Black-Scholes Model - Solutions
Throughout this exercise you may use assume (where appropriate) the following results without proof
d1
=
d2
=
N (x)
=
log (S=E) + r D +
p
T t
log (S=E) + r D
p
T t
Z x
2
1
=2
p
e
d
2
1
1
2
2
1
2
2
(T
t)
(T
t)
,
and
where S
0 is the spot price, t
T is the time, E > 0 is the strike, T > 0
the expiry date, r
0 the interest rate, D is the dividend yield and is the
volatility of S.
1. The Black–Scholes formula for a European call option C (S; t) is given by
C (S; t) = S exp( D(T
t))N (d1 )
E exp( r(T
t))N (d2 ):
a) By di¤erentiating with respect to S and show that the delta and vega
are given by
r
d12
T t ( D(T t))
2
= e( D(T t)) N (d1 ) ; and
=
Se
e
:
2
Note:
@d1
@d2
=
@S
@S
and
@d2
@d1
=
@
@
p
T
t
So
=
@C
@S
( D(T
t))
( D(T
t))
= e
= e
= e(
D(T
t))
( D(T
N (d1 ) + Se
t))
1
p e
2
1 @d1
N (d1 ) + p
Se(
2 @S
|
D(T
d12
2
t))
e
@d1
@S
E exp( r(T
d12
2
Ee(
{z
r(T
t))
=0
N (d1 ) because the term in the bracket above is zero.
1
e
1
t)) p e
2
!
d 2
2
2
}
d22
2
@d2
@S
=
@C
@
= Se(
D(T
t))
= Se(
D(T
t))
=
=
r
r
T
t
2
T
t
2
Se(
d12
2
1
p e
2
d12
2
1
p e
2
D(T
( D(T
Se
@d1
@
t))
t))
e
e
Ee(
r(T
@d2 p
+ T
@
d12
2
+
d12
2
t))
d22
2
1
p e
2
t
2
1
p Ee(
2
@d2 1 4 ( D(T
p
Se
|
@
2
r
T t ( r(T
=
Ee
2
t))
@d2
@
r(T
e
t))
d22
2
e
d12
2
{z
Ee(
=0
t))
e
d22
2
!
2. Given that S is de…ned by the SDE
dS = a (S; t) dt + b (S; t) dW
(2.1)
where a and b are given functions of S and t; show using Itô’s lemma that
any function V (S; t) satis…es the SDE
dV =
@V
1 @2V
@V
+a
+ b2 2
@t
@S
2 @S
dt + b
@V
dW
@S
where we have assumed that all partial derivatives exist.
Hence derive the partial di¤erential equation
@V
1 @2V
+ b2 2 = r V
@t
2 @S
S
@V
@S
(2.2)
for the fair price of an option based on a security S which satis…es (2:1)
with r the risk-free interest rate.
Show (by substitution) that V (S; t) = e t S 2 is a solution of (2:2) provided
b2 = (
r) S 2
and is a constant.
The …rst part of this problem is trivial. Follow the derivation of the
BSE as done in the notes. The only di¤erence here is that a (S; t) and
b (S; t) replace S and S: For the second part simply substitute V (S; t) =
e t S 2 in (2:2) ; the following terms are needed
@V
=
@t
e
t
e
1
S 2 + b2
2
t
S2;
2e
@V
= 2e
@S
t
S 2 + b2
2
t
= r e
=
@2V
= 2e
@S 2
S;
t
S2
2e
rS 2 ! b2 = (
t
t
S2
r) S 2 :
@d2
@
r(T
t))
e
d22
2
}
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5
3. The Black–Scholes formula for a European call option C (S; t) is
C (S; t) = S exp( D(T
t))N (d1 )
E exp( r(T
t))N (d2 )
From this expression, …nd the Black–Scholes value of the call option in
the following limits:
a. (time tends to expiry) t ! T , > 0 (this depends on S=E);
exp ( r(T t)), exp( D(T t)) ! 1
We know
d12
d12
log (S=E) + r D 21 2 (T t)
p
T t
r D 12 2 (T t)
log (S=E)
p
p
=
+
T t
T t
1 2 (T t)
log (S=E)
p
p
+ r D
=
2
T t
T t
1 2 p
p
r
D
log (S=E)
log (S=E)
2
p
+
+O
=
T t= p
T t
T t
T t
8
8
< S E S>E
< 1 S>E
p
log (S=E)
0
S=E
0 S=E
! p
+O
so C !
T t !
:
:
T t
0
S<E
1 S<E
=
b. (volatility tends to zero) ! 0+ , t < T ; (this depends on S exp( D(T
t))=E exp( r(T t)))
Again start with what we know, i.e.
d12
=
=
=
log (S=E) + r D 21 2 (T
p
T t
log (S=E) (r D) (T t)
p
p
+
T t
T t
log (S=E) (r D) (T t)
p
p
+
T t
T t
t)
1 2 (T t)
p
2
T t
p
1
T t
2
log (S=E) + (r D) (T t)
p
+O( )
T t
Now we employ a small trick in the …rst quotient
d12 !
t))=E exp( r(T t)))
p
+O( )
T t
8
Se( D(T t)) > Ee( r(T t))
< 1
!
0
Se( D(T t)) = Ee( r(T t))
:
1
Se( D(T t)) < Ee( r(T t))
h
i
! max Se( D(T t)) Ee( r(T t)) ; 0
=
so C
log (S exp( D(T
3
4. Suppose S evolves according to the SDE
dS = Sdt + S dW
where
and
are positive constants. Given that the interest rate is
zero, derive the corresponding Black–Scholes partial di¤erential equation
(PDE) for the option based upon this asset S (you are not required to
solve any equation). Write this PDE in terms of the greeks.
We know that if S evolves according to the stochastic di¤erential equation
(SDE)
dS = a (S; t) dt + b (S; t) dW
and V = V (S; t) then Itô gives
@V
1
@2V
@V
+ S
+ S2
@t
@S
2
@S 2
dV =
Then set up a portfolio
d = dV
dS:
So
d
=
=V
dt + S
@V
dW
@S
S ) in one time-step (we hold
@V
@V
1
@2V
+ S
+ S2
@t
@S
2
@S 2
@V
dW
@S
dt + S
…xed)
( Sdt + S dW )
therefore we take = @V
@S to eliminate the risk associated with the portfolio (to cancel out terms with dX), which gives
d
=
1
@V
@2V
+ S2
@t
2
@S 2
dt
portfolio now riskless. No arbitrage tells us that we are guaranteed return
at risk free rate, so
1
@V
@2V
+ S2
@t
2
@S 2
and r = 0 so
Using greeks
dt = r dt
@V
1
@2V
+ S2
= 0.
@t
2
@S 2
@V
@t
=
and
@2V
@S 2
=
allows us to write this pde as
1
+ S2
2
= 0.
5. The value of an option V (S; t) satis…es the Black–Scholes equation. Write
the option value in the form
V (S; t) = exp( r(T
4
t))q(S; t):
(5.1)
Show that the function q(S; t) satis…es the equation
@q 1
+
@t
2
2
S2
@2q
+ (r
@S 2
D)S
@q
= 0:
@S
This is the backward Kolmogorov equation, used for calculating the expected value of stochastic quantities.
Substitute
@V
@t
@V
@S
@2V
@S 2
=
exp( r(T
=
exp( r(T
=
exp( r(T
@
q(S; t) + rV (S; t);
@t
@q
t))
&
@S
@2q
t)) 2
@S
t))
from (5:1) into the BSE, all the exponentials cancel out and the above
equation is left.
Thus the value of an option can be expressed in the form
V (S; t) = exp ( r(T
t)) E [Payo¤(S)]
where E[x] means the expected value of x . This is not a real expectation,
but taken under the risk-neutral random walk (so r replaces ) and forms
the basis of Monte Carlo methods applied to …nance. More on this later.
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