The document summarizes key concepts related to the Black-Scholes partial differential equation. It introduces Black-Scholes, which revolutionized finance by finding the fair price of derivatives. The formula was derived from the heat equation and allowed investors to earn maximum profits without risk. It discusses the variables in the Black-Scholes equation like stock price, exercise price, volatility and risk-free rate. An example valuation of a call and put option is shown. The document also covers fundamental concepts like interest rates, probability, expected value, and continuous random variables.

1. SENIOR YEAR EXPERIENCE
BLACK & SCHOLES PARTIAL DIFFERENTIAL EQUATION
WRITTEN BY: MERIH OZUGUZEL
ADVISOR: DR. JIM DEFRENZA

2. I Introduction
One of the most significant developments in the history of economics and Finance is Black
Sholes Equation. It was developed in 1973 by Robert Merton and Myron Scholes, who
were nominated as the Nobel Prize winners in 1997 to honor their contributions to Pricing
of Options. Unfortunately, Fisher Black passed away two years prior to the nomination.
Black Sholes formula has revolutionized the financial industry by finding the fair price of
a financial instrument such as option or other financial derivatives, which fluctuates over
time. It originally developed for stock options, however future options were the perfect
application for this mathematical formula.
A mathematical formula which helped Mr. Black and Scholes to win millions of dollars day
after day was actually derived from a mathematical formula called the Heat Equation.
The formula enabled investors to earn maximum profit without taking any risk at all.
There has been studies over years to see if it was possible to predict markets and at the
end, it was concluded that markets moved randomly and a money manager could beat
the market over several years just by random luck. For this reason, taking the risk factor
away from the markets enabled Black and Scholes to win no matter what happens. Before
we derive the heat equation from Black Scholes formula it is crucial to understand the
variables in the black Scholes equation that determines the fair value of the stock.
Solving the Black Scholes Equation:
The Black Scholes formula which caused the market crash considers the current and the
target price as two critical variables in valuing an option. A call option gives the right
to buy a stock at a pre-agreed target price within a specific time period, no matter what
the future value of the stock is. In this formula volatility is one of the key concepts
to consider. The implication for investors is that more volatile stock prices will pay
higher price to issue options to employees. Related to this concept, higher interest
rates will increase the value of a call option. For instance when the Federal Reserve
increases interest rates, this will make the stock option grants more expensive for issuing
companies. Financial terms such as portfolio, arbitrage, risk-free interest rate,
dividend, volatility will have an impact on the end result of the value of an option.
Considering all the steps, let us give an example how to value a call option and a put
option.
Assume that Netflix is currently trading for $273,40 per share. Today is June 4th and
the following options are available on the market to investors.
1

3. • June 270 call at $9.70
• June 280 Call at $4.90
• July 270 Call at $16.20
• July 280 Call at $11.30
• July 270 Put at $12.50
• July 270 Put at $17.60
As stated before, option prices are a function of five variables after simplification: Stock
Price, exercise price, time to expiration, volatility of the Stock and the Risk-free rate.
So Value of a call option is given by the formula:
Vc =P0Nd1
− X
ekRF t Nd2
d1=
[ln
P0
X +(kRF +.5σ2
)t]
σ
√
t
where t =number of years, sigma is denoted by decimal points and X
ekRF t is the present
value of the stock.
d2=d1 − σ
√
t
Given these values, we also now the following values about this stock:
• Stock Price = $62.oo
• Exercise Price(X)=$60.oo
• Time to expiration =40 days= 40/365
• Volatility (σ) 32 % = .32
• Risk-Free Rate =4 % = .04
d1= ln(62/60)+[.04+.05(.32)2
](4065)
.32
√
(40/365)
= .404
d2=.404 − .32 (40/365)=.30
By using the standard Normal Table we obtain;
N(.40) = .6554 and N(.30) = .6179
Finally, by using the Black Scholes equations Value of the call option for this specific
stock becomes;
Vc = (62)(.6554) − [(60)/(e.04(40/365)
)](.6179) = $ 3.72
2

4. Similarly, value of the put option is;
VP = VC + X
ekRF t −P0
Vp = $3.72 + [(60/(e.04(40/365)
) − $62.oo = $1.46
As we already know the stock price is $62.oo and the exercise price is $60.oo. So,this call
option is in the money, has an intrinsic value of $2.oo and has a speculative premium of
$1.72. The put option on the other hand is out of the money. In order for this to have
some intrinsic value, the stock has to fall down by $2.00 since the put option gives the
investor right to sell at $ 60.oo. So it is not going to be valuable at expiration unless
the stock is trading below $60.oo per share. Thus, right now there is no intrinsic and
all the value is speculative premium which is $1.46. We need to keep in mind the fact
that this type of calculation by hand takes a lot of time and in the mean time the price
of the option will change. That is why the Black Scholes formula was already uploaded
to calculators for floor traders in the early 1980s and was widely used since then. The
problem for arbitrage traders was that more and more people started to use the formula
which reduced the profit per investor who already had access to this formula. Finally in
2008, Myron Scholes and Fisher Black’s formula which was once the holy grail of every
investor, was known as the mathematical equation that caused the banks to crash.
II Fundamentals of the Black & Scholes PDE
(1)INTEREST RATE
In order to understand the Black Scholes formula, an investor should grasp the idea of
interest, a fractional payment from banks in exchange for the use of investors/depositors
money.
Simple Interest: A = P(1 + r)t
, where r= interest rate, t= number of years
Compound Interest: A = P(1 + r/n)nt
There is also the notion of effective simple interest rate where compounding interest rate
is equal to the simple interest rate.
Continuous compound interest:
One way to visualize the continuous compounding concept is making the compounding
period infinitely small by taking the limit of n to infinity.
limn→∞(1 + r/n)n
By evaluating the process using the natural logarithm and l‘Hopital‘s Rule we obtain the
following equation: limn→∞ lny =limn→∞
d
dn (ln(1+r/n))
d
dn (1/n)
Hence, limn→∞ y = er
and A = Pert
3

5. Future and Present Value: One of the very useful mathematical tools to calculate
the future and present values is the geometric series. Let us assume that
S = 1 + a + a2
+ · · · + an
where n is a positive whole number. As the next step, if we multiply both sides by a and
subtract it from from the equation we would not change the equality.
S − aS=1 + a + a2
+ · · · + an
− (a + a2
+ a3
+ · · · + an+1
)
S=1−an+1
1−a , where a = 1
By the help of this simple formula, it is easier to show the relationship between the interest
rate, the length of the loan, compounding frequency , principal amount borrowed and the
payment amount as seen in the following equation:
P=xn
r (1 − [1 + r
n ]−nt
)
Inflation and the rate of Return: The increase in the amount of money circulating
without having an increase in the amount of goods is known to be inflation. It is a crucial
concept in the world of finance in order to find the real value of an asset compared to
effects of the current economic environments such as depreciation, inflation and intrinsic
value.
Rate of return is the gain or loss on an investment over a specific period of time, and it
is given as a percentage gain or loss over the initial investment amount.
The formula is again constructed on the initial amount of money, and the interest rate as
follows: P= A(1 + r)−1
or r=A
P − 1
(2)DISCRETE PROBABILITY
In this chapter, we will focus on the stochastic model of the market by introducing Discrete
Random Variable.
Random Variable: a numerical quantity whose value is not known until an experiment
is conducted.
Event: Set of outcomes of an experiment. Experiment: any activity that generates an
observable outcome. i.e. flipping a coin, rolling a pair of dice
Discrete: the outcomes will be from a set whose members are isolated from each other
by gaps. Flipping a coin, only outcomes are heads or tails.
Probability of an event: is the likelihood of that event occurring as the outcome of an
experiment.
Conditional probability: The probability that one event occurs given that another
event has occurred is called conditional probability.
Probability of an even A occurs: P(A)=P(A)+P(B)-P(A∩B)
Multiplication Rule: P(A ∩ B) = P(A)P(B|A)
In an experiment, if A has no effect on event B then A and B are said to be independent.
4

6. Probability Distribution is a function which assigns a probability to each element in
the sample space of outcomes of an experiment.
Bernoulli random variable: which takes on only one of the two possible values, often
thought of as true or false.
A Bernoulli random is particularly simple, f(1) = P(X = 1) = p where 0 ≤ p ≤ 1 and
f(0) = 1 − p
A binomial random variable: is parameterized by the number of successful outcomes
out of n independent Bernoulli random variable.
n
x = n!
x!(n−x)! By using the Addition and multiplication rules, the probability of x
successes in n trials is represented by the function:
P(X = x)= n
x px
(1 − p)n−x
= n!
x!(n−x)! px
(1 − p)n−x
Expected Value: of a random variable is the typical outcome of an experiment per-
formed an infinite number of times whereas the statistical mean is calculate based on a
finite collection of observations of the outcome of an experiment.
If X is a discrete random variable with P(X) probability distribution then the E[X] is
represented as:
E[X]= x(X · P(X))
Joint Probability Function:
f(x, y)=P(X = x, Y + y)
Marginal Probability: The sum of the joint probability of X and Y where Y is allowed
to take on each of its possible values is called Marginal Probability of X.
Variance: of a random variable is a measure of the spread of values of the random
variable about the expected value of the random variable.
Standard Deviation: Since by definition, the standard deviation is the square root of
the variance. It is denoted by σ(X) and thus
σ(X)= V ar(X).
(3)CONTINUOUS PROBABILITY
Continuous Random Variable: Can take on infinite number of variables. limn→∞
limn→∞
1
n+1 =0
A random variable X has a continuous distribution if ∃ a non-negative function f: R0→
R | for an interval [a,b]
P(a≤ X ≤ b)=
b
a
f(x)dx. The function f which is also the probability distribution
function must have the following property as well,
∞
−∞
f(x)dx = 1
Uniformly distributed Random Variable: A continuous random variable has a
uniform distribution if all the values belonging to its support have the same probability
5

7. density.
f(x)= 1
b−a if a ≤ x ≤ b, 0 otherwise.
Expected Value of Continuous Random Variable:
By definition E[X]=
∞
−∞
xf(x)dx.
Marginal Distribution:
By definition, two
∞
−∞
continuous random variables are independent if the joint distribu-
tion factors as the product of the marginal distributions of Y and X.
f(x, y) = g(x)h(y) Variance and Standard deviation: Variance of a continuous
random variable is a measure of spread of values of the random variable about the expected
value of the random variable. Var(X)= E[(X-µ)2
] =
∞
−∞
(x − µ)2
f(x)dx
where µ = [x] and f(x) is the probability distribution function of X.
Theorems:
Let X be a continuous random variable with the probability distribution f(x) and let a,b
∈ R, then
V ar(aX + b) = a2
V ar(X)
Let X1, X2, · · · Xk be pairwise independent continuous random variables with joint prob-
ability distribution f(x1, x2, x3, · · · , xk), then:
Var(X1 + X2+· · · +Xk)=Var(X1)+Var(X2)+· · · +Var(Xk) By definition stan-
dard deviation of a continuous random variable is the square root of its variance.
Normal Random Variables: Continuous Random variable obeying a normal distribu-
tion is said to be bell curved random variable.
Normal Probability Distribution with mean µ and variance σ 2
is
f(x)= 1
σ
√
2π
e−
(x−µ)2
2σ2
when µ=0 and σ = 1, this becomes a standard normal probability distribution.
Central Limit Theorem: Central limit theorem explains why many distributions
tend to be really close to normal distribution. The key factor in this theorem is that
the random variable should be the sum or mean of many independent and identically
distributed random variables.
Theorem: If random variables X1, X2..., Xn form a random sample of size n from a
probability distribution with mean µ and standard deviation σ then for all x,
limn→∞ P (
√
n(X−µ)
σ ≤ x) = θ(x).
Log-normal Random Variables:
A log-normal distribution is a continuous probability distribution of a random variable
whose logarithm his normally distributed. For instance, if X is a random variable with a
normal distribution, then we can conclude Y=exp(X) has a log-normal distribution.
6

8. P(Y<y )= 1
σ
√
2π
y
−∞
2−(t−µ)2/2σ2
dt.
Expected Value of a security:
If an investor purchases a security with price worth X for price K, the excess profit of this
transaction is denoted by X-K. Mathematically, we will write this as;
(X − K)+
=max{0, X − K}
(4)ARBITRAGE
The financial definition of arbitrage is the simultaneous purchase and sale of an instrument
in order to receive profit from the difference in prices. This concept exists as a result of
market inefficiencies and could be quite profitable in the short run. The Arbitrage theorem
states that the probabilities of the outcomes are such that all bets are fair or has a betting
scheme which produces a positive gain independent of the outcome of the experiment.
Let us take look at the arbitrage concept from a probabilistic point of view. If the odds of
a particular stock value going up and going down are quoted as 2:1 against. We can think
that there are three outcomes to the experiment and in two of them the desired outcome
does not occur. So, the probability of the desired outcome arising is1/3. In general if the
outcome is n:1, then the probability of the outcome is 1/(n+1). Now, let us consider a
strategy game called ‘Hex’[5]. Suppose the odds against player 1 defeating player 2 in this
game are 3/2:1 and the odds against player 2 defeating player 1 are 3/7:1. By using the
previous stock example, we can conclude that player 1 defeats player 2 with probability
.4 while player 2 defeats player 2 with probability .7. As seen, the probabilities does not
add up to 1. This is where Arbitrage theorem kicks in, according to this theorem there is
a betting strategy which generates a positive gain regardless of the outcome of the match.
Duality of Linear Programming:
The following equation will help us to understand the Black Scholes formula since it
represents a linear product of a cost function.
The vector x is feasible by definition if xi ≥ 0 for i=1,2,· · · , n and x satisfies the set
of constraint equations Ax = b. In this case, A is an m x n matrix while b ∈ Rmxn
is
another column vector. The inner product is:
c· x= c1x1 + c2x2+· · · +cnxn
The Fundamental Theorem of Finance:
Consider an experiment with m possible outcomes numbered 1 through m. Suppose we
can place n wagers on the outcomes. Let rij be the return for a unit bet on wager i when
the outcome of the experiment is j. The vector x=(x1, x2 · · · , xn) is called a betting
strategy.
7

9. (5)RANDOM WALKS AND BROWNIAN MOTION
Random Walk concept using the probabilistic models is crucial in the world of finance
in order to capture the behavior of stocks, securities, options or any indexes. This concept
can be explained by flipping a coin. Imagine a person standing on a number line at the
origin and any time he the coin lands on heads, he takes one step to the right in the
positive direction and anytime it lands on tails he takes a step left. This concept is
called a random walk. According to this theory the changes in the stock price have the
same distribution and they are independent of each other. As most hedge funds would
say: ”The past movement of a stock price or market cannot be used to predict its future
movement.”
First Step Analysis:
Let us denote the current value of a stock as S. This stock can move at discrete intervals
(for instance a day) up or down by one unit. So, if the value on day n is S(n),then
tomorrow the value will be either S(n + 1)=S(n) + 1 or S(n + 1)=S(n) − 1. Let us also
assume that the probability of the stock going either way is p = 1/2, which implies this
random walk is symmetric. To generalize this random variable, Nth
state of the random
walk is the partial sum where N>0.,
S(N) = S(0) + X1 + X2 + · · · + XN
If we assume that the random variables Xi and Xj are identically distributed and indepen-
dent for i= j then the transition between states in the random walk, S(N)−S(N−1) = XN
are independent random variables. Thus, when n ≥ 0 and the stock moves in the positive
direction (k ≥ 0) or in the negative direction (n − k ≥ 0) then,
P(S(n)=S(0)+k-(n-k))=P(S(n)-S(0)=2k-n)
n
k (1
2 )n
.
In this case, assume the current value of the stock is some positive value, if the state of
the random walk reaches the lower boundary in a finite number of steps, then the state
remains at the boundary value. This is also known as the absorbing boundary condition.
This concept makes perfect sense in the stock market since a stock with a value of $ 0 is
worthless and will trade in the pink sheet after the boundary level.
Example:
The graph below represents a stock which was trading at $ 10 per share since its IPO.
Since the boundary at 0 is absorbing we must keep track of the smallest value which S(n)
takes on. So, let us define mn=minS(k) : 0 ≤ k ≤ n. So if i is the smallest non-negative
value such that mi = 0, then by the absorbing boundary condition S(K) = 0 ∀ k ≥ i.
8

10. Sample random walk of the price of a stock. [1a]
Suppose a random walk has the form where Xifori = 1, 2,· · · are independent, identically
distributed random variable taking on the values +/- 1 with probability p=.5. Suppose
further that the boundary at 0 is absorbing, then A,i > 0,
P(S(n)=A ∧mn>0|S(0) = i)
= [ n
(n+A−i)/2 - n
(n−A−i)/2 ] (1
2 )n
.
Example: For a symmetric random walk with initial state S(0) = 10, what is the proba-
bility that S(50) = 16 and S(n) >0 for n = 0, 1, · · · , 50?
So by using the formula, P(S(50) = 16 ∧ m50 > |S(0) = 10) = [ 50
28 - 50
12 ] 2−50
=
.0787178
Let us call the first time that the random walk S(n) equals A the stopping time, ωA. Given
the initial state of the random walk is S(0) =i >0, parameter A>0, and the boundary at
0 is absorbing, we will see the probability that ωA takes on the various natural number
values.
Assume that the initial sate of the random walk S(0) =i>A>0, then because of the spatial
homogeneity of the random walk,
P(S(n)=A∧mn≥ A —S(0) =i) =P(S(n)=0 ∧mn≥ 0 — S(0) =i-A).
ω0=n if an only if S(n − 1) = 1mn−1> 0 and Xn = −1. It follows as
P(ω0 = n|S(0) = i − A)
=P(Xn = −1 ∧ S(n − 1) = 1 ∧ mn−1>0|S(0) = i − A)
=1/2 P(S(n-1)= 1∧mn1>0|S(0) = i − A)
=1/2 f1,(i−A)(n − 1).
Stochastic Processes:
Before getting into Stock Market examples of the Random Walk, we will examine the
deterministic process of exponential growth and decay. Mathematical expression of the
rate of change of a non negative quantity P is;
9

11. dP
dt =µP
where µ is the proportionality constant and t represents time. When µ > 0, the propor-
tionality constant is called the growth rate. Similarly, when µ <0, it is called the decay
rate. If the value of P is known at a specific value of t(usually when t=0), then this DFQ
has a solution P(t) =P(0) eµt
. This model is called deterministic since there is no room
for random events to show themselves in the model. Now, let X = lnP, then dX = µdt.
It follows as, X(t + ∆t)-X(t)=∆ X = µ ∆t. As seen this equation is still deterministic,
however becomes stochastic when we introduce a random element. Let us take random
variable dz(t) with a mean of zero and standard deviation of one.
So; ∆ X=µ ∆ t+σdz(t)
√
∆t
Random variable in the equation above is written as dzand the t dependency of dz is
associated with z taking on a random value at each time t. As a common knowledge
in finance, σ is referred as the volatility of the stock or any financial instrument. The
product of σdz(t)
√
∆t could be thought as the normal random variable with a mean zero
and a standard deviation of σ
√
∆t.
Consider the cumulative change in X over several times each of size ∆. Let ti<tj =ti +
(j − i)∆. Then;
∆ XX = X(tj) − X(ti)
=
j−1
k=i
(X(tk + ∆t) − X(tk)) =
j−1
k=i
(µ ∆ t+ σ dz(tk)
√
∆t.
But all of the random variables {dz(tk)}j−1
k=i are independent for tk = tk‘ and identically
distributed with a mean zero and average value of ∆X is µ(tj − ti).
Now, let us consider the variance of X= X(tj) − X(ti).
Var(∆X) = V ar[
j−1
k=i
(µ∆t + σ dz (tk) ∆t)] = σ2
∆t(j − i) =σ2
(tj − ti).Now it is clear
that random portion of the stochastic model contains
√
∆t which makes the variance of
∆X proportional to ∆t. Without this assumption, the variance of ∆ X would contain
nonlinear dependence on ∆t. We will be using normal variable with mean zero and
variance
√
∆t, so dW(t) will represent this random variable from now on.
The difference between X(ti + 1) and X(ti) will generate µ ∆ t + σdW (ti). This follows
as;
X(ti+1)= X(ti) + µ∆t + σdW(ti)
lnP(ti+1)= lnP(ti) + µ∆t + σdW(ti) = P(ti)eµ∆t+σdW (ti)
From a stock price point of view, let µ = .01, σ = .05,∆t = .004 and P(0) = 10. The
random walk generated by the stochastic equation is shown below.
10

12. [1b]
Random Walk of a stock using the variables above.
The equation that we have used before ∆ X=µ ∆ t+σdz(t)
√
∆t is a perfect example of a
generalized Wiener process. But this equation can also be described by a straight forward
extension of a differential equation. This stochastic differential equation is obtained by
adding a normally distributed random variable to the ordinary differential equation.
dX= a dt +b dW(t) where a and b are constants and dW(t) is a normal random
variable. In order to find the random variable X, we will take the integral of both sides.
X(t) =X(0) +at+
t
0
b dW(T)
At this point, it is crucial to write and understand the chain rule since we will use it
to develop the Black Scholes partial differential equation. In standard calculus when
someone makes the assignment X = ln P then it is understood as dX = dP/P because
of the chain rule. When this is applied on a generalized Wiener process, we obtain;
dX = adt + bdW(t)
dP = aPdt + bPdW(t)
Since in the first equation a and b are mean and standard deviation respectively, this
does not mean that a and be are mean and standard deviation of the underlying variable
P in Ito‘s equation. For this reason we will need a stochastic calculus version of the chain
rule for differentiation.
Ito‘s Lemma:
In this section, we will derive the multi-variable version of Taylor‘s Theorem in order to
change variable in a stochastic differential equation. Here is an example of single variable
Taylor‘s theorem.
f(x) = f(x0)+f‘
(x0)(x−x0)+ f“
(x0)
2! (x−x0)2
+· · ·+ f(n)
(x0)
n! (x−x0)n
+ f(n+1)
(θ)
(n+1)! (x−x0)n+1
The last term shown above is the Taylor remainder formula and known as Rn+1. θ is
between x and x0 and the other terms form a polynomial in x of degree at most n which
can be used as an approximation for f(x) in an epsilon neighborhood of x0 .This single
variable form of Taylor‘s formula will be used to derive the two variable form.
11

13. Assume that the function F(y, z) has partial derivatives on an open disk containing the
point with coordinates (y0, z0). Since we know that f is a function of a single variable
then we can use the single variable form of Taylor‘s theorem with x0 = 0 and x = 1 in
order to obtain the equation below up to order 3;
f(1) = f(0) + f‘
(0)x + 1
2 f“
(0)x2
+ R3.
By using the chain rule we have;
f‘
(0) = hFy(Y0, z0) + kFz(y0, z0)
f“
(0) = h2
Fyy(y0, z0) + 2hkFyz(y0, z0) + k2
Fzz(y0, z0).
As seen above we used the fact that Fyz = Fzy for this function. Now by substituting
F‘
(0) and f“
(0) into our first equation containing R3, we will obtain;
∆F = f(1)−f(0) = F(y0+h, z0+k)−F(y0, z0) = 1/2(h2
Fyy(y0, z0)+2hkFyz(y0, z0)+
k2
Fzz(y0, z0)) + R3.
In order to derive the Ito‘s Lemma we need to take on more step.
dX = a(X, t)dt + b(x, t)dW(t)
where dW(t) is a normal random variable and a and b are both function of X and t.
Let Y = F(X, t) be another random variable depended on X and t. Given X, we will
determine the Ito process which describes Y . Again, by using the Taylor series expansion
for Y ;
∆Y = FX∆X + Ft∆t + 1/2FXX(∆X)2
+ FXt∆X∆t + 1/2Ftt(∆t)2
+ R3=
Fx(a∆t+bdW(t))+Ft∆+1/2FXX(a∆t+bdW(t))2
+FXt(a∆t+bdW(t))∆t+1/2Ftt(∆t)2
+
R3.
Since the term R3 contains terms of order(∆t)k
wherek≥ 2, ∆t becomes really small.
∆y ≈ Fx(a∆t + bdW(t)) + Ft∆t + (1/2!)FXXb2
(dW(t))2
.
But dW(t) is a normal random variable with expected value of zero and variance of ∆t.
So, by definition;
E[(dW(t))2
] = ∆t which is used for an approximation to (dW(t))2
∆Y ≈ FX(a∆t + bdW(t)) + Ft∆t + (1/2!)FXXb2
∆t.
Now, if we get back to the question about stochastic process followed by a stock price
P. If X = lnP and dX = µ dt +σdW(t). then by letting F(X, t) = eX
= P and using
Ito‘s Lemma;
dP =(µeX
+ 0 + 1
2 σ2
eX
)dt + σeX
dW(t)
=(µ + 1
2 σ2
)Pdt + σPdW(t).
12

14. III Options
In the financial world options play a huge role for investors to make profit. An option
is a financial derivative that represents a contract between the option writer and option
holder. The key concept is the fact that contract offers the buyer the right, but not the
obligation, to buy(call option) or sell(put option) a particular financial asset at an
agreed price(the strike price) within a specific period of time(exercise date). There
is different kinds of options such as European option which can only be exercised at
maturity, and American option which can be exercised at or before the strike time. In
practice, American options are more commonly traded, however from the mathematical
stand point we will be focusing on European option which is easier to understand. In order
to visualize the concept of options, let us assume that a particular stock X is trading at
$ 100.oo
per share. An investor may not want to own this stock today due to volatility in
the market, however he would like to own this stock in the near future, say three months
from now. In order to reduce the risk the investor can buy a European call option on
this stock with three month strike time and a strike price of $ 100.oo
. As stated before,
the investor have the right to buy the stock but not obligated. If after three months the
stock price goes higher that $100.oo
, it is beneficial for the option holder to buy the stock
at $ 100.oo
as agreed before. However, if the stock price is lower than $100.oo
the investor
does not need to exercise the option and buy the stock at the market price.
As stated in the previous section, Arbitrage, options can be a huge opportunity for bargain
hunters since the options themselves have some value and they might be mis-priced. We
also need to note that in the financial world, an investor can sell the stocks first without
owning them and then buy them which is also referred as short selling or shorting
a stock. Similar to a long position where investor believes that a particular company‘s
stock price will go higher than the current price due to some catalyst, another investor
can have short position if he believes that the price is going to go down in the future
because of various reasons.
Properties of Options
Ca=value of an American call option
Ce=Value of a European call option
K=Strike price of an option
Pa=Value of American call put option
Pe=Value of European put option
r=risk free interest rate
S=Price of a share of a security
T=exercise time of an option
13

15. Let us consider the theory of arbitrage. In the absence of arbitrage, an American option
must be worth at least as much as its European counterpart. For instance Ca ≥ Ce and
Pa ≥ Pe. Other things also being equal(strike times, underlying securities etc. ) To
the contrary, suppose Ca<Ce. Assuming that every investor has the perfect information
and knowing that the American option has all the characteristics of the European option
with increased flexibility that it may be exercised early, no one would invest in European
call option. Since Ce − Ca >, the investor may also purchase a risk free bond paying
interest at rate r compounded continuously. At the expiration date, the bond will reach a
value (Ce − Ca)erT
. So if the European option holder wishes to exercise the option, the
investor insures this is possible by exercising his own American option. If the owner of the
European option lets it sit and not exercise, then the investor can do the same with the
American option. So, in both cases the investor makes a risk free profit of (Ce − Ca)erT
which is a perfect example of arbitrage.
Put- Call Parity Formula:
Assume that Pe +S = Ce +Ke−rT
where the left hand side represents portfolio A and the
right hand side represents portfolio B. What this parity implies is that in an arbitrage-free
setting these portfolios should have the same value.
Now suppose portfolio A is less valuable than portfolio B. Pe + S<Ce + Ke−rT
. An
investor can borrow at an interest rate of r an amount equal to Pe + S − Ce. This allows
the investor to purchase the European put option and to sell the European call option. At
the strike date, the investor must repay the principal amount and interest in the amount
of (Pe + S − Ce)rT
. Now, if the security if worth more than K at time T, the put expires
worthless and the call will be exercised. So, the investor must sell the security for K.
K-(Pe+S−Ce)rT
>0. This inequality is equivalent to Pe+S<Ce+Ke−rT
. Similarly, if the
call expires worthless and the put will be exercised , the net proceeds of this transaction
are the same as in the previous example. So, there is a risk-free profit to be realized if
portfolio A is worth less than portfolio B.
Pricing of an Option Using a Binary Model:
Let us try to find the correct price of an option by the following question. Assume that
a stock is currently trading at $100.00
a share. After a single unit of time T, the price of
the share will be either $50.00
with probability p or $200.00
with probability 1 − p. An
investor can purchase European call option whose value is C. The exercise time is T and
the strike price is $150.00
. Assuming that there is no arbitrage, what is the value of C.
Since the option or stock might be purchased at the beginning of the time, we must
calculate the present value of any potential profit. Let us call the interest rate per T unit
of time r. Suppose the investor purchases the stock initially. At time T, his net gain is
14

16. −100+200(1+r)−1
with probability 1−p. In this no arbitrage environment the expected
value of this gain is zero.
0 = (−100 + 200(1 + r)−1
)p + (−100 + 50(1 + r)−1
)(1 − p)
0 = −100(1 + r) + 150p + 50
p = 1+2r
3
On the other hand, assuming that the investor purchases the option initially, at time T,
his net gain is −C + (200 − 150)(1 + r)−1
with probability p. Once again, the expected
gain is 0 in arbitrage free environment.
0 = (−C + 50(1 + r)−1
p + (−C)(1 − p)
0 = 50p(1 + r)−1
− C
C = 50+100r
3(1+r)
Hence, we can conclude that if (p(r), C(r)) deviate from the cost function of interest rate,
then risk free profit can be made.
Black Scholes Partial Differential Equation:
All the concepts that are mentioned above such as arbitrage, stochastic process, present
and future value and mathematical equations will be helpful for us to derive the Black
Scholes partial differential equation in this section.
Let us assume that S is the current value of a security and it obeys a stochastic process
of the form;
dS = (µ + σ2
2 )Sdt + σSdW(t)
If F(S, t) is the value of an option, then by using Ito‘s Lemma, F obeys the following
stochastic process as well.
dF = ([µ + σ2
2 ]SFS = 1
2 σ2
S2
FSS + Ft)dt + σSFS, dW(t)
Suppose that a portfolio with value P is created by selling the option and buying ∆ units
of the security So, the value of the portfolio is P = F − ∆S. Since we know that the
portfolio is a linear combination of the option and the security, then the stochastic process
is;
dP = d(F − ∆S)=dF − ∆dS
=([µ + σ2
2 ]SFS + 1
2 σ2
S2
FSS + Ft)dt + σSFSdW(t) + ∆((µ + σ2
2 )Sdt + σSdW(t))
=([µ + σ2
2 ]S[FS − ∆] + 1
2 σ2
S2
FSS + Ft)dt + σS(FS − ∆)dW(t).
As seen above the coefficient of the normal random variable dW(t) contains the factor
FS − ∆. So in order to simplify the equation let ∆ = FS. Notice that the randomness is
not entirely eliminated since the value of the security S remains and is stochastic. The
new version of the equation is as follows;
dP = (1
2 σ2
S2
FSS + Ft)dt
We already know that in an arbitrage free setting, the difference in returns from the
portfolio above or investing an exactly equal amount in a risk free bond paying interest
15

17. at rate r should be zero. So;
0 = rPdt − dP
0 = rPdt-dP=(1
2 σ2
S2
FSS + Ft)dt
rP=(1
2 σ2
S2
FSS + Ft)
r(F − ∆S) =1
2 σ2
S2
FSS + Ft)
rF=1
2 σ2
S2
FSS + Ft) +r∆S
rF=Ft + rSFS + 1
2 σ2
S2
FSS.
So, the equation above is an example of partial differential equation. It is important to
note that PDEs are often described by their order. For instance Black Scholes PDE is a
second order equation,because the highest order derivative of the unknown function F is
the second derivative, which is parabolic type. The best known example for second order
PDE is the heat equation in physics. By the separation of variables Black Scholes PDE,
we can obtain the Heat Equation by appropriate changes in variables.
IV Solution of the Black Scholes Equation
Changing Variables in the Black-Scholes Partial Differential Equation:
As shown in the last chapter, for the partial differential equation and its side conditions
for a European call option, we have the following equations;
rF=Ft + rSFS + 1
2 σ2
S2
FSS. where t ∈ [0, T), and S ∈ (0, ∞)
F(S, T) = (K − S)+
for S ∈ (0, ∞) which indicates when the expiry date for the
option arrives, the call option will not worth anything if the value of the stock is less than
the strike price.
F(0, t) = 0, F(S, t) → S as S → ∞ for t ∈ [0, T) indicates the stock itself is worthless
before maturity, the call option is also worthless. An investor might choose to buy the
stock for nothing and let the call option expire. Knowing these equations, let us convert
the Black Scholes partial differential equation to the Heat Equation.
Suppose F, S, and t are defined in terms of the new variables v,x, and θ.
S = Kex
↔ x=ln S
K
t=T- 2θ
σ2 ↔ θ = σ2
2 (T − t)
F(S, t) = Kv(x, θ)
Hence, FS =(Kv(x, θ))S= K(vxxS + vttS)= K(vx
1
S + 0) =e−x
vx
Ft = −Kσ2
2 vθ and FSS = e−2x
K (vxx − vx)
By substituting these equations into the Black Scholes equation, we obtain;
vθ = vxx + (k − 1)vx − kv where k= 2r
σ2
The final condition for F is converted by this change of variable into an initial condition
since when t = T and θ = 0. The initial condition v(x, 0) is then;
16

18. Kv(x, 0) = F(S, T) = (S − K)+
= K(ex
− 1)+
v(x, 0) = (ex
− 1)+
Since limS→0+ x=-∞,
0=limS→0+ F(S, t) =limx→=∞ Kv(x,θ)−→ limx→=∞ v(x,θ) = 0.
Similarly, since as limS→∞
F(S,t)=limx→∞Kv(x,θ)=Kex →limx→∞v(x,θ)=ex
.
Now that we have derived a pair of boundary condition for the partial differential equation,
the original Black Scholes initial, boundary value problem has been recast in the form of
the following.
vθ = vxx + (k − 1)vx − kv for x ∈ (−∞, ∞), θ ∈ (0, T σ2
2 ) ψ
v(x, 0) = (ex
− 1)+
forx ∈ (−∞, ∞)
v(x, θ) → 0asx → −∞ and
v(x, θ) → ex
asx → ∞, θ ∈ (0, T σ2
2 )
It is important to note that x can take any real number. Now, in order to introduce the
new dependent variable u, α and β constants are needed.
v(x,θ) = eαx+βt
u)x, θ)
vx = eα
x + βt(αu(x, θ) + ux)
vxx = eαx+βt
(α2
u(x, θ) + 2αux + uxx)
vθ = eαx+βt
(βu(x, θ) + uθ)
By substituting the expressions above into equation ψ we obtain;
uθ = (α2
+ (k − 1)α − k − β)u + (2α + k − 1)ux + uxx
Let arbitrary constants equal to 0 in order to simplify the equation. 0 = α2
+(k−1)α−k−β
0 = 2α + k − 1
α = (1 − k)/2 and β =-(k + 1)2
/4. The initial condition for u can be derived from the
initial condition.
v(x,0)=(ex
− 1)+
eαx
u(x, )) = e(k−1)x/2
(ex
− 1)+
=e(k−1)x/2 ex
− 1 if x>0
0 if x ≤ 0
.
=e(k−1)x/2 ex
e(k − 1)x/2 − e(k − 1)x/2 − ek−1)x/2
if x>0
0 if x ≤ 0
.
=e(k−1)x/2 e(
k + 1)x/2 − e(k − 1)x/2 − ek−1)x/2
if x>0
0 if x ≤ 0
.
=(e(k−1)x/2
− e(k−1)x/2)
)+
To summarize the Black Scholes partial differential equation, initial, and boundary
conditions have been converted to the following system of equations.
uθ = uxx for x ∈ (−∞, ∞) and θ ∈ (0, Tσ2
/2)
The equation above is the well-known heat equation of mathematical physics.
u(x, 0) = (e(k−1)x/2
− e(k−1)x/2)
)+
for x ∈ (−∞, ∞)
u(x, θ) → 0asx → +/ − ∞ for θ ∈ (0, Tσ2
/2). ψ
17

19. References
[1] J. Robert Buchanan. An Undergraduate Introduction to Financial Mathematics.
World Scientific Publishing Co. Pte. Ltd., New Jersey, 2006.
[2] Robert A. Jarrow, Andrew Rudd. Option Pricing. Richard D. Irwin Inc, 1983.
[3] Paul Wilmott Sam Howison Jeff Dewynne. The Mathematics of Financial Derivatives.
Cambridge University Press, New York, 1995.
[4] Franois Coppex. Solving the Black-Scholes equation: a demystification.
http://www.francoiscoppex.com, unpublished, 2009.
[5] Yohei Yamasaki. Theory of Division Games. 1978.
18