We will start the section on the partial differential equation approach to derivative asset
pricing exploring the derivatives of the Black-Scholes model. These derivatives are
important in their own right. The delta, gamma and theta of an option are crucially
important in risk management applications.
Our principal purpose will be to identify the methods necessary to validate the
fundamental partial differential equation. A partial differential equation is a relationship
between first and higher order derivatives of a function. If we are lucky the differential
equation and boundary conditions can be used to recover the function. If we are not so
lucky numerical methods will be necessary to evaluate the function through the
differential equation.
  ( x  u) 2 
1
exp 
Normal density function: f ( x) 

2
 2
 2

Black-Scholes Model:
C 0  S 0 N (d1 )  ke rT N (d 2 )
P0  ke rT N (d 2 )  S 0 N (d1 )
N  N (0,1)
S
S
ln( 0 )  (r  1  2 )T ln( 0  rT )
k
ke
2
d1 

 1 T
2
 T
 T
S
S
ln( 0 )  (r  1  2 )T ln( 0  rT )
k
ke
2
d 2  d1   T 

 1 T
2
 T
 T
Partial derivatives of the Black-Scholes Model
d i  f (S 0 , k , r, 2 , T )

  d 
1
N (d i )
d
 N (d i ) i  (2 ) 2 exp(  1 d i2 )  i 
2
v
v 
  v 
Delta = =
C
:
S 0
d
d
C
 N (d1 )  S 0 N (d1 ) 1  ke rT N (d 2 ) 2
S 0
S 0
S 0


d i
1
 S 0 T
S 0
1
2
 ln( S 0
)
S
2 
ke rT 
1 2
d1  
  ln( 0  rT )  4  T
ke
 T




2
 ln( S 0
)
S
2 
ke rT 
1 2
d2  
  ln( 0  rT )  4  T
ke
 T




1
N (d1 )  2  2 * exp  1 ( A  ln( S 0 ke rT )  B)
2
1
N (d 2)  2  2 * exp  1 ( A  ln( S 0 ke rT )  B)
2
ln N (d1 )  ln N (d 2 )   ln( S 0 ke rT )
C0
d
)  ln N (d1 )  ln( S 0 )  ln N (d1 )  ln 1  ln ke rT  ln N (d 2 )  ln d 2 
S 0
S 0 
S 0 
C
ln( 0 )  ln N (d1 )
S 0
ln(
C 0
S 0
   N (d1 )  0
1
 2C0
   (2 ) 2 * exp(  1 d12 ) * ( S 0 T ) 1  0
2
S 0 2
T = time to maturity = (-t). 
C 0
t
1
2
1
2T
C 0
C
 0
T
t
S 0 exp(  1 d12 )  rKe  rT N (d 2 )  0
2
The partial differential equation developed later for European derivatives is a linear,
second-order equation of the parabolic type. If the Black-Scholes model is a solution to
this partial differential equation, , ,  should satisfy the PDE.
Additional derivatives of the Black-Scholes model are important in their own right and
deserve to be mentioned.
C 0
 Vega  S 0 T N (d1 )  0

C 0
 rho  KTe  rT N (d 2 )  0
r
C 0
 e  rT N (d 2 )  0
K
2
1. Varian 1987 shows that the second derivative of the option’s price with
respect to the exercise price is the state price. Derive the expression for the
second-partial derivative of the European call option’s price with respect to
the strike price.
2. Calculate the value of the partial derivatives wrt (S0,k,r,t, ) of C0=f(S0,
k,r,T,) and the second-partial derivative with respect to S0 and k for the
Black-Scholes model using:
S0
k
T
r

Call option 1
$50
$50
0.5 years
3%
20%
Call option 2
$50
$45
0.5 years
3%
20%
Call option 3
$50
$55
0.5 years
3%
20%
3. Derive the expression for the partial derivative of a European put option
with respect to the price of the underlying asset and the second-partial of a
European put option with respect to the price of the underlying asset.
Development of the partial differential equation approach to derivative valuation.
Ito’s Lemma: Chp 10 Neftci page 240:
Let F(St,t) be a twice-differentiable function of t and of the random process St:
dS t  at dt   t dWt then
2
dFt  F dS t  F dt  1  F  t2 dt after substituting for dSt
S t
t
2 S 2
t


2
dFt   F at  F  1  F  t2  dt  F  t dWt
t
2 S 2
S t
 S t

t
Neftci Chp 12 page 276: Forming Risk Free portfolios:
Let dSt be the SDE (equation of motion) for the underlying asset. Because Ito’s lemma
indicates the SDE of the derivative is a function of the same stochastic process, dWt, it
will in general be possible to form a portfolio of the derivative and underlying asset such
that the stochastic component’s cancel out. To prevent arbitrage opportunities the hedge
portfolio must earn the risk free rate of return.
Given portfolio weights, 1 , 2 the value of the portfolio is then; Pt = 1F + 2S
Assuming the portfolio weights are constant (independent of S,t);
3
dPt = 1dF + 2dS. Choosing weights such that 1 = 1, 2 = -Fs makes the portfolio
stochastic components, dWt, cancel out.

dP   F

dPt  Fs at  1 Fss  t2  Ft dt  Fs t dWt  Fs at dt   t dWt 
2
1
2
t
2
ss  t

 Ft dt
To prevent arbitrage this portfolio must earn the risk free rate; dPt = rPtdt .


rPt dt  1 Fss t2  Ft dt
2


rF  Fs S dt  1 Fss t2  Ft dt
2
Fundamental PDE for a European derivative asset on non-payout asset;
 rF  rFs S  Ft  1 Fss  t2  0
2
s.t.F ( ST , T )  G ( ST , T )
Fs  , Ft  , Fss  
The fundamental PDE for European derivative asset on a non-payout asset is a linear
second-order parabolic PDE. Each derivative is unique in the specification of the
dependence of the terminal payoff on the ST, G(ST , T).
Stochastic processes important in financial modelingNeftci page 176-177:
A standard Wiener process (Brownian Motion) is appropriate when the random events
being described can only change continuously and during a small time interval small
changes in the process are generally observed.
A Wiener process, Wt , is the limiting process for a binomial process that takes values
Wt  h , h as the interval 0,……,T is divided into n subintervals of length h, i.e.
n
h  T . If the increments are independent then the sum, Wt n   Wt i will converge
n
i 1
weakly to a Weiner process in the limit as n. The Wiener process is obtained as the
limit of a sum of independent, identically distributed random variables.



A Wiener process is identical to a Brownian motion
Wt is a martingale.
Wt has uncorrelated increments
4



Wt is continuous
Wt has zero mean
Wt has variance t
W (t  1)  W (t )   (t  1)
 ~ iid
N (0,1)
W (t  1)  W (t  1)  W (t )   (t  1)
Define dt to be the smallest positive real number such that dt   0,   1 then
dWt  W (t  dt )  W (t ) and the multiplication rules apply to manipulation of functions of
Wt : dWt2  dt
dWt dt  0 dt 2  0 .
Arithmetic Brownian Motion: dX t 1  dt  dWt 1
 Xt >=< 0


For T>t, X T X t ~ N ( X t   (T  t ),  2 (T  t ))
Forecast variance tends to  as T .
Geometric Brownian Motion: dX t 1  X t dt  X t dWt 1
 Xt has an absorbing barrier at 0

For T>t, ln( X T ) ln( X t ) ~ N (ln( X t )   (T  t )  1  2 (T  t ),  2 (T  t ))
2

Forecast variance tends to  as T .
Mean Reverting Process: dX t 1   (   X t )dt  X  dWt 1
t



Xt > 0 for X0>0 and >0
As X0, dX>0 and volatility vanishes
Forecast variance tends to  as T .
For   1 : X T X t ~ non  central 2
2
E ( X T )  ( X t   ) exp(  (T  t ))  
VAR( X T )  X t (  )(exp(  (T  t )  exp( 2 (T  t ))   ( 2 )(1  exp(  (t  t ))) 2
2
2
5