Chapter 5 An Introduction to
Stochastic Processes and
Applications
5.1.1 Stochastic Processes: A Brief
Introduction
 A stochastic process is a sequence of random variables
Xt defined on a common probability space (,,P) and
indexed by time t
 The values of Xt() define the sample path of the
process leading to state . The terms X(,t) and
Xt() are synonymous.
 A discrete time process is defined for a finite set of time
periods; a continuous time process that is defined over
an infinite number of periods.
 The state space is the set of values in process {Xt}:
X = {Xt() for some    and some t}
A Brief Introduction: Continued
 The set of possible events associated with the random variable Xt is
called its σ-algebra ℱ𝑡 .
 A sequence of filtrations ℱt for a σ-algebra ℱ is a sequence of σalgebras with the property:
ℱ0  ℱ1  ℱ2  ⋯  ℱ
for discrete processes.
 Similarly, continuous processes will have the following property:
ℱ𝑠 ℱ𝑡  ℱ
𝑠<𝑡
 for all 0  s  t
 Filtrations can be characterized as the information set required for
valuation at any time or state.
Illustration: Filtrations in a Two timeperiod Random Walk
 Suppose a stock has an initial price of X0 at time 0, which, at time 1, will
either increase or decrease by 1. Independently, the same potential price
changes will occur at time 2. All of the possible distinct price 2-period
change outcomes for the stock to time 2 are (1,1), (1.-1), (-1,1), and (-1,-1).
 The sample space for this process is:
Ω = {(1,1), (1,-1), (-1,1), (-1,-1)}.
 ℱ 0 = {, },
 ℱ 1 = {, {(1,1), (1,-1)}, {(-1,1), (-1,-1)}, },
ℱ2 =
{∅
1,1
1, −1
−1,1
−1, −1
1,1 , 1, −1
1,1 , −1,1
1,1 , −1, −1
1, −1 , −1,1
1, −1 , −1, −1
−1,1 , −1, −1
1,1 , 1, −1 , −1,1
1,1 , 1, −1 , −1, −1
1,1 , −1,1 , −1, −1
1, −1 , −1,1 , −1, −1
Ω}
5.1.2 Random Walks and Martingales
 A discrete Markov Process or discrete random walk is a
non-continuous stochastic process whose state at time t
depends only on its immediately prior state at time t-1
and not on the remainder of its history:
𝑃 𝑋𝑡 𝑋𝑡−1 , 𝑋𝑡−2 , … , 𝑋0 = 𝑃 𝑋𝑡 𝑋𝑡−1 .
 We define a continuous Markov Process by the
condition:
𝑃 𝑋𝑡 𝑋0 , 𝑋1 , … , 𝑋𝑠 = 𝑃 𝑋𝑡 𝑋𝑠 ∀𝑠 < 𝑡
Martingales
 A martingale process is a stochastic process Xt with the
properties:
1. 𝐸 𝑋𝑡 𝑋0 , 𝑋1 , … , 𝑋𝑡−1 = 𝑋𝑡−1 (discrete case)
E[Xt | Xi, 0 ≤ i ≤ s] = Xs for s < t (continuous case)
2. 𝐸 𝑋𝑡 < ∞
for all t =1, 2, 3, … in the discrete case and for all t > 0 in the
continuous case.
 A martingale is a process whose future variations have no
specific direction based on the process history (X0, X1,…, Xt1).
 A martingale is said to be a "fair game" and will not exhibit
consistent trends either up or down.
Submartingales and Super Martingales
 A submartingale with respect to probability measure ℙ is a
stochastic process Xt in which the first property to be a
martingale is replaced with:
Eℙ[XtX0, X1, X2, ..., Xt-1]  Xt-1
in the discrete case and the analogous requirement in the
continuous case.
 A submartingale will tend either to trend upward over time or
is a martingale. The definition of a supermartingale replaces
the greater than or equal inequality above with a less than or
equal inequality.
 A supermartingale will tend to trend downward over time or is
a martingale.
5.1.3 Equivalent Probabilities and
Equivalent Martingale Measures
 Prices of assets need not be functions of physical
probabilities or expected values based on physical
probabilities.
 Risk neutral probabilities arise from the strongest of
financial forces - arbitrage.
 Risk-neutral probabilities do not strictly measure
opinions of likelihood in the way that physical
probabilities do.
 However, they are essential to arbitrage-free pricing,
and they can still be treated as probabilities for
relative pricing purposes.
Numeraires
 Thus far we have expressed all asset values in terms of
dollars and returns relative to dollars. Thus, the dollar
served as the numeraire.
 However, we can just as easily express values in terms of
other currencies, other securities such as pure securities
as we did in Chapter 3, or riskless bonds as we will
shortly.
 Flexibility in selecting the numeraire and an associated
equivalent martingale measure affords us the ability of
being able to use valuation techniques that otherwise
would not be available or would be more complicated.
Equivalent Probability Measures
 A probability measure ℙ is a mapping from an event space such that






all events  have probabilities p[]  [0, 1].
An equivalent probability measure ℚ has the same null space as ℙ.
Two probability measures are said to be equivalent (ℙ ~ ℚ) if the set
of events that have probability 0 under measure ℙ is the same as
that set under the second measure ℚ .
This implies that ℚ << ℙ (ℚ is absolutely continuous with respect
to ℙ, which means that p() = 0  q() =0) and ℙ << ℚ (ℙ is
absolutely continuous with respect to ℚ; q() = 0  p() =0).
This implies that an equivalent probability measure is consistent
with respect to which outcomes are possible.
However, the actual non-zero probabilities assigned to events might
differ.
If ℚ << ℙ and ℙ << ℚ, then ℚ ~ ℙ.
Equivalent Martingale Measures
 A probability measure ℚ is an equivalent martingale measure (risk-neutral




measure) to ℙ in a complete market if ℚ and ℙ are equivalent probability
measures, and the price of every security in the market (using the riskless
bond as the numeraire) is a martingale with respect to the probability
measure ℚ.
In a complete market in which there are no arbitrage opportunities, there
will always exist a unique equivalent martingale measure. This measure can
be used to obtain risk-neutral pricing for every security in that market.
For the finite outcome case, we will define a complete market to be one in
which every security in the market can be expressed as a portfolio of a finite
number of pure securities.
Since the equivalent martingale measure ℚ is unique, one is free to choose
any set of n linearly independent securities that form a basis for the market
and that have already been priced in the market in order to construct the
measure ℚ.
A market has a unique equivalent martingale measure if and only if it is
both complete and arbitrage-free.
The Riskless Bond as the Numeraire
 To use the riskless bond as the numeraire is
equivalent to discounting the security by the riskless
rate.
 Since the discounted security price is a martingale,
its expected value has the same return as the return
on the riskless bond. If this were not the case, there
would be an opportunity for arbitrage.
Pricing with Submartingales
 We generally expect that securities will price as
submartingales with respect to money because of the
time value of money as investors demand compensation
for giving up alternative uses of their initial investments:
𝐸ℙ 𝑆𝑡+∆𝑡 > 𝑆𝑡
 If there exists an equivalent probability measure ℚ such
that discounted security prices can be converted into
martingales, this will simplify valuation:
𝑆𝑡+∆𝑡
𝐸ℚ
= 𝑆𝑡
∆𝑡
1+𝑟
5.2 Binomial Processes: Characteristics
and Modeling
 A Bernoulli trial is a single random experiment with
two possible outcomes (e.g., 0, 1, or u or d) whose
outcomes depend on a probability.
 We will define a binomial process to be any
stochastic process based on a series of n statistically
independent Bernoulli (0,1) trials, all with the same
outcome probability.
Binomial Processes
 Consider the Markov Process: Xt = X0 + aZ1 + aZ2
+…+ aZt with




Zt() = 1 if  = u with probability p
Zt() = -1 if  = d with probability 1-p
a is a positive constant
Random variables Zt are pairwise independent.
 This process is like a random walk starting at X0,
taking a step of length a right with probability p and
of length a left with probability 1 – p.
Binomial Processes (Continued)
 Under probability measure ℙ, the expected value of
Xt at time t is:
𝐸ℙ 𝑋𝑡 = 𝐸ℙ 𝑋0 + 𝑎𝑍1 + 𝑎𝑍2 + ⋯ + 𝑎𝑍𝑡
= 𝑋0 + 𝑎𝐸ℙ 𝑍1 + 𝑎𝐸ℙ 𝑍2 + ⋯ + 𝑎𝐸ℙ 𝑍𝑡
= 𝑋0 + 𝑎 2𝑝 − 1 𝑡
 Observe that Xt is a martingale when p = ½, a
submartingale when p > ½, and a supermartingale
when p < ½.
 The process described here does not involve
multiplicative factors or compounded returns.
Binomial Returns Process
 Applies to returns rather than to prices
 Allows for compounding of returns
 Defined as follows:
 Implies that:
𝑆𝑡
= (1 + 𝑎𝑍𝑡 )
𝑆𝑡−1
𝑆𝑡 = 1 + 𝑎𝑍1 1 + 𝑎𝑍2 ⋯ 1 + 𝑎𝑍𝑡 𝑆0
 The probability that exactly k upjumps occurred from
time zero to time t equals:
𝑡 𝑘
𝑝 (1 − 𝑝)𝑡−𝑘 .
𝑘
Expected Value of the Binomial Returns
Process
 Thus, the expected value of St given S0 is:
𝑡
𝐸ℙ 𝑆𝑡 =
(1 + 𝑎)𝑘 (1 − 𝑎)𝑡−𝑘 𝑆0
𝑘=0
𝑡
= 𝑆0
𝑘=0
𝑡 𝑘
𝑝 (1 − 𝑝)𝑡−𝑘
𝑘
𝑡
[𝑝(1 + 𝑎)]𝑘 [ 1 − 𝑝 1 − 𝑎 ]𝑡−𝑘
𝑘
 Which, from the Binomial Theorem, implies:
𝐸ℙ 𝑆𝑡 = 𝑆0 𝑝(1 + 𝑎) + (1 − 𝑝 (1 − 𝑎)]𝑡
= 𝑆0 [1 + 𝑎(2𝑝 − 1)]𝑡
 And with multiplicative jumps u and d:
𝐸ℙ 𝑆𝑡 |𝑆𝑠 = 𝑆𝑠 [𝑝𝑢 + (1 − 𝑝)𝑑]𝑡−𝑠
5.2.2
Illustration: Binomial Outcome
and Event Spaces
u,u
22.5
15
u,d
d,u
7.5
10
5
d,d
2.5
Time 0
Time 1
Time 2
Figure 1: Two Time Period Binomial Model - Current and Potential Stock Prices
The Sample Space
Ω = {𝑢𝑢, 𝑢𝑑, 𝑑𝑢, 𝑑𝑑}
ℱ 0 = {, }
ℱ 1 = {, {uu, ud}, {du,dd}, }
ℱ2 = {, uu , ud , du , dd , uu, ud , uu, du , uu, dd , ud, du , uu, dd , du, dd ,
uu, ud, du , 𝑢𝑢, 𝑢𝑑, 𝑑𝑑 , 𝑢𝑢, 𝑑𝑢, 𝑑𝑑 , 𝑢𝑑, 𝑑𝑢, 𝑑𝑑 , }.
Space Pure Securities and Market
Securities
 Next, we will discuss Arrow-Debreu (pure) security prices, one
associated with each starting and stopping node combination
in our 2-time period model (See Table 1 below). First, we will
list spot (time 0) prices for investments extending from time 0
to time 1. Let 0,1;u be the time zero price of a pure security
that is worth 1at time 1 if and only if an upjump occurs at time
1. The first subscript 0 refers to the initial time 0, the second
subscript 1 refers to the next time 1, and the third subscript u
refers to the fact that there is an upjump (at time 1). Let0,1;d
be the time zero price of a pure security that is worth 1at time
1 if and only if a downjump occurs at time 1. The third
subscript d in this case refers to the fact that there is a
downjump (at time 1). Subscripts before the semicolon refer
to time periods, and subscripts after the semicolon refer to the
occurrence of upjumps or downjumps.
Space Pure Securities and Market
Securities (Continued)
 The vector [1 0]T will denote the payoff of the pure
security identified with an upjump occurring at time
1. The vector [0 1]T will denote the payoff of the
pure security identified with a downjump occurring
at time 1. To purchase the stock at time 0 at a price of
10 and hold to it until time 1 means that at time 1 the
portfolio of the owner takes the form of the vector [15
5]T because it will pay 15 if an upjump occurs and
will pay 5 if a downjump occurs.
Valuing Pure Securities
 Since an investor is willing to pay 0,1;u at time 0 for a
payoff vector [1 0]T and pay 0,1;d at time 0 for a payoff
vector [0 1]T and since the value of the stock portfolio
[15 5]T is equal to 10 at time 0, then:
𝜓0,1;𝑢
15𝜓0,1;𝑢 + 5𝜓0,1;𝑑 = 15 5
= 10
𝜓0,1;𝑑
 Also, if an investor has the riskless portfolio [1 1]T, then
she is guaranteed to be paid 1 at time 1. As this is
equivalent to holding the bond that pays 1 at time 1, and
since the time 0 price of the portfolio [1 1]T is .8, then:
𝜓0,1;𝑢
𝜓0,1;𝑢 +𝜓0,1;𝑑 = 1 1
= .8
𝜓0,1;𝑑
Valuing Pure Securities (Continued)
 The two previous equations can be expressed by the single
matrix equation:
10
15 5 𝜓0,1;𝑢
=
.
𝜓
.8
1 1
0,1;𝑑
 Before we discuss the 2-period spot prices, we also have time 1
forward prices for pure securities that pay off in time 2. Let
1,2;u,u be the time 1 price of a pure security that pays 1 at time
2 if and only if upjumps occur at both times 1 and 2 (Outcome
u,u). The remaining forward prices 1,2;u,d, 1,2;d,u, and 1,2;d,d
are defined similarly. Note that two paths lead to a time 2
price of 7.5. The same procedure can used to calculate pure
security prices as we did going from time 0 to 1. These forward
pure security prices are calculated from the following:
Valuing Pure Securities (Continued)
𝜓1,2;𝑢,𝑢 = .6
22.5 7.5 𝜓1,2;𝑢,𝑢
15
=
;
𝜓1,2;𝑢,𝑑 = .2
.8
1
1 𝜓1,2;𝑢,𝑑
𝜓1,2;𝑑,𝑢 = .6
7.5 2.5 𝜓1,2;𝑑,𝑢
5
=
;
𝜓
𝜓1,2;𝑑,𝑑 = .2
.8
1
1
1,2;𝑑,𝑑
 We can calculate the 2-period pure security spot
prices from the 1-period spot and forward prices. Let
0,2;u,u be the time zero price of a pure security that
pays 1at time 2 if and only if there are consecutive
upjumps (denoted by the outcome u,u). Thus, the
time zero pure security price associated with time 2
outcome u,u is 0,2;u,u.
Valuing Pure Securities (Continued)
 The remaining 2-period spot prices 0,2;u,d, 0,2;d,u,
and 0,2;d,d are defined similarly. Internal pricing
consistency (no-arbitrage pricing) requires that
0,2;u,u = 0,1;u1,2;u,u = .6  .6 = .36 and 0,2;d,d =
0,1;d1,2;d,d= .2  .2 = .04. Also, no-arbitrage pricing
requires that 0,2;u,d = 0,1;u1,2;u,d = .6  .2 = .12 and
0,2;d,u = 0,1;d1,2;d,u= .2  .6 = .12. For the
recombining part of our lattice, the pure security
price for the set of outcomes {u,d, d,u} is 0,2;u,d +
0,2;d,u = .12+.12 = .24. These no-arbitrage
relationships are captured in the following pure
security prices matrix equation:
Valuing Pure Securities (Continued)
 These no-arbitrage relationships are captured in the
following pure security prices matrix equation:
0,2;𝑢,𝑢
1,2;𝑢,𝑢
1,2;𝑢,𝑑
(0,2;𝑢,𝑑 + 
)
=
0,2;𝑑,𝑢
0,2;𝑑,𝑑
0
0
0,1;𝑢
1,2;𝑑.𝑢
0,1;𝑑
1,2;𝑑,𝑑
0,2;𝑢,𝑢
.6 0
. 36
.
6
(0,2;𝑢,𝑑 + 
) = .2 .6
= . 24
0,2;𝑑,𝑢
.2
0,2;𝑑,𝑑
0 .2
. 04
Spot Market Bond Prices and Yields
 Spot market bond prices and yields are computed as
follows:
𝐵0,1 = 𝑑0,1 = 0,1;𝑢 + 0,1;𝑑 = .8;
1
𝑟0,1 =
− 1 = .25
0,1;𝑢 + 0,1;𝑑
𝐵0,2 = 𝑑0,2 = 0,2;𝑢,𝑢 + 0,2;𝑢,𝑑 + 0,2;𝑑,𝑢 + 0,2;𝑑,𝑑 = .64
𝑟0,2 =
1
− 1 = .25
0,2;𝑢,𝑢 + 0,2;𝑢,𝑑 + 0,2;𝑑,𝑢 + 0,2;𝑑,𝑑
Forward Market Bond Prices and Yields
 Forward market bond prices and yields are computed as
follows:
𝐵1,2;𝑢 = 𝑑1,2;𝑢 = 1,2;𝑢,𝑢 + 1,2;𝑢,𝑑 = .8 ;
1
𝑟1,2;𝑢 =
− 1 = .25
1,2;𝑢,𝑢 + 1,2;𝑢,𝑑
𝐵1,2;𝑑 = 𝑑1,2;𝑑 = 1,2;𝑑,𝑢 + 1,2;𝑑,𝑑 = .8;
1
𝑟1,2;𝑑 =
− 1 = .25
1,2;𝑑,𝑢 + 1,2;𝑑,𝑑
 Note that these prices and yields are contingent on time
1 outcomes.
The Equivalent Martingale Measure
 Physical probabilities matter only to he extent that
we can obtain their equivalent martingale measures.
 Risk neutral probabilities are computed as follows:
𝑡,𝑇;𝑖 𝑡,𝑇;𝑖
𝑞𝑡,𝑇;𝑖 =
=
𝑑𝑡,𝑇
𝐵𝑡,𝑇
𝑡,𝑇;𝑖
𝑞𝑡,𝑇;𝑖 = 𝑛
𝑗=1 𝑡,𝑇;𝑗
 For our example, q0,1;u= 0,1;u /B0,1=.6/.8=.75,
q1,2;u,d= 1,2;u,d /B1,2=.2/.8=.25, and
q0,2;d,d= 0,2;d,d /B0,2=.04/.64=.0625
Change of Numeraire and Martingales
 The numeraire is the unit in which values are expressed.
 We calculate risk neutral probabilities and the expected
value of the stock with respect to our numeraire (the
bond) as follows:
𝑞0,1;𝑢 = .75
15 5 𝑞0,1;𝑢
12.5
=
;
𝑞
𝑞0,1;𝑑 = .25
1 1 0,1;𝑑
1
Eℚ[S0,1|S0/B0] = 15𝑞0,1;𝑢 + 5𝑞0,1;𝑑 = 12.5 = S0/B0

Since the time-zero expected value of the stock at time 1 (given the time-zero stock value
in bond numeraire) under the probability measure ℚ equals the time-zero stock value (in
bond numeraire), then the stock price has the martingale property at time 1. We can
demonstrate similar martingale properties going from time 1 to time 2, and from time 0
to time 2. Thus, in this two time period process, the probability measure ℚ is an
equivalent martingale measure to ℙ.
Change of Numeraire and Martingales
(Continued)
 Also note that if we convert from the bond numeraire
back to the original currency (such as dollars), we get
the right values at time 0. The time-zero value of the
stock is 12.5 times that of the bond. The bond is
worth .8 dollars at time zero, therefore the stock has
a time zero value equal to 10 dollars:
𝐸ℚ 𝑆0,1
15𝑞0,1;𝑢 + 5𝑞0,1;𝑑
𝑆0 =
=
= 10
∆𝑡
(1 + 𝑟)
1 + .25
𝐸ℚ 𝐵0,1
1𝑞0,1;𝑢 + 1𝑞0,1;𝑑
𝐵0 =
=
= .8
∆𝑡
(1 + 𝑟)
1 + .25
Binomial Pricing, Change of Numeraire
and Martingales
 To summarize, we price the stock and the bond as
follows:
𝑆0,1;𝑢
1
𝑆0
𝑆0,1;𝑑 𝜓0,1;𝑢
= 𝐵
𝜓0,1;𝑑
1
0,1
Binomial Option Pricing
 The Binomial Option Pricing Model is based on the
assumption that the underlying stock price is a Bernoulli trial
in each period.
 We will work with the following:
X = Exercise price of the stock
S0 = Initial stock value
u = Multiplicative upward stock price movement
d = Multiplicative downward stock price movement
cu = MAX[0,uS0 - X]; Value of call if stock price increases
cd = MAX[0,dS0 - X]; Value of call if stock price decreases
α = Hedge ratio
r = Riskless return rate
One-Time Period Case
We value the one-period call as follows, based on risk neutral
probability q:
Eℚ[c1] = [𝑐𝑢 𝑞 + 𝑐𝑑 (1 − 𝑞 )] = 𝑐0 (1 + 𝑟)
Solving for c0 gives the price:
c0 =
𝑐𝑢 𝑞 +𝑐𝑑 (1−𝑞 )
1+𝑟
We value shares of the stock relative to the bond as follows:
Eℚ [S1 ] = 𝑢𝑆0 𝑞 + 𝑑𝑆0 1 − 𝑞 = 𝑆0 (1 + 𝑟),
which implies that:
𝑞 =
1+𝑟−𝑑
𝑢−𝑑
Multi-time Period Binomial Model
 We value the multi-period call as follows:
c0 =
𝑇!
𝑇
𝑗 (1−𝑞 )𝑇−𝑗 𝑀𝐴𝑋
𝑞
𝑗 =0 𝑗 ! 𝑇−𝑗 !
(1+𝑟)𝑇
𝑢 𝑗 𝑑 𝑇−𝑗 𝑆0 −𝑋,0
 The term a, defined as the minimum number of upjumps required for exercise, is
defined as follows, resulting in the call value following it:

  X

 ln 
S0 d T



a  INT  MAX

u

 ln  
d 


  
  
 ,0  1
 
 
 
j T j
T

T!
X T
T!
T j u d
T j 
j
j
c0  S 0 
q 1  q 

q 1  q  
T 
T 
1  r   1  r   j a j!(T  j )!
 j a j!(T  j )!

The Dynamic Hedge
 One Period Hedge:
 In a one-time period binomial framework where
there exists a riskless asset, we can hedge the call
against the stock such the resultant portfolio with
one call and  shares of stock produces the same
cash flow whether the stock increases or decreases:
cu  uS 0  cd  dS 0  (c0  S 0 )(1  r )
 Solving for the hedge ratio :
cu  c d

S 0 (d  u )
Dynamic Hedge Illustration: One Period
 S0 = 10
cu = 6
u = 1.5
cd = 0
d = .5
r = .25
X=9
1 + 𝑟 − 𝑑 1 + .25 − .5
𝑞 =
=
= .75
𝑢−𝑑
1.5 − .5
cu  c d
60


 .6
S 0 (d  u ) 10(.5  1.5)
c0 =
𝑐𝑢 𝑞 +𝑐𝑑 (1−𝑞 )
1+𝑟
=
6.75 + 0 (1−.75)
1+.25
= 3.6
Eℚ[c1] = [𝑐𝑢 𝑞 + 𝑐𝑑 (1 − 𝑞)] = 6 × .75 + 0(1 − .75) = 4.5
Two-Period Binomial Model
uuS0= 22.5, c2;u,u = 13.5
uu
uS0=15
u
ud
S0=10
du
d
dS0=5
dd
Time 0
udS0 = 7.5, c2;u,d = 0
duS0 = 7.5, c2;d,,u = 0
Time 1
ddS0 = 2.5, c2;d,d = 0
Time 2
Figure 2: Two Period Binomial Model with Option Values
Dynamic Hedge Illustration: Two Periods
0 
1,u
1,d 
𝐜𝟎 =
=
2
𝑗=0 𝑗!
𝑐2;𝑢𝑢 𝑞 2
1+𝑟 2
c1;u  c1;d
S 0 (d  u )

60
 .6
10(.5  1.5)
c2;u ,u  c2;u ,d
13.5  0


 .9
uS 0 (d  u) 15(.5  1.5)
c2;d ,u  c2,d ,d
dS 0 (d  u)

00
0
7.5(.5  1.5)
(no hedge)
2!
.75 𝑗 (1 − .75)2−𝑗 𝑀𝐴𝑋 1.5𝑗 .752−𝑗 10 − 9, 0
2−𝑗 !
(1 + .25)2
13.5.5625
=
= 4.86
1.5625
Brownian Motion and Itô Processes
 A process Zt is a standard Brownian motion process
if:




Changes in Zt over time are independent over disjoint intervals
of time; that is, COV(Zs-Zt, Zu-Zv,) = 0 when s > t > u > v.
Changes in Zt are normally distributed with E[Zs-Zt] = 0 and
E[(Zs-Zt)2] = s - t for s > t. Thus, Zs - Zt ~ N(0, s-t) with s > t.
Zt is a continuous function of t.
The process begins at zero, Z0 = 0.
Brownian Motion Processes
 Standard Brownian motion is a martingale because:
𝐸 𝑍𝑡 ℱ𝑠 = 𝐸 𝑍𝑡 − 𝑍𝑠 + 𝑍𝑠 ℱ𝑠
= 𝐸 𝑍𝑡 − 𝑍𝑠 ℱ𝑠 + 𝐸 𝑍𝑠 ℱ𝑠 = 𝑍𝑠
Since Zt ~ N(0, t),
𝐸 𝑍𝑡
=
1
∞
1
−2𝑡𝑧 2
|𝑧|𝑒
2𝜋𝑡 −∞
𝑑𝑧 =
=
1
∞
2
−2𝑡𝑧 2
𝑧𝑒
𝑑𝑧
2𝜋𝑡 0
∞
−2 𝑡 − 1 𝑧 2
2 𝑡
𝑒 2𝑡
=
2𝜋
2𝜋
0
< ∞.
Brownian Motion Processes (Continued)
 Consider a security that follows an (arithmetic)
Brownian motion process of the form: 𝑆𝑡 = 𝑆0 + 𝜎𝑍𝑡 .
 This process has a variance that is scaled by a
multiple 𝜎 2 from standard Brownian motion so that
the change in the process St from time t to time t+∆t
takes the form:
∆𝑆𝑡 = 𝜎∆𝑍𝑡 = 𝜎 𝑍𝑡+∆𝑡 − 𝑍𝑡 ~𝑁 0, 𝜎 2 ∆𝑡
𝑆𝑡 −𝑆0 𝑍𝑡
= =𝑍
𝜎 𝑡
𝑡
Brownian Motion Processes (Continued)
t
t
t
Brownian Motion: Illustration
A stock price St follows a Brownian motion process
with 𝑆0 = $50, 2 = 4.
Find the probability that the S3 < $56.
Solution:
Since St ~ N(50,4t), then S3 ~ N(50,12).
Using a standard z-Table, we find this probability :
𝑆2 − 50 56 − 50
𝑃 𝑆3 < 56 = 𝑃
<
= 𝑃 𝑍 < 1.73
12
12
= .9582
Stopping Times
 A stopping time is the length of time required for a
stochastic process to first satisfy some condition.
 A stopping time for a stochastic process Xt is a
random variable τ. For example, let τS be the first
time that Brownian motion St hits fixed value S*. We
will derive the probability that the stopping time is
less than or equal to t.
Deriving the Stopping Time Probability
for Brownian Motion
 First The probability that St ≥ S* equals:
The probability that the hitting time for S* is not more
than t, multiplied by the probability that St  S*,
conditional on the hitting time already having been
reached
 plus the probability that the hitting time for S* exceeds
t, multiplied by the probability that St  S*, conditional
on the hitting time not already having been reached :

𝑃 𝑆𝑡 ≥ 𝑆 ∗ = 𝑃 𝜏𝑆 ∗ ≤ 𝑡 𝑃 𝑆𝑡 ≥ 𝑆 ∗ 𝜏𝑆 ∗ ≤ 𝑡
+𝑃 𝜏𝑆 ∗ > 𝑡 𝑃 𝑆𝑡 ≥ 𝑆 ∗ 𝜏𝑆 ∗ > 𝑡
Deriving the Stopping Time Probability
for Brownian Motion (Continued)
𝑃(𝑆𝑡 ≥ 𝑆 ∗ ) = .5𝑃 𝜏𝑆 ∗ ≤ 𝑡
P 𝜏𝑆 ∗ ≤ 𝑡 = 2𝑃 𝑆𝑡 ≥ 𝑆 ∗
𝑆 ∗ − 𝑆0
= 2𝑃 𝑍 >
𝜎 𝑡
=2 1−𝑃 𝑍 ≤
=2 1−𝑁
𝑆 ∗ −𝑆0
𝜎 𝑡
𝑆 ∗ − 𝑆0
𝑡
Stopping Time Illustration
A stock whose price follows a Brownian motion
process, with S0 = 3 and unit variance 2 = 4. What is
the probability that the hitting time tS* = t5 for S* = 5 is
less than 8?
P 𝜏5 ≤ 8 = 2𝑃 𝑍 >
5−3
4 8
=2 1−𝑃 𝑍 ≤
= .72367
1
8
The Optional Stopping Theorem
 Any gambling rule based on some stopping time  has an
expected payoff equal to zero if any one of the following hold:
1.
2.
3.
𝑋𝑡 ≤ 𝑐 for all t ≤ τ
τ≤c
𝑙𝑖𝑚 𝑠𝑢𝑝 𝐸 𝑋𝑚𝑖𝑛 𝜏,𝑠 ≥ 𝑎 = 0,
𝑎→∞
𝑠≥0
 That is, the expected value of the random variable Xτ is
E[Xτ]=X0.
 The theorem roughly holds that any gambling rule based on
some stopping time  has an expected payoff equal to zero, as
long as the gambler either has limited funds or limited time to
gamble.
High and Low Hitting Times
 Suppose we are interested in the probability that our
Brownian motion process St hits a lower barrier S*Low
before an upper barrier S*High :
∗
∗
𝑃[𝜏𝐿𝑜𝑤
< 𝜏𝐻𝑖𝑔ℎ
]
 Solving algebraically for our Brownian motion, we have:
𝑆0 = 𝐸 𝑆𝜏 = 𝑆 ∗ 𝐿𝑜𝑤 𝑃 𝜏 ∗ 𝐿𝑜𝑤 < 𝜏 ∗ 𝐻𝑖𝑔ℎ + 𝑆 ∗ 𝐻𝑖𝑔ℎ 𝑃 𝜏 ∗ 𝐿𝑜𝑤 > 𝜏 ∗ 𝐻𝑖𝑔ℎ
= 𝑆 ∗ 𝐿𝑜𝑤 𝑃 𝜏 ∗ 𝐿𝑜𝑤 < 𝜏 ∗ 𝐻𝑖𝑔ℎ + 𝑆 ∗ 𝐻𝑖𝑔ℎ 1 − 𝑃 𝜏 ∗ 𝐿𝑜𝑤 < 𝜏 ∗ 𝐻𝑖𝑔ℎ
∗
∗
P 𝜏𝐿𝑜𝑤
< 𝜏𝐻𝑖𝑔ℎ
∗
𝑆𝐻𝑖𝑔ℎ
− 𝑆0
= ∗
∗
𝑆𝐻𝑖𝑔ℎ −𝑆𝐿𝑜𝑤
Illustration: Stock Price Hitting Times
with Arithmetic Brownian Motion
 An investor who purchased a stock whose price
follows a Brownian motion process, with S0 = 3 and
2 = 4. The investor has placed a limit order at 5.
Now, suppose that the investor also placed a stop
order at 2. What is the probability that the stock
price hits 2 before it hits 5?
P
∗
𝜏𝐿𝑜𝑤
<
∗
𝜏𝐻𝑖𝑔ℎ
=
∗
𝑆𝐻𝑖𝑔ℎ
− 𝑆0
∗
∗
𝑆𝐻𝑖𝑔ℎ
−𝑆𝐿𝑜𝑤
5−3
=
= .667
5−2
Expected Stopping Time
 The expected stopping time for a process with
stopping time τ = MIN[τ*Low, τ*High]:
𝐸𝜏 =
∗
∗
(𝑆𝐻𝑖𝑔ℎ
− 𝑆0 )(𝑆0 − 𝑆𝐿𝑜𝑤
)
𝜎2
 From our example above, the expected selling time
is:
𝐸
∗
∗
𝑀𝐼𝑁[𝜏𝐿𝑜𝑤
, 𝜏𝐻𝑖𝑔ℎ
]
=
5−3 3−2
4
=.5
Brownian Motion Processes with Drift
 Standard models for Brownian motion with Drift:
dSt  adt  bdZ t
dSt / St  dt  dZ t
 The second is called geometric Brownian motion (or
geometric Wiener process)
 The following is an Itô process:
dS t  a(St , t )dt  b( St , t )dZ t
Option Pricing: A Heuristic Derivation of
Black Scholes
 The expected future value of the call is:
𝐸 𝑐𝑇 = 𝐸 𝑀𝐴𝑋 𝑆𝑇 − 𝑋 , 0]
∞
=
𝑆𝑇 − 𝑋 𝑝 𝑆𝑇 𝑑𝑆𝑇
𝑋
∞
=
∞
𝑆𝑇 𝑝 𝑆𝑇 𝑑𝑆𝑇 − 𝑋
𝑋
∞
=
𝑝 𝑆𝑇 𝑑𝑆𝑇
𝑋
𝑆𝑇 𝑝 𝑆𝑇 𝑑𝑆𝑇 − 𝑋𝑃(𝑆𝑇 > 𝑋)
𝑋
5.4.1 Estimating Exercise Probability in a
Black-Scholes Environment
 Assume that stock prices follow geometric Brownian
motion:
𝑆𝑇
𝑙𝑛
= 𝜇𝑇 + 𝜎𝑍𝑇
𝑆0
 The probability of option exercise is calculated as follows:
𝑃 𝑆𝑇 > 𝑋 = 𝑃 𝑆0 𝑒 𝜇 𝑇+𝜎
ln
=𝑃 𝑍>
𝑇𝑍
> 𝑋 = 𝑃 𝜇 𝑇 + 𝜎 𝑇𝑍 > ln
𝑋
− 𝜇𝑇
𝑆0
𝜎 𝑇
=𝑃 𝑍>−
ln
𝑋
𝑆0
𝑆0
+ 𝜇𝑇
𝑋
𝜎 𝑇
𝑆
1
𝑆
1
ln 𝑋0 + (𝑟 − 2 𝜎 2 )𝑇
ln 𝑋0 + (𝑟 − 2 𝜎 2 )𝑇
=𝑃 𝑍<
=𝑁
= 𝑁 𝑑2
𝜎 𝑇
𝜎 𝑇
The Expected Expiry Date Call Value
 The expected value of the call contingent on its exercise:
∞
∞
𝑆𝑇 𝑝 𝑆𝑇 𝑑𝑆𝑇 = 𝑆0
𝑋
∞
𝑧2
−
𝑒 𝜇𝑇+𝜎 𝑇𝑧 𝑒 2
−𝑑2
∞
𝑆𝑇 𝑝 𝑆𝑇 𝑑𝑆𝑇 = 𝑆0
𝑒 𝑟𝑇
𝑒
2
1
−2 𝑧−𝜎 𝑇
−𝑑1
𝑋
𝑑𝑧
2𝜋
𝑑𝑧
2𝜋
Which is rewritten as follows:
∞
∞
𝑆𝑇 𝑝 𝑆𝑇 𝑑𝑆𝑇 = 𝑆0
1
− 𝑦2
𝑒 2
𝑒 𝑟𝑇
𝑋
−𝑑2 −𝜎 𝑇
𝑑2 +𝜎 𝑇
= 𝑆0
1
− 𝑦2
𝑒 2
𝑒 𝑟𝑇
−∞
𝑑𝑦
2𝜋
𝑑𝑦
2𝜋
= 𝑆0 𝑒 𝑟𝑇 𝑁(𝑑1 )
The Expected Expiry Date Call Value
(Continued)
 With d1 = d2 + σ 𝑇, the expected expiry date call
value equals:
∞
𝐸 𝑐𝑇 =
𝑆𝑇 𝑝 𝑆𝑇 𝑑𝑆𝑇 − 𝑋𝑃 𝑆𝑇 > 𝑋
𝑋
= 𝑆0 𝑒 𝑟𝑇 𝑁(𝑑1 ) − 𝑋𝑁 𝑑2
 And we discount to get the Black-Scholes model:
X
c0  S 0 N d1   rT N d 2 
e
Observations Concerning N(d1), N(d2) and
c0
 The probability that the stock price at time T will
exceed the exercise price X of the call is
P[ST > X] = N(d2)
 The expected value of the stock conditional on the
stock's price ST exceeding the exercise price of the
call is:
∞
𝑟𝑇
𝑆
𝑝
𝑆
𝑑𝑆
𝑆
𝑒
𝑁(𝑑1 )
𝑇
𝑇
𝑇
0
𝑋
𝐸 𝑆𝑇 𝑆𝑇 > 𝑋 = ∞
=
𝑁 𝑑2
𝑝 𝑆𝑇 𝑑𝑆𝑇
𝑋
Observations Concerning N(d1), N(d2) and
c0 (Continued)
 The expected expiry date call value is simply the product
of the probability of call exercise and the expected value of
the stock conditional on the stock's price exceeding the
exercise price of the call, minus the expected exercise
value paid at time T:
𝐸 𝑐𝑇 = EST ST  X  PST  X  − 𝑋𝑃 𝑆𝑇 > 𝑋
𝑆0 𝑒 𝑟𝑇 𝑁(𝑑1 )
=
𝑁 𝑑2 − 𝑋𝑁 𝑑2 = 𝑆0 𝑒 𝑟𝑇 𝑁 𝑑1 − 𝑋𝑁 𝑑2
𝑁 𝑑2
 The present value of the call is simply the discounted
value of its expected future value:
𝑐0 = 𝐸 𝑐𝑇 𝑒 −𝑟𝑇 = 𝑒 −𝑟𝑇 𝑆0 𝑒 𝑟𝑇 𝑁 𝑑1 − 𝑋𝑁 𝑑2
= 𝑆0 𝑁 𝑑1 − 𝑋𝑒 −𝑟𝑇 𝑁 𝑑2
The Tower Property
 The Tower property suggests that the expected value
now (time s) of tomorrow's (time t) expected value for
the following day's (time T) value equals the expected
value today for time T.
 More precisely, the Tower property says:
𝐸[𝐸[𝑋𝑇 |ℱ𝑡 ]| ℱ𝑠 ] = 𝐸 𝑋𝑇 ℱ𝑠
where T ≥ t ≥ s. This means that our expected value at time s
(today) is our expected value at time T (the day after
tomorrow) based on what we expect our information to be at
time t (tomorrow).