White R.E. Computational Mathematics: Models, Methods, and Analysis with MATLAB and MPI
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220 CHAPTER 5. EPIDEMICS, IMAGES AND MONEY
5.5.2 Application
We will focus on a particular option contract called an American put option.
This option contract obligates the writer of the contract to buy an underlying
asset from the holder of the contract at a particular price, called the exercise
or strike price,
H. This must be done within a given time, called an expiration
date,
W . The holder of the option contract may or may not sell the underlying
asset whose market value,
V, will vary with time. The value of the American
put option contract to the holder will vary with time, w> and V. If V gets large
or if w gets close to W , then the value of the American put option contract,
S (V> w), will decrease. On the other hand, if V gets small, then the value of the
American put option contract will increase towards the exercise price, that is,
S (V> w) will approach H as V goes to zero. If time exceeds expiration date, then
the American put option will be worthless, that is,
S (V> w) = 0 for w A W=
The objective is to determine the value of the American put option contract
as a function of
V, w and the other parameters H, W , u ( the interest rate) and
(the market volatility), which will be described later. In particular, the holder
of the contract would like to know when is the "best" time to exercise the
American put option contract. If the market value of the underlying contract
is well above the exercise price, then the holder may want to sell on the open
market and not to the writer of the American put option contract. If the market
price of the underlying asset continues to fall below the exercise price, then at
some "point" the holder will want to sell to the writer of the contract for the
larger exercise price. Since the exercise price is fixed, the holder will sell as
soon as this "point" is reached so that the money can be used for additional
investment. This "point" refers to a particular market value V = V
i
(w)> which
is also unknown and is called the optimal exercise price.
The writers and the holders of American put option contracts are motivated
to enter into such contracts by speculation of the future value of the underlying
asset and by the need to minimize risk to a portfolio of investments. If an
investor feels a particular asset has an under-valued market price, then entering
into American put option contracts has a p ossible value if the speculation is
that the market value of the underlying asset may increase. However, if an
investor feels the underlying asset has an over-priced market value, then the
investor may speculate that the underlying asset will decrease in value and may
be tempted to become holder of an American put option contract.
The need to minimize risk in a portfolio is also very important. For example,
suppose a portfolio has a number of investments in one sector of the economy.
If this sector expands, then the portfolio increases in value. If this sector con-
tracts, then this could cause some significant loss in value of the portfolio. If
the investor becomes a holder of American put option contracts with some of
the portfolio’s assets as underlying assets, then as the market values of the
assets decrease, the value of the American put option will increase. A proper
distribution of underlying investments and option contracts can minimize risk
to a portfolio.
© 2004 by Chapman & Hall/CRC
5.5. OPTION CONTRACT MODELS 221
5.5.3 Model
The value of the payo from exercising the American put option is either the
exercise price minus the market value of the underlying asset, or zero, that is,
the payo is max(H V> 0). The value of the American put option must almost
always be greater than or equal to the payo
. This follows from the following
risk free scheme: buy the underlying asset for
V, buy the American put option
contract for
S (V> w) ? max(H V> 0)> and then immediately exercise this option
contract for
H, which would result in a profit H S V A 0= As this scheme is
very attractive to investors, it does not exist for a very long time, and so one
simply requires
S (V> w) max(H V> 0)= In summary, the following conditions
are placed on the value of the American put option contract for w W . The
boundary conditions and condition at time equal to
W are
S (0> w) = H (5.5.1)
S (O> w) = 0 for O AA H (5.5.2)
S (V> W ) = max(H V> 0)= (5.5.3)
The inequality constraints are
S (V> w) max(H V> 0)
(5.5.4)
S
w
(V> w) 0 and S
w
(V
i
+> w) = 0 (5.5.5)
S (V
i
(w)> w) = H V
i
(w) (5.5.6)
S (V> w) = H V for V ? V
i
(w) (5.5.7)
g
gw
V
i
(w) A 0 exists. (5.5.8)
The graph of
S (V> w
The partial derivative of S (V> w) with respect to V needs to be continuous
at
V
i
so that the left and right derivatives must both be equal to -1. This
needs some justification. From (5.5.7) S
V
(V
i
> w) = 1 so we need to show
S
V
(V
i
+> w) = 1= Since S (V
i
(w)> w) = H V
i
(w)>
g
gw
S
(V
i
(w)> w) = 0
g
gw
V
i
(w)
S
V
(V
i
+> w)
g
gw
V
i
(w) + S
w
(V
i
+> w) =
g
gw
V
i
(w)
S
w
(V
i
+> w) = (1 + S
V
(V
i
+> w))
g
gw
V
i
(w)=
Since S
w
(V
i
+> w) = 0 and
g
gw
V
i
(w) A 0> 1 + S
V
(V
i
+> w) = 0.
In the region in Figure 5.5.1 where
S (V> w) A max(H V> 0)> the value of
the option contract must satisfy the celebrated Black-Scholes partial dierential
equation where
u is the interest rate and is the volatility of the market for a
particular underlying asset
S
w
+
2
2
V
2
S
VV
+ uVS
V
uS = 0= (5.5.9)
© 2004 by Chapman & Hall/CRC
) for fixed time should have the form giv en in Figure 5.5.1.
222 CHAPTER 5. EPIDEMICS, IMAGES AND MONEY
Figure 5.5.1: Value of American Put Option
The derivation of this equation is beyond the scope of this brief intro duction to
option contracts. The Black-Scholes equation di
ers from the partial dierential
equation for heat di
usion in three very important ways. First, it has variable
co e
!cients. Second, it is a backward time problem where S (V> w) is given at a
future time
w = W as S (V> W ) = max(H V> 0)= Third, the left boundary where
V = V
i
(w) is unknown and varies with time.
Black-Scholes Model for the American Put Option Contract.
S (V> W ) = max(H V> 0) (5.5.10)
S (0> w) = H (5.5.11)
S (O> w) = 0 for O AA H (5.5.12)
S (V
i
(w)> w) = H V
i
(w) (5.5.13)
S
V
(V
i
±> w) = 1 (5.5.14)
S (V> w) max(H V> 0) (5.5.15)
S = H V for V V
i
(w) (5.5.16)
S
w
+
2
2
V
2
S
VV
+ uVS
V
uS = 0 for V A V
i
(w)= (5.5.17)
The volatility
is an important parameter that can change with time and
can be di
!cult to approximate. If the volatility is high, then there is more
uncertainty in the market and the value of an American put option contract
should increase. Generally, the market value of an asset will increase with time
according to
g
gw
V
= V=
© 2004 by Chapman & Hall/CRC
5.5. OPTION CONTRACT MODELS 223
The parameter
can be approximated by using past data for V
n
= V(nw)
V
n+1
V
n
w
=
n
V
n
=
The approximation for is given by an average of all the
n
=
1
N
N1
X
n=0
n
=
1
Nw
N1
X
n=0
V
n+1
V
n
V
n
= (5.5.18)
The volatility is the square root of the unbiased variance of the above data
2
=
1
(N 1)w
N1
X
n=0
µ
V
n+1
V
n
V
n
w
¶
2
= (5.5.19)
Thus, if
2
is large, then one may expect in the future large variations in the
market values of the underlying asset. Volatilities often range from near 0=05
for government bonds to near 0
=40 for venture stocks.
5.5.4 Method
The numerical approximation to the Black-Scholes model in (5.5.10)-(5.5.17)
is similar to the explicit method for heat diusion in one space variable. Here
we replace the space variable by the value of the underlying asset and the
temperature by the value of the American put option contract. In order to
obtain an initial condition, we replace the time variable by
W w= (5.5.20)
Now abuse notation a little and write S(V> ) in place of S (V> w) = S (V> W )
so that
S
replaces S
w
in (5.5.17). Then the condition at the exercise date in
(5.5.10) becomes the initial condition. With the boundary conditions in (5.5.11)
and (5.5.12) one may apply the explicit finite dierence method as used for the
heat di
usion model to obtain S
n+1
l
approximations for S (lV> (n + 1))=
But, the condition in (5.5.15) presents an obstacle to the value of the option.
Here we simply choose the
S
n+1
l
= max(H V
l
> 0) if S
n+1
l
? max(H V
l
> 0)=
© 2004 by Chapman & Hall/CRC
224 CHAPTER 5. EPIDEMICS, IMAGES AND MONEY
Explicit Method with Projection for (5.5.10)-(5.5.17).
Let
(@(V)
2
)(
2
@2)=
S
n+1
l
= S
n
l
+ V
2
l
(S
n
l
1
2S
n
l
+ S
n
l
+1
)
+ (
@V) uV
l
(S
n
l
+1
S
n
l
) uS
n
l
= V
2
l
S
n
l
1
+ (1 2V
2
l
(@V) uV
l
u)S
n
l
+(V
2
l
+ (@V) uV
l
)S
n
l
+1
S
n+1
l
= max(S
n+1
l
> max(H V
l
> 0))=
The conditions (5.5.13) and (5.5.14) at V = V
i
do not have to be explicitly
implemented provided the time step is suitably small. This is another
version of a stability condition.
Stability Condition.
(@(V)
2
)(
2
@2)
1
2V
2
l
(@V) uV
l
u A 0=
5.5.5 Implementation
The MATLA B code bs1d.m is an implementation of the explicit method for the
American put option contract model. In the code the array
{ corresponds to
the value of the underlying asset, and the array x corresponds to the value of
the American put option contract. The time step,
gw, is for the backward time
step with initial condition corresponding to the exercise payo
of the option.
Lines 1-14 define the parameters of the model with exercise price
H = 1=0. The
payo obstacle is defined in lines 15-20, and the boundary conditions are given
in lines 21 and 22. Lines 23 -37 are the implementation of the explicit scheme
with projection to payo
obstacle given in lines 30-32. The approximate times
when market prices correspond to the optimal exercise times are recorded in
lines 33-35. These are the approximate points in asset space and time when
(5.5.13) and (5.5.14) hold, and the output for time versus asset space is given
in figure(1) by lines 38 and 39. Figure(2) generates the value of the American
put option contract for four di
erent times.
MATLAB Code bs1d.m
1. % Black-Scholes Equation
2. % One underlying asset
3. % Explicit time with projection
4. sig = .4
5. r = .08;
6. n = 100 ;
7. maxk = 1000;
8. f = 0.0;
9. T = .5;
10. dt = T/maxk;
© 2004 by Chapman & Hall/CRC
5.5. OPTION CONTRACT MODELS 225
11. L = 2.;
12. dx = L/n;
13. alpha =.5*sig*sig*dt/(dx*dx);
14. sur = zeros(maxk+1,1);
15. % Define the payo
obstacle
16. for i = 1:n+1
17. x(i) = dx*(i-1);
18. u(i,1) = max(1.0 - x(i),0.0);
19. suro(i) = u(i,1);
20. end
21. u(1,1:maxk+1) = 1.0; % left BC
22. u(n+1,1:maxk+1) = 0.0; % right BC
23. % Use the explicit discretization
24. for k = 1:maxk
25. for i = 2:n
26. u(i,k+1) = dt*f+...
27. x(i)*x(i)*alpha*(u(i-1,k)+ u(i+1,k)-2.*u(i,k))...
28. + u(i,k)*(1 -r*dt) ...
29. -r*x(i)*dt/dx*(u(i,k)-u(i+1,k));
30. if (u(i,k+1)
?suro(i)) % projection step
31. u(i,k+1) = suro(i);
32. end
33. if ((u(i,k+1)
Asuro(i)) & (u(i,k)==suro(i)))
34. sur(i) = (k+.5)*dt;
35. end
36. end
37. end
38. figure(1)
39. plot(20*dx:dx:60*dx,sur(20:60))
40. figure(2)
41. %mesh(u)
42. plot(x,u(:,201),x,u(:,401),x,u(:,601),x,u(:,maxk+1))
43. xlabel(’underlying asset’)
44. ylab el(’value of option’)
45. title(’American Put Option’)
values of the American put option contracts are increasing with respect to
(decreasing with respect to time w). Careful inspection of the curves will verify
that the conditions at
V
i
in (5.5.13) and (5.5.14) are approximately satisfied.
at time
w = 0=5 and variable volatilities = 0=4> 0=3> 0=2 and 0=1= Note as the
volatility decreases, the value of the option contract decreases towards the payo
value. This monotonicity prop erty can be used to imply volatility parameters
based on past market data.
© 2004 by Chapman & Hall/CRC
Figure 5.5.2 contains the output for the above code where the curves for the
Figure 5.5.3 has the curves for the value of the American put option contracts
226 CHAPTER 5. EPIDEMICS, IMAGES AND MONEY
Figure 5.5.2: P(S,T-t) for Variable Times
Figure 5.5.3: Option Values for Variable Volatilities
© 2004 by Chapman & Hall/CRC
5.5. OPTION CONTRACT MODELS 227
Figure 5.5.4: Optimal Exercise of an American Put
Figure 5.5.4 was generated in part by figure(1) in bs1d.m where the smooth
curve represents the time when
V equals the optimal exercise of the American
put option contract. The vertical axis is
= W w> and the horizontal axis
is the value of the underlying asset. The non smooth curve is a simulation of
the daily market values of the underlying asset. As long as the market values
are above
V
i
(w)> the value of the American put option contract will be worth
more than the value of the payo
, max(H V> 0)> of the American put option
contract. As soon as the market value is equal to
V
i
(w)> then the American
put option contract should be exercised This will generate revenue H = 1=0
where the
w is about 0.06 before the expiration date and the market value
is about
V
i
(w) = 0=86= Since the holder of the American put option contract
will give up the underlying asset, the value of the payo
at this time is about
max(H V> 0) = max(1=0 0=86> 0) = 0=14=
5.5.6 Assessment
As usual the parameters in the Black-Scholes model may depend on time and
may be di
!cult to estimate. The precise assumptions under which the Black-
Scholes equation models option contracts should be carefully studied. There are
a numb er of other option contracts, which are di
erent from the American put
option contract. Furthermore, there may be more than one underlying asset,
and this is analogous to heat di
usion in more than one direction.
The question of convergence of the discrete model to the continuous model
needs to be examined. These concepts are closely related to a well studied
© 2004 by Chapman & Hall/CRC
228 CHAPTER 5. EPIDEMICS, IMAGES AND MONEY
applied area on "free boundary value" problems, which have models in the
form of variational inequalities and linear complementarity problems. Other
applications include mechanical obstacle problems, heat transfer with a change
in phase and fluid flow in earthen dams.
5.5.7 Exercises
1. Experiment with variable time and asset space steps.
2.
3. Experiment with variable interest rates
u.
4. Experiment with variable exercise values
H=
5. Experiment with variable expiration time W> and examine figure(1) gen-
erated by bs1d.m.
5.6 Black-Scholes Model for Two Assets
5.6.1 Introduction
A portfolio of investments can have a number of assets as well as a variety of
option contracts. Option contracts can have more than one underlying asset
and di
erent types of payo
The Black-Scholes two assets model is similar to the heat diusion model in
two space variables, and the explicit time discretization will also be used to
approximate the s olution.
5.6.2 Application
Consider an American put option contract with two underlying assets and a
payo
function max(H V
1
V
2
> 0) where H is the exercise price and V
1
and
V
2
are the values of the underlying assets. This is depicted in Figure 5.6.1
where the tilted plane is the positive part of the payo
function. The value of
the put contract must be ab ove or equal to the payo
function. The dotted
curve indicates where the put value separates from the payo
function; this
is analogous to the
V
i
(w) in the one asset case. The dotted line will change
with time so that as time approaches the expiration date the dotted line will
move toward the line
H V
1
V
2
= 0 = S . If the market values for the two
underlying assets at a particular time are on the dotted line for this time, then
the option should be exercised so as to optimize any profits.
5.6.3 Model
Along the two axes where one of the assets is zero, the model is just the
Black-Scholes one asset model. So the boundary conditions for the two as-
set mo del must come from the solution of the one asset Black-Scholes model.
© 2004 by Chapman & Hall/CRC
Duplicate the calculations in Figures 5.5.2 and 5.5.3.
functions such as illustrated in Figures 5.6.1-5.6.3.
5.6. BLACK-SCHOLES MODEL FOR TWO ASSETS 229
Figure 5.6.1: American Put with Two Assets
Let
S (V
1
> V
2
> w) be the value of the American put option contract. For positive
values of the underlying assets the Black-Scholes equation is
S
w
+
2
1
2
V
2
1
S
V
1
V
1
+
1
1
12
V
1
V
2
S
V
1
V
2
+
2
2
2
V
2
2
S
V
2
V
2
+uV
1
S
V
1
+uV
2
S
V
2
uS = 0=
(5.6.1)
The following initial and boundary conditions are required:
S (V
1
> V
2
> W ) = max(H V
1
V
2
> 0) (5.6.2)
S (0> 0> w) = H (5.6.3)
S (O> V
2
> w) = 0 for O AA H (5.6.4)
S (V
1
> O> w) = 0 (5.6.5)
S (V
1
> 0> w) = from the one asset model (5.6.6)
S (0> V
2
> w) = from the one asset model. (5.6.7)
The put contract value must be at least the value of the payo
function
S (V
1
> V
2
> w) max(H V
1
V
2
> 0)= (5.6.8)
Other payo
functions can be used, and these will result in more complicated
boundary conditions. For example, if the payo
function is pd{(H
1
V
1
> 0=0)+
pd{(H
2
V
2
> 0=0)> then S (O> V
2
> w) and S (V
1
> O> w) will be nonzero solutions of
two additional one asset models.
Black Scholes Model of an American Put with Two Assets.
Let the payo function be max(H V
1
V
2
> 0)=
Require the inequality in (5.6.8) to hold.
The initial and boundary condition are in equations (5.6.2)-(5.6.7).
Either
S (V
1
> V
2
> w) = H V
1
V
2
or S (V
1
> V
2
> w) satisfies (5.6.1).
© 2004 by Chapman & Hall/CRC