic calculus is the Black-Scholes Model, a tool for the
pricing of options. An option is a contract that gives the person a right to either
buy or sell an asset (such as a stock) at a predetermined price (called the strike
price) at or before a future date (called the expiration date). A call option gives
the holder the right to buy the asset while a put option gives the holder the right
to sell the asset. A European option can only be exercised at the expiration date,
while an American option can be exercised at or before the expiration date.
Example 4.1. Consider a European call option on a stock with strike price K and
expiration date T . We denote the price of the stock at time t as St . If at time
T the stock price is higher than the strike price, the holder of the option makes a
profit equal to the difference from exercising the option and immediately selling for
ST . Otherwise, the option will not be exercised. Therefore, the value of the option
at time T is
F (ST ) = (ST − K)+ = max(0, ST − K).
Similarly, the value of a European put option would be
F (ST ) = (K − ST )+ = max(0, K − ST ).
An arbitrage is a strategy that allows an investor to have a positive probability
of making money without a risk of losing money. Options are generally priced to
prevent arbitrage, and this arbitrage-free price is not necessarily the same as the
price based on expected value. The following example shows that pricing an option
based on its expected value may lead to arbitrage opportunities.
Example 4.2. As a simple example, consider a two-period model. Suppose there
is a stock with price $200 at time 0, and after one time period, the stock will either
be worth $170 or $230, with equal probability. Suppose there is also a risk-free
bond with interest rate 10%.
Let us consider pricing the call option at its expected value. The expected value
of a call option with strike price $200 and expiration date 1 would be 0.5($30) +
16
BRIAN CHEN
0.5($0) = $15. So, if we were to price the option based on its expected value, we’d
set the price at $13.64, discounting the future value of $15 by the available 10%
interest rate (the $13.64 could be invested in a bond to yield $15).
Let us suppose the option is priced at $13.64. Then, we could buy 2 call options
and short sell 1 share of stock. At time 0, we spend $27.28 on options and make
$200 on the stock. The difference of $172.72 is then invested in bonds.
Now suppose that by time 1, the stock has risen to $230. The option would be
exercised, so we’d make $60 off the option. However, because we sold the stock at
time period 0, we now lose $230 off of that sale. The money we have invested in
the bond is now worth $189.99 as a result of the 10% interest rate.
If the stock drops to $170, then we would not exercise the option and therefore
make no money. As in the previous case, because we sold the stock earlier, we now
lose $170. As a result of the interest, the bond is now worth $189.99. In both of
these cases, we get a profit of $19.99.
These results are summarized in the table below:
Time 0 Time 1, S1 = 230 Time 1, S1 = 170
Buying Call Options -$27.28
$60
$0
Selling Stock
$200
-$230
-$170
$172.72
$189.99
$189.99
Bond
Profit
$19.99
$19.99
In this example, there are arbitrage opportunities as there exists a strategy that
yields a non-zero probability of making money but zero probability of losing money.
Thus, pricing an option at its expected value may lead to arbitrage opportunities.
To develop the arbitrage-free price of an option for a stock, we will want the
price of the option Ft to equal the value of a portfolio Vt that is maintained in
order to have a value of F (ST ) at time T . Let us assume that the stock price St
satisfies the stochastic differential equation
(4.3)
dSt = St (m(t, St )dt + σ(t, St )dBt )
and that there is a risk-free bond whose value Rt satisfies
(4.4)
dRt = r(t, St )Rt dt.
The value of the hedging portfolio Vt at time t is given by
(4.5)
Vt = aSt + bRt ,
where a and b are the number of units of stocks and bonds held at time t, respectively. We assume that the portfolio is self-financing, that is, the change in value of
the portfolio comes only from changes in the value of the assets. This is given by
(4.6)
dVt = at dSt + bt dRt .
Using (4.3)-(4.6), we get:
(4.7)
dVt = at St m(t, St )dt + at St σ(t, St )dBt + bt Rt r(t, St )dt
= [at St m(t, St ) + (Vt − aSt )r(t, St )]dt + at St σ(t, St )dBt
We let f (t, St ) be the desired price of the option at time t, assuming the stock
price at that time is St . We set the price of the option equal to the value of the
STOCHASTIC CALCULUS AND AN APPLICATION TO THE BLACK-SCHOLES MODEL 17
hedging portfolio, so f (t, St ) = Vt . Taking the derivative and using Theorem 3.10
along with (4.3), we get
(4.8)
1
dVt =[δx f (t, St )St m(t, St ) + δt f (t, St ) + St2 (σ(t, St ))2 δxx f (t, St )]dt
2
+ δx f (t, St )σ(t, St )St dBt .
By equating (4.7) and (4.8), we get:
(4.9)
δx f (t, St ) = at
and
1
δt f (t, St ) + St2 (σ(t, St ))2 δxx f (t, St ) = (Vt − at St )r(t, St ).
2
Plugging (4.9) into (4.10) and re-arranging gets us the Black-Scholes equation,
a stochastic differential equation for the arbitrage-free price of an option:
1
δt f (t, St ) = r(t, St )f (t, St ) − r(t, St )St δx f (t, St ) − St2 (σ(t, St ))2 δxx f (t, St ).
2
(4.10)
5. The Girsanov Theorem
In the following section, we will deal with changing probability measures. We
will state the Radon-Nikodym Theorem and the Girsanov Theorem. An application
of the Girsanov Theorem to the Black-Scholes Model will be presented in the next
section.
Definition 5.1. Take two measures µ and v on (Ω, F). We say v is absolutely
continuous with respect to µ if for all A ∈ F, if µ(A) = 0, then v(A) = 0. This is
denoted as v << µ. Two measures µ and v are said to be equivalent or mutually
absolutely continuous if for every set A ∈ F, µ(A) = 0 if and only if v(A) = 0.
We introduce the Radon-Nikodym Theorem, which demonstrates the existence
of a way to move from one measure to one that is absolutely continuous to it.
Theorem 5.2. (Radon-Nikodym) Let µ and v be two measures on (Ω, F) such that
v << µ. Suppose furthermore that there are countable sets An such that Ω = ∪∞
n An
and µ(An ) is finite for all n. Then, there exists a function f such that the for all
E ∈ Ω,
Z
v(E) =
f dµ.
E
The function f is called the Radon-Nikodym derivative and is denoted by
dv
dµ .
A proof of the Radon-Nikodym Theorem can be found in [3].
Now suppose P and Q are probability spaces such that Q << P . Then, the
Radon-Nikodym Theorem tells us that if P is σ finite, then Q(A) = EP [ dQ
dP 1A ],
where EP denotes the expectation under probability measure P and 1A is the
indicator function (i.e., 1A = 1 if the event A occurs and 0 otherwise).
Theorem 5.3. (Girsanov Theorem I) Let Mt be a non-negative martingale satisfying dMt = At Mt dBt , M0 = 1. We define a probability measure
R t Q such that for
Ft measurable events V , Q(V ) = E[Mt 1V ]. Then, Wt = Bt − 0 As ds is a standard
Brownian motion under probability measure Q.
18
BRIAN CHEN
Remark 5.4. Note that the conclusion to Theorem 5.3 is equivalent to the statement
that dBt = dWt + At dt, where Wt is a standard Brownian motion under Q.
The solution to the stochastic differential equation dMt = At Mt dBt , M0 = 1 is
Z t
Z
1 t 2
As dBs −
Mt = exp
A ds .
(5.5)
2 0 s
0
Rt
Rt
To see this, we let Yt = 0 As dBs − 21 0 A2s ds. Thus, Yt satisfies dYt = At dBt −
1 2
2
2 At dt. Using 3.13, hY it = At . By applying Theorem 3.10, we see that dMt =
At Mt dBt .
Before we begin the proof of Theorem 5.3, several facts must be established.
First, we claim that for a random variable X, the characteristic function φX (λ) =
E[eiλX ] determines the distribution of the random variable. We also claim that the
characteristic function of a normal random variable with mean µ and variance σ 2
is given by φ(λ) = exp{iµλ − 12 σ 2 λ2 }.
Next, we will want to show that if Mt is a continuous martingale with M0 = 0
and hM it = t, then Mt is a Brownian motion. We first prove the following lemma:
Lemma 5.6. Let Mt be a continuous martingale such that M0 = 0 and hM it = t.
Then, for all λ ∈ R,
2
E[eiλMt ] = e−λ
t/2
.
Proof. Let f (x) = eiλx . We proceed similarly as in the proof of Theorem 3.6:
n−1
X
f (Mt ) − f (M0 ) =
f (M(i+1)t/n ) − f (Mit/n )
i=0
We denote ∆Mi to be M(i+1)t/n − Mit/n . By Taylor’s Theorem,
1
f (M(i+1)t/n ) − f (Mit/n ) = f 0 (Mit/n )∆Mi + f 00 (Mit/n )(∆Mi )2 + o((∆Mi )2 ),
2
where lim∆Mi →0
o((∆Mi )2 )
∆Mi
= 0. Then
n−1
X
1
f 0 (Mit/n )∆Mi + f 00 (Mit/n )(∆Mi )2 + o((∆Mi )2 )
n→∞
2
i=0
Z t
Z t
1 00
=
f 0 (Ms )dMs +
f (Ms )ds.
0
0 2
f (Mt ) − f (M0 ) = lim
The last step is obtained following the same argument as in the proof of Theorem
3.6.
Then,
R t by following the proof of the Martingale Property in Theorem 3.3, we see
that 0 f 0 (Ms )dMs is a martingale.
Therefore, by taking the expectation,
Z t
Z t
λ2
1
00
f (Ms )ds = − E
f (Ms )ds ,
E[f (Mt ) − f (M0 )] = E
2
2
0
0
and similarly, for r < t,
1
E[f (Mt ) − f (Mr )] = E
2
Z
r
t
Z t
λ2
f (Ms )ds = − E
f (Ms )ds
2
r
00
STOCHASTIC CALCULUS AND AN APPLICATION TO THE BLACK-SCHOLES MODEL 19
Because |f (Ms )| is bounded, we can then apply the Dominated Convergence Theorem to get that
Z
λ2 t
(5.7)
E[f (Mt ) − f (Mr )] = −
E[f (Ms )]ds.
2 r
2
We let G(t) = E[f (Mt )]. Then, from (5.7), G0 (t) = − λ2 G(t) and G(0) = 1. The
2
solution to this is given by G(t) = e−λ t/2 .
Theorem 5.8. If Mt is a continuous martingale adapted to Ft , M0 = 0, and
hM it = t, then Mt is a standard Brownian motion.
Proof. We are given that Mt is continuous and that M0 = 0. Therefore, we just
need to check that Mt has independent normal increments. We can show this by
proving that for s < t, the conditional distribution of Mt − Ms given Fs is normal
with mean 0 and variance t − s.
By Lemma 5.6, the characteristic function of Mt − Ms is
E[eiλ(Mt −Ms ) |Fs ] = e−λ
2
(t−s)/2
.
Since this is also the characteristic function of a N (0, t − s) random variable, the
conditional distribution of Mt − Ms given Fs is normal with mean 0 and variance
t − s.
Using Theorem 5.8, we can try to prove Theorem 5.3. We begin a partial proof
here:
Proof of Theorem 5.3. We begin the proof by showing that Wt is a Q-martingale.
To prove this, we need to show that EQ (Wt |Fs ) = Ws . Take V to be any Fs measurable event and multiply both sides of the equation by 1V . Then, we have
EQ [1V Wt |Fs ] = 1V Ws .
We then take the expectation of both sides and apply the Tower Property [Prop
1.22] to get that
EQ [1V Wt ] = EQ [1V Ws ].
This is then equivalent to
EP [1V Ws Ms ] = EP [1V Wt Mt ].
Proceeding in the opposite direction, we apply the Tower Property:
EP [EP (1V Ws Ms |Ft )] = EP [EP (1V Wt Mt |Ft )]
This simplifies to
EP [1V Ws Ms ] = EP [1V Wt Mt ],
and since this is true for all Ft -measurable events V , this implies that
EP [Ws Ms ] = EP [Wt Mt ].
Therefore, to prove that Wt is a Q-martingale, it suffices to show that Wt Mt is a
P -martingale.
We note that dWt = dBt − At dt and dMt = At Mt dBt . We let Zt = Wt Mt and
apply Theorem 3.14 to get that
dZt = (Wt At + 1)Mt dBt .
20
BRIAN CHEN
Therefore, Zt is a local P -martingale. It is in fact a P -martingale; however, it is
rather difficult to prove this.
Proving Zt is a P -martingale then implies that Wt is a Q-martingale. Furthermore, hW it = t and Wt is continuous with probability one under probability
measure P (and therefore under Q as well because P and Q are equivalent measures). Then, by Theorem 5.8, Wt is a Brownian motion under probability measure
Q.
As mentioned before, the solution to Mt = At Mt dBt is not necessarily a martingale. Therefore, it will be useful to introduce a version of the Girsanov Theorem
that applies to Mt that are not martingales. Recall that solutions to Mt = At Mt dBt
are local martingales, so we now introduce a version of the Girsanov Theorem that
applies for local martingales. To do so, we will introduce a stopping time T .
Theorem 5.9. (Girsanov Theorem II) Suppose Mt = eYt satisfies dMt = At Mt dBt ,
M0 = 1. We define a probability measure Q such that for Ft -measurable events V ,
Q(V ) = E[Mt 1V ]. Let Tn = inf{t : Mt + hY it = n} and let T = limn→∞ Tn . Then,
Rt
Wt = Bt − 0 As ds for t < T is a standard Brownian motion under probability
measure Q.
Furthermore, if any of the following hold,
(1) Q(hY it < ∞) = 1
(2) EP [Mt ] = 1
hY i
(3) EP [exp( 2 t )] < ∞,
then for 0 ≤ s ≤ t, Ms is a martingale.
Proof. Let us first consider the case where there exists a K such that Mt < K and
hY it < K, with probability one.
As in the proof of Theorem 5.3, we let Zt = Wt Mt . Proving that Zt is a P martingale then implies that Wt is a Q-martingale. If this is true, since hW it = t
and Wt is continuous, Wt is a Brownian motion under Q by Theorem 5.8. Therefore,
we will prove that Zt is a P -martingale. As shown before, Zt satisfies
dZt = (Wt At + 1)Mt dBt ,
Rt
so Zt is a local martingale. Then, because Mt < K and hY it = 0 A2s ds < K,
Z t
Z t
2
2
2
E[(Ws As + 1) Ms ]ds < K
(5.10)
E[2(Ws2 A2s + 1)]ds.
0
Rt
A2s ds
0
Clearly, if
some M . Then,
0
< K, then
E[Ws2 A2s ] = E[(Bs −
Rt
0
Z
As ds is also bounded, so let |
Rt
0
As ds| < M for
s
Ar dr)2 A2s ] < E[(Bs2 + 2Bs M + M 2 )A2s ]
0
Therefore, continuing from (5.10),
Z t
Z t
E[(Ws As + 1)2 Ms2 ]ds < 2K 2
E[(Bs2 + 2Bs M + M 2 )A2s + 1]ds
0
0
2
< 2K (tM + M 3 + t) < ∞.
If Mt and hY it are not both bounded, then we take Tn = inf{t : Mt + hY it = n}
and let T = limn→∞ . Then Wt is a Q-martingale for t < T .
STOCHASTIC CALCULUS AND AN APPLICATION TO THE BLACK-SCHOLES MODEL 21
The proof that any of the three conditions imply that Mt is a martingale can be
found in [1].
Example 5.11. Suppose Xt satisfies dXt = Xt [m(t, Xt )dt + σ(t, Xt )dBt ], where
σ(t, Xt ) > 0 for all x and t and Bt is the standard Brownian motion under probability measure P . We want to find a probability measure Q equivalent to P such
that Xt is a martingale.
To do so, take
(Z
2 )
Z t
m(s, Xs )
1 t m(s, Xs )
−
Mt = exp
dBs −
ds ,
σ(s, Xs )
2 0
σ(s, Xs )
0
t)
which satisfies dMt = −m(t,X
σ(t,Xt ) Mt dBt , M0 = 1, by (5.5). We take Q to be the
probability measure defined such that for all Ft -measurable events V , Q(V ) =
E(1V Mt ), where the expecation is still under probability measure P . Because Mt
is strictly positive, it is easy to see that Q(V ) = 0 if and only if P (V ) = 0, so P
and Q are equivalent.
Then, the Girsanov Theorem [Theorem 5.3] states that
(5.12)
dBt = −
m(t, Xt )
dt + dWt ,
σ(t, Xt )
where Wt is the standard Brownian motion under probability measure Q.
Because dXt = Xt [m(t, Xt )dt + σ(t, Xt )dBt ],
(5.13)
dBt =
−m(t, Xt )
dXt
dt +
.
σ(t, Xt )
Xt σ(t, Xt )
From (5.12) and (5.13), we see that Xt satisfies
dXt = Xt σ(t, Xt )dWt .
Note that Xt is a martingale by Theorem 3.3.
In the next section, we will use the fact that a continuous non-negative local
martingale is a supermartingale. We present it as a theorem:
Theorem 5.14. Continuous non-negative local martingales are supermartingales.
Proof. Let Mt be a continuous non-negative local martingale and let τn be an
increasing sequence of stopping times. Then, Mt∧τn is a martingale. Let s < t and
let V be a Fs -measurable event. Define Vk = V ∩ {τk > s}.
Then, because Mt∧τn is a martingale,
E[1Vk Mt∧τn ] = E[1Vk Ms∧τn ] = E[1Vk Ms ].
We then use Fatou’s Lemma to show that
E[1Vk Ms ] = lim inf E[1Vk Mt∧τn ] ≥ E[1Vk Mt ].
n→∞
We let k → ∞ and use the Monotone Convergence Theorem to get that E[1V Mt ] ≤
E[1V Ms ]. Using the Tower Property, E[E[1V Mt |Fs ]] ≤ E[1V Ms |Fs ]. Since V is
a Fs -measurable event, E[1V E[Mt |Fs ]] ≤ E[1V Ms ]. Since this holds for all Fs measurable events V , E[Mt |Fs ] ≤ Ms . Therefore, Mt is a supermartingale.
22
BRIAN CHEN
6. The Black-Scholes Model for Pricing Other Claims
In this section, we seek to extend the theory developed in Section 4 when dealing
with the Black-Scholes model. First, we define a claim.
Definition 6.1. A claim V is a FT measurable random variable, where T is a
future time.
The profit from a European call option V = F (ST ) = (K − ST )+ = max(0, K −
RT
ST ) is a claim. As another example, V = 0 St dt would also be a claim.
As before, we assume that the stock price St satisfies
dSt = St [mt dt + σt dBt ],
where mt and σt are adapted to Ft . We also assume there exists a risk-free bond
with value Rt that satisfies
dRt = rt Rt dt,
where rt is also adapted to Ft . We will want a portfolio consisting of at units of
stock and bt units of the bond. Then, the value of the portfolio at time t is given
by Vt = at St + bt Rt . Because the portfolio should replicate the claim, at time T ,
we need VT = V . As before, we assume the portfolio is self-financing, so we assume
that
(6.2)
dVt = at dSt + bt dRt .
St
Vt
We let S̃ = R
and Ṽ = R
be the discounted stock price and discounted value,
t
t
respectively. Next, we let Q be an equivalent probability measure such that S̃ is a
martingale. Then, under some assumptions, we can find the arbitrage-free price of
the claim.
First, we need to define a contingent claim.
Definition 6.3. A contingent claim is a claim V that is non-negative such that
EQ [Ṽ 2 ] < ∞.
Intuitively, a contingent claim is a non-negative claim that, with probability
one, remains at a finite value. A call option is an example of a contingent claim,
assuming that the stock price remains finite, with probability one.
Now, we state the following theorem for pricing contingent claims.
Theorem 6.4. Let V be a contingent claim. We let Q be the probability measure
such that for all Ft measurable events V , Q(V ) = E[1V Mt ], where Mt satisfies
t
dMt = rt −m
σt Mt dBt , M0 = 1 and the expectation is taken under probability measure
P . We assume that the stock price St satisfies dSt = St [mt dt + σt dt], where mt and
σt are adapted to Ft and that there exists a risk-free bond with value Rt satisfying
dRt = rt Rt dt, where mt , σt , and rt are adapted to Ft . Furthermore, suppose the
following assumptions are true:
t
(1) Mt satisfying dMt = rt −m
σt Mt dBt , M0 = 1 is a martingale.
St
(2) The discounted stock prices S̃t = R
is a Q-martingale.
t
Vt
(3) There exists At such that Ṽt = Rt satisfies dṼt = At dWt , where Wt =
Rt
R t As
Rt
At
s
Bt − 0 rs −m
σs ds, and that Vt = 0 σ S̃ dSs + 0 (Ṽt − σt )dRs is well-defined.
s
s
STOCHASTIC CALCULUS AND AN APPLICATION TO THE BLACK-SCHOLES MODEL 23
Then, the arbitrage-free price of the claim is given by
Vt = Rt EQ (ṼT |Ft )
Proof. We are given that Mt satisfies dMt =
know that
Z
Mt = exp
0
t
rt −mt
σt Mt dBt , M0
= 1. From (5.5), we
rt − mt
1 rt − mt 2
dBs − (
) ds ,
σt
2
σt
and therefore Mt is strictly positive. Because Q(V ) = EP [1V Mt ] and Mt is always
positive, Q(V ) = 0 if and only if P (V ) = 0. Therefore, Q and V are equivalent
St
,
probability measures. Also, by assumption 1, Mt is a martingale. Because S̃t = R
t
using Theorem 3.14, we perform the following calculation:
D
E
dSt = Rt dS̃t + S̃t dRt + R, S̃
t
(6.5)
1
=⇒ dS̃t =
(dSt − S̃t dRt )
Rt
1
[St (mt dt + σt dBt ) − S̃t rt Rt dt]
=
Rt
= S̃t [(mt − rt )dt + σt dBt ]
Mt is a martingale, we can apply Theorem 5.9. Thus, Wt = Bt −
R t Because
rs −ms
ds
is a standard Brownian motion under probability measure Q. Then,
σt
0
t
dWt = dBt − rt −m
σt dt. Multiplying by σt S̃t and using (6.5) gives us that
(6.6)
σt S̃t dWt = σt S̃t dBt + S̃t (mt − rt )dt = dS̃t .
As we have shown earlier, this implies that S̃t is a local Q-martingale. Furthermore, by Assumption 2, S̃t is a Q-martingale.
Let Ṽt = EQ [Ṽ |Ft ]. Note that Ṽt is then a martingale under Q, as for t < s,
EQ [Ṽs |Ft ] = EQ [EQ [Ṽ |Fs ]|Ft ] = EQ [Ṽ |Ft ] = Ṽt .
Using
Assumption 3, there exists an adaptedR process At such that ṼtR = Ṽ0 +
Rt
t
t
2
A
dW
s
s , or equivalently, dṼt = At dWt . If 0 EQ [As ]ds < ∞, then 0 As dWs
0
is a square-integrable martingale, by Theorem 3.4. Note that Vt is also a squareintegrable martingale. In fact, it can be proven that Assumption 3 is always true:
The Martingale Representation Theorem states that any continuous martingale can
be expressed as a stochastic integral (See [5]). Furthermore, the theorem does not
state what At is. For our purposes, we will need to know At to find the replicating
portfolio.
24
BRIAN CHEN
With this assumption, we can make the following computation:
D
E
dVt = Rt dṼt + Ṽt dRt + R, Ṽ
t
= Rt At dWt + Ṽt dRt (by Assumption 3)
= Rt At
dS̃t
+ Ṽt dRt (by (6.6))
σt S̃t
At
[dSt − S̃t dRt ] + Ṽt dRt (By using Theorem 3.14 on St = Rt S̃t )
σt S̃t
At
At
=
dSt + (Ṽt −
)dRt
σt
σt S̃t
=
t
Comparing this to (6.2), we let at = σAS̃t and bt = Ṽt − A
σt . Note that then
t t
at St + bt Rt = Vt . Therefore, we have found a self-financing portfolio that replicates
the contingent claim V , whose value at time t is given by Vt = Rt EQ (Ṽt |Ft ).
We will now prove that Vt gives the minimum value at time t of a self-financing
portfolio that replicates V .
Consider a different, self-financing hedging portfolio with value Vt∗ at time t. At
time T , we want the portfolio to have a value of at least VT , so VT∗ ≥ VT . As before,
V∗
St
be the discounted value and discounted stock prices,
we let Ṽt∗ = Rtt and S̃t = R
t
respectively. The portfolio consists of a∗t units of stock and b∗t units of bond. Thus,
dVt∗ = a∗t dSt + b∗t dRt = a∗t (S̃t dRt + Rt dS̃t ) + b∗t dRt = a∗t Rt dS̃t + (a∗t S̃t + b∗t )dRt .
Using the stochastic product rule,
dVt∗ = Ṽt∗ dRt + Rt dṼt∗ = (a∗t S̃t + b∗t )dRt + Rt dṼt∗ .
Then, equating coefficients gets us that dṼt∗ = a∗t dS̃t = at σt S̃t dWt . Therefore, Ṽt∗
is a local Q-martingale. By Theorem 5.14, Ṽt∗ is a supermartingale, so
EQ [ṼT∗ |Ft ] ≤ Ṽt∗ ,
and since ṼT∗ ≥ ṼT ,
Ṽt = EQ [ṼT |Ft ] ≤ EQ [ṼT∗ |Ft ] ≤ Ṽt∗ .
When dealing with specific cases, the conditions listed in Theorem 5.9 can be
used to check whether Assumptions 1 and 2 are met.
Acknowledgments. I would like to thank my mentor, Bowei Zheng, for all his
help throughout the REU with this paper. I would also like to thank Peter May for
running the program as well as the numerous instructors and teaching assistants
who lectured during the REU. I have learned a lot over these past few weeks and
it would not have been possible without all of your help.
References
[1] Gregory Lawler. Unpublished Book Draft.
[2] Fima C. Klebaner. Introduction to Stochastic Calculus with Applications. Imperial College
Press. 2012.
[3] Rick Durrett. Probability: Theory and Examples Cambridge University Press. 2010.
[4] Sheldon Ross. A First Course in Probability. Pearson. 2009.
STOCHASTIC CALCULUS AND AN APPLICATION TO THE BLACK-SCHOLES MODEL 25
[5] Bernt Oksendal. Stochastic Differential Equations:
Springer. 1998.
An Introduction With Applications.