Rensselaer Polytechnic Institute.
MEAE 6630: Conduction Heat Transfer Project
Authors: Joe Ortoleva & Naushir Bala
April 13, 2000.
Introduction
This project will show that the problem of 1-D conduction heat transfer and the
problem of the pricing of stock options are analogous problems. This is accomplished by
first providing the mathematical formulation of each problem using their respective
governing equations and associated boundary conditions, and then solving these
formulations to provide the analytical solution in either case. Finally, each problem is
examined using a numerical method.
Definition of terms in the Stock Option Pricing problem.
Since the stock option pricing problem is the lesser known of the two problems it
is important to become familiar with its associated jargon.
The word stock/share of company X denotes a unit of ownership in company X.
Each share has an underlying value associated with it. The value of the share increases
and decreases daily, as shares of company X, are traded, i.e. bought and sold. The value
of the share at any instant is a reflection of the market’s view of company X at that
instant.
While trading shares is like buying/selling one’s ownership in the company, the
trading of stock options is different. Buying a stock option of company X does not mean
buying stock of company X, but rather the right to buy stock in company X at an agreedupon price and on an agreed-upon date. The stock in this case, or more generally, the
asset in question, is sometimes referred to as the underlying, because it is the underlying
value of this asset at some future date, that is being speculated upon. The agreed-upon
price is called the exercise price or strike price, and the agreed-upon date is referred to as
the expiry date, expiration date, or the date on which the option is said to become due.
The person buying the option is the holder of the option, and the person from
whom the option is bought is the writer of the option. Note that the option is a contract
which gives the holder the right to buy; it is not an obligation to buy. The choice of
whether or not to exercise the option is up to the holder. However, should the holder
choose to exercise the option, the writer then has a potential obligation; he must sell the
asset in question. An option that gives its holder the right to buy is a Call Option,
whereas an option that gives its holder the right to sell is a Put Option.
There are two main classes of options. Options that may be exercised only on the
expiration date are called European options, whereas options that may be exercised at
any time between the purchase of the option and the expiry date, are called American
options.
Most of the options that are traded on exchanges are American.
However,
European options are generally easier to analyze than American options, and some of the
properties of an American option are frequently deduced from those of its European
counterpart. American options are more interesting to mathematicians because not only
do they pose the problem of how should the option be priced but also pose the problem of
when is it optimum to exercise the option. Note that this classification has nothing to do
with the country of origin, but only refers to the difference in the type of contract
between the holder and the writer.
There are two sides to every option contract. On one side is the investor/holder
who has bought the option and has taken the long position. On the other side is the writer
of the option who has taken the short position by agreeing to sell should the holder
choose to buy (a call), and by agreeing to buy should the holder decide to sell (a put). The
writer receives cash up front for selling/writing the option but faces potential liabilities
later on. The writer’s profit or loss is the reverse of that for the holder of the option.
Thus, there are four possible option positions, the long position and short position for
each type of option, i.e. put and call.
Finally, there is the question of how much an option should be worth, and that
poses the original problem of the pricing of an option. The price of an option is a
function of,
1. the value of the underlying, and
2. the time to expiry.
Asset Name
Asset Value
Exercise Prices
Expiry Dates
Call Values
C(S, t)
Put Values
P(S, t)
Mathematical Formulation and analytical solution
One of the governing equations in the option-pricing problem is the Black-Scholes
equation:
V 1 2 2  2V
V
  S

rS
 rV  0
t 2
S
S 2
The terms are explained as follows:

V is the value of the option at time t, when the value of the underlying is S. V(S,t) is
denoted by C(S,t) for a call and by P(S,t) for a put,

 is the volatility of the underlying. It is a measure of the uncertainty of the future
value of the underlying. Large values of , for a particular S, reflect an increased
chance that S will either do very well or very poorly.
For the owner of the
underlying, S, this uncertainty has an equal effect on both outcomes. However, for
the owner of a call option on S the effect of  is unequal, because if S did well, the
owner would benefit to the extent of the price increase, but if S fared poorly, the
owner‘s loss is only limited to the price of the option. The exact opposite holds for a
put. The values of both calls and puts therefore increase as volatility increases.

r is the risk-free interest rate,

E is the exercise price or strike price.
Finally, an imporant note on upper and lower bounds for options. The value of a call
option can never be worth more than the stock price. In effect, S is an upper bound for a
call and C(S,t)  S is always true. Similarly, the value of a put option can never be worth
more than the exercise price, E, regardless of how low S becomes, therefore, E is an
upper bound for a put, and P(S,t)  E is always true. At expiry, the value of a call or a put
option to the holder (sometimes referred to as payoff) is:
C ( S , t )  max( S  E ,0)
P( S , t )  max( E  S ,0)
The value for a call is given by:
C (S , t )  SN (d1 )  Ee r (T t ) N (d 2 )
while the value for a put is given by:
P(S , t )  Ee r (T t ) N (d 2 )  SN (d1 )
The function N() is the cumulative distribution of a normal distribution function:
1
2
N ( x) 
x
e
1
 y2
2
dy

and the terms d1 and d2 are:
1 

ln S / E    r   2 T  t 
2 

d1 
 T t
and,
1 

ln S / E    r   2 T  t 
2 

d2 
 d1   T  t
 T t
where (T-t) is the time to expiration
Compare to the general 1-D heat conduction equation:
  T 
T
0
k
  g  x, t   C P
x  x 
t
For the case where thermal conductivity, k, is constant, i.e. k  k(x), and there is no heat
generation, g(x,t), the equation reduces to,
 2 T 1 T

 0 , for x, t > 0,
x 2  t
where:

T is the temperature at a point x in the medium at time t

 is the thermal diffusivity of the medium,

 is the density,

Cp is the specific heat at constant heat.
The solution for the above 1D heat conduction equation is,

T  x, t    c m X  m , x e  m t
2
m 1
where X(m,x) is the eigenfunction with eigenvalues, m, and
cm 
1
L
N  m  0
X  m , x'F  x'dx' , and the norm N(m) is defined as,
L
N  m    X  m , x  dx
2
0
The Black-Scholes equation and the 1D heat conduction are both second order, linear,
partial differential equations. The are both solved by their respective sets of boundary
conditions.
For the Black-Scholes equation governing the option pricing problem, the initial
conditions are the price, S, of the asset at time t = 0, the volatility , and the risk-free
interest rate r, at time t = 0. The only other parameter that the option price is a function
of is the time to expiry, i.e. t = T. However, at expiry, the another set of conditions must
be met and those are the upper and lower bounds that were mentioned earlier, i.e.
C ( S , t )  max( S  E ,0)
P( S , t )  max( E  S ,0)
for a call and a put respectively. There are additional constraints surrounding the option
pricing model depending on whether we are talking of European or American option, but
those involve the time-value-of-money concept, which is not really relevant to our goal.
For the 1D heat conduction problem, the boundary conditions are generally of 3 types;
the first is one where the temperature at a boundary is specified, the second is one where
the heat flux at the boundary is specified, and the third is one where a convective
condition at a boundary is specified. Depending on the type of problem being solved, the
appropriate eigenfunctions and eigenvalues may be obtained from tables that describe the
various boundary conditions surrounding an object in a medium. A later section will
describe a particular heat conduction problem whose boundary conditions are similar to
that of an option pricing problem.
Numerical Solution of an American Put Option:
Analytical solutions to the Black-Scholes exist for European Calls, European Puts, and
American Calls, however there is no exact solution for pricing American Put options.
This is due to certain boundary conditions that relate to American Put options. To find
the pricing for American Put options, finite difference methods can be used to solve the
Black-Scholes differential equation.
This process of using finite difference methods can be demonstrated by solving an option
pricing for a specific example. The example will be solving pricing values for an
American Put option. Let’s say that we are given an American Put option with an
exercise price of $50, with an expiry date in 5 months, an interest rate of 10%, and an
asset volatility of 40%. We wish to calculate the option pricing for the given information.
E = $50.00
T = 5 months (0.4167 year)
 = 0.40
r = 0.10
The black-scholes equations governs this relationship:
f
f 1 2 2  2 f
 rS
  S
 rf
t
S 2
S 2
If a scale between 0 and 100 is used for S, we know that the maximum value of the price
will be $50 when S=0, because this is for an American put option. And we also know
that the minimum value of the price will be $0 as the price S approaches infinity (For
simplicity, we can say when S=$100).
When the time t equals the expiry date T, we know what the value pricing should be. For
example when S=0, The option price will be 50, and when S=50 or higher the option
pricing will be $0.
Essentially, this all translates to all the boundary conditions for our problem:
Option value: p(S,t) = 0
when S = 100
Option value: p(S,t) = 50
when S = 0
Option value: p(S,t) = 50-S
when 0  S  50
Option value: p(S,t) = 0
when S > 0
Now that we know the b.c.’s, we can use finite difference methods to solve for everything
in between. We can use the following finite difference representation in the BlackScholes equation:
f i n 1  f i n
f

t
t
n
f i n  f i 1
f

S
S
f i n  2 f i n1  f i n 2
2 f

S 2
(S ) 2
Therefore the Black-Scholes equation becomes:
f i n1  f i n
f n  f i n1  2 S 2 ( f i n  2 f i n1  f i n2 )
 rS i

 rf i n
2
t
S
2
(S )
Solving for f i n we get:
f i n 1 rS n 1  2 S 2


f i 1 
(2 f i n1  f i n 2 )
2
t
S
2 (S )
fi n 
 1 rS 1  2 S 2

  

 r 
2
 t S 2 (S )

where we can say: S = 5 and t = 0.04167 (every ½ month)
A matrix chart can be created using Excel and the above formula for f i n . The result is
figure (2) on the next page. This shows the pricing values for a American Put when the
Expiry date is in 5 months and the Expiry price is $50.
Table pricing, p(S,t), for an American Put
E = $50
S= 100
S= 95
S= 90
S= 85
S= 80
S= 75
S= 70
S= 65
S= 60
S= 55
S= 50
S= 45
S= 40
S= 35
S= 30
S= 25
S= 20
S= 15
S= 10
S= 5
S= 0
5
0.00
0.01
0.04
0.09
0.13
0.30
0.48
1.07
1.60
1.99
2.21
7.14
12.13
17.13
22.12
27.12
32.16
37.24
42.16
46.28
50.00
4.5
0.00
0.02
0.02
0.07
0.15
0.21
0.41
0.96
1.50
1.90
2.02
6.92
11.92
16.91
21.90
26.91
31.94
37.00
41.94
46.22
50.00
4
0.00
0.02
0.03
0.04
0.08
0.17
0.37
0.85
1.33
1.75
1.85
6.70
11.70
16.70
21.69
26.69
31.71
36.76
41.72
46.10
50.00
3.5
0.00
0.00
0.02
0.03
0.08
0.15
0.34
0.78
1.16
1.45
1.69
6.48
11.48
16.48
21.48
26.48
31.49
36.53
41.51
46.02
50.00
T = 5 months
r = 10%
 = 40%
Time to Maturity (Months)
3
2.5
2
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.00
0.00
0.05
0.03
0.01
0.11
0.09
0.08
0.28
0.25
0.26
0.70
0.61
0.55
1.11
1.03
0.85
1.39
1.40
1.12
1.55
1.44
1.36
6.27
6.06
5.84
11.27 11.06 10.84
16.27 16.05 15.84
21.27 21.05 20.84
26.27 26.05 25.84
31.27 31.06 30.84
36.30 36.08 35.85
41.29 41.07 40.86
45.89 45.77 45.70
50.00 50.00 50.00
Figure (2)
1.5
0.00
0.00
0.00
0.00
0.00
0.02
0.09
0.18
0.49
0.82
1.15
5.63
10.63
15.63
20.63
25.63
30.63
35.64
40.64
45.65
50.00
1
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.11
0.25
0.45
0.92
5.42
10.42
15.42
20.42
25.42
30.42
35.43
40.42
45.30
50.00
0.5
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.03
0.21
0.63
5.21
10.21
15.21
20.21
25.21
30.21
35.21
40.21
45.05
50.00
0
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
45.00
50.00
Analogy of an American Put Example to a Heat Conduction Example
In the last section an example was given for solving pricing options for an American Put
using numerical methods. Because of the close similarity between the Black-Scholes
equation and the heat conduction equation, it can be said that our American Put example
is analogous to a real-life heat conduction problem. The analogy would simply involve
replacing option pricing terms with heat conduction terms. And also, the boundary
conditions would remain the same for both types of examples.
The switch between pricing option terms with heat conduction terms can be made by
letting:
p=T
American Put option price = Temp
S=x
stock price = x-coordinate
E=E
Expiry price = Expiry Temp
t= t
time = time
T=T
Expiry time = final time
The formulation equations would be the same between the two examples with the
exception of the Black-Scholes equation being replaced by the heat conduction equation.
The boundary conditions are the same for our American Put example and our heat
conduction example. So essentially the solution to our heat conduction example will be
exactly the same as the solution to our American Put example. This means that the
resulting solution for our new heat conduction problem will be the same as figure (3). By
observing figure (3) we can describe the formulation equation and boundary conditions
for our heat conduction problem. They are:
 2 T ( x, t ) 1 T ( x, t )

 t
x 2
when 0  t < T
Between 0 < x  d (t)
T(x,t) = 0
when 0  t  T
Between d (t) < x <2 E
T(x,t) = E
when 0  t < T
At x = 0
T(x,t) = 0
when 0  t < T
At x = 2 E
T(x,t) = F(x)
when t = T
Between 0 < x  d (t)
Where d (t) is a changing distance, as a function of time.
The above equations govern the matrix chart of figure (3) if we pretend the chart to be for
a heat conduction problem.
One heat conduction problem, which obeys the figure (3) and the governing equations, is
a simple cylindrical rod being slowly dipped into an ice bath reservoir figure (4). Figure
(4) shows this example where at first t = 0 and later when t = T.
figure (1)
At t = 0, the cylindrical rod is barely touching an ice bath reservoir which happens to be
at 0 temp and has h =  because of a large flow rate. At the x = 0 section of the rod a
heat source is applied to provide a constant temp of E, or as in our case 50. The rest of
the exterior is surrounded by insulating foam when it lies above the water, so that no heat
is transferred to or from the outer diameter.
The temperature distribution has been written on the cylindrical rod to better understand
how it changes as the rod is lowered into the ice bath.. This temperature distribution
follows the temperature distribution found in figure (3) for our option pricing example.
Therefore, the comparison between our temperature distribution for the rod is directly
analogous to that of a pricing option for an American Put. If we had used an American
Call for our comparison, the only change in our heat conduction analysis would have
been to have the x-coordinate start at the opposite end of the cylindrical rod.
Sources
1. Dewynne, Howison & Wilmott, The Mathematics of Financial Derivatives, A Student
Introduction, Cambridge University Press, 1997.
2. Hull, John C., Options, Futures, and other Derivative Securities, 2nd Ed., PrenticeHall, NJ, 1993.
3. Ozisik, M. Necati, Heat Conduction, 2nd Ed., John Wiley & Sons Inc., 1993
4. The Wall Street Journal, March 17, 2000.