Convert the Black-Scholes Equation to the Heat equation ** you MUST use the change of variables specified in the problem!!!

Problem - 3
Start with the Black-Scholes equation:
d 1 2 2 d2 d
V ( S , t) + S V ( S , t) + r V ( S , t) - r V ( S , t) = 0
dt 2 2 dS
dS
And it's peripheral details:
Boundry Condition: V ( 0 , t) = 0
Terminal Condition: V ( S , t) = max ( S - E , 0)
Range of S: 0 S <
Instead of S & t, introduce a pair of new independent variables:

X = ln
S
= T-t & = tao T and t - are just t
E
Since the above is equivalent to:
X
S = E e & t= T-

Any function of S and t can be re-expressed as a function of X and
, and vice versa. Thus, it makes sense to speak interchangeably

about V ( S , t) and V ( X , ).

Define the new quantity ( )
V X, = .
V
Then we have:
E

(
V ( S , t) = E V X , .)
Show that the original Black-Scholes System can now be recast into
2 d
2
the P.D.E: d
V X, =(
)
d2
V X , + r - ( )
V X , - r V X , ( ) ( )
d 2 dX2 2 dX



Hint: Remember the multi-variable chain rule from Calculus III:

d d V d X + d V d & d V = d V d X + d V d
V=
dt d X dt d dt dS d X dS d dS
_________________________________________________________________________________



d 1 2 2 d2 d
V ( S , t) + S V ( S , t) + r V ( S , t) - r V ( S , t) = 0
dt 2 2 dS
dS

S
X = ln = T-t
E

X
S = E e t= T-



This is as far as I've gotten!


Homework Set # 6 Page - 8 of 13
I'm pretty much lost from here on out. I've allready found other
examples of how to change the Black-Scholes eq. to the heat
equation with other - even with similar - changes of variables. But I
was just unable to follow the math, so in your awnser PLEASE BE
VERY SPECIFIC!!!! - If you're not EXPLICIT in your awnser - it will be
of no help to me.

Thanks for your time.

- also, if you feel this question warrents more credit's than I've
assigned to the problem - *(and I'm happy with the way the awnser is
presented), just tell me so, and I'll offer you more credits. Just let me
know how much work you feel you did and I'll credit you appropriatly. -
keep in mind however, that this is ONLY if I am able to CLEARLY
follow your awnser!

If on the other hand I don't recieve a professional - ie. pdf, or word
generated answer, I will revoke the creidts for the problem and
resubmit it! - So don't bother doing the problem if you're not sure what
I'm asking or can't present a professional answer!




Homework Set # 6 Page - 9 of 13