1. Proprietary & Confidential
For Use by Permission Only
Guided Tour Inside the Random Walk
with
Declensions of Black, Scholes & Merton
Frederic Siboulet
fsiboulet(at)nyu(dot)edu
28 March 2016 1

2. Proprietary & Confidential
For Use by Permission Only
Objective
• The objective of this presentation is to compare and
contrast some of the theories and techniques that may be
used to demonstrate either the Black-Scholes-Merton
partial differential equation (BSME), or the Black-Scholes-
Merton option pricing formulae (BSMF) – or both.
• Considering the extensive work of many scholars, there are
at least 12 methods that can lead to the BSME or the BSMF.
We limit our presentation to 5 methods, and refer the
reader to specialized text books on the matter if required
(see for example Paul Wilmott’s Frequently Asked Questions
in Quantitative Finance or CQF classes).
• This work borrows extensively from many research papers,
finance classes and other conversations. In particular, it is
largely inspired by Riaz Ahmad, Peter Carr, Raphael
Douady, Randeep Gug, Seb Leo, Fabio Mercurio, Zari
Rachev, Paul Wilmott – and many more, all of whom I am
grateful to.
28 March 2016 2

3. Proprietary & Confidential
For Use by Permission Only
TOC
1. Partial Differential Equation
2. Binary Tree and Cox Ross Rubinstein
3.1 Martingales and Probabilities
3.2 Martingales and Change of Numeraire
4. Kolmogorov and Fokker Planck Equations
28 March 2016 3

4. Proprietary & Confidential
For Use by Permission Only
General Introduction
• BSM: there are at least 12 methods (see Paul Wilmott’s
FAQQF) expressed either in
– Equation (Partial Differential Equation, infinitesimal) or
– Formulae (Option Pricing, integral)
BSM Equation is more general than BSM Formulae
• For derivatives pricing on traded (Equities, Ccies, Futures) or
non-traded underlying assets (Interest Rate, Credit Spread)
• Generic or specific formulations (i.e. may or may not
be easily generalized)
• In continuous time or discrete time
• With the Gaussian distribution assumed - or not
• With or without incremental or friction costs, such as bid∆ask
spread, dividend, carry, transaction cost, liquidity premium
• With constant, deterministic or stochastic parameters (e.g.
volatility)
• Some solutions focus solving for the drift μ, others for the
diffusion σ.
28 March 2016 4

5. Proprietary & Confidential
For Use by Permission Only
1/ Partial Differential Equation:
Classic BSM Equation and Formulae
28 March 2016 5

6. Proprietary & Confidential
For Use by Permission Only
Our Assumptions (see Black and Scholes, or Markowitz)
• Short selling, Fractional trading always possible
• No market friction (such as BΔA spread, transaction fee,
dividend, tax, illiquidity)
• Continuous trading and dynamic hedging assumed
• At this stage
– Constant rate r for the RFA At=ert
– Constant stock return μ
– Constant volatility σ
• The only derivative payoff is at the horizon T (V(T), Euro-
style) with no path dependency
• Underlying stock asset (S) follows a GBM and assumed to
be traded. In that complete market, any such contingent
claim Vt (i.e. derivative) can be replicated @t with a
replicating portfolio of 1.Stock (St) & 2.Risk Free Asset (At).
28 March 2016 6
1/ Partial Differential Equation

7. Proprietary & Confidential
For Use by Permission Only
28 March 2016 7
1/ Partial Differential Equation
The risky term (not to be confused with credit risk) of the
PDE is the Brownian Motion (BM), aka Weiner Process (WP). A
WP is a Random Process (RP), aka a time dependent Random
Variable (RV) iff it has the four following properties:
[ An alternative definition, the Levy Characterization, states
that a RP is a WP iff it is an almost surely continuous
martingale, with zero origin and quadratic variation [W]t =t ]

 

 
 
     
   
 
   
is continuous
2*
2*
(1) X(0) 0
=1
Xis continuous in time
(2) /
w/proba one , ,
0,
(3) , 0, X's Variance is the time interval
,
(4) , independent
' 0,
t X t
t T
t s X N t s
t s
X t s X t
t t





 


     

 
   
 
 
 

 


 
Markovian
of '
property
X t















8. Proprietary & Confidential
For Use by Permission Only
1/ Partial Differential Equation
1. Assume a Stock S following a GBM – S is the time dependent
RV and the underlying of the derivative, where X is the WP:
2. Construct the portfolio Π, long 1xOption and short Δx stocks –
∆ tbd:
3. Move forward in time, from t to t+δt, assuming Δ constant:
4. Itô Lemma (or Taylor expansion order o(dt), with δS2~δt~dt):
28 March 2016 8
     , , ,S t V S t S S t   
   , ,t t t tS t V S t S     
Sdt dXdS S  
 
 
 
2
2
2 2 2
2
2
2
2
2
2 2
2
2
2
dt
1 1
d
2
t
2
t
1
d
2 2
Tay
d
lo
tX
r
d
o
V V
dt dt
dS
dV dS o
dS
V
dS
t
S
dS
S
V
dS
S
dt S dt
V V
dt
t S
V S V
d
t
t dt
t S
S
S
dX
o
 






   

   

  
 

 
 
 



 






9. Proprietary & Confidential
For Use by Permission Only
1/ Partial Differential Equation
5. Neglecting the high order terms (HOT), dΠ is s.t. (w/Δ=cte):
6. Now, pick Δ so that Π is risk-free over dt (dynamic hedging
argument). There should be no arbitrage, therefore the return
of that portfolio over dt should be that of the RFR:
7. And therefore over dt, (5)=(6) gives:
28 March 2016 9
 d r dt r V S dt

   

 3
2 2 2
2
2
2
dV
d dV dS O
V
d
V S
S dS
S
V
dt d
S
tt
t


 
 
      


  

 
 
 
risk free, determini
2 2 2
2
2 2 2
2
2 2 2
ti
2
s
2
2
2
dS dS
V S V
dt dt r V S dt
t S
V S V
dt Sdt dt Sdt r V S dt
t S
V S V
r V S S S dt
t S
SdX SdX
V
d
S
S
V
S
S dS
V


  

 
 
 
 
 
 
 
 
  
  

   
    
    


   
            






 
risky, random termcterm
0
V
S dX
S


 
 
 
 
 


10. Proprietary & Confidential
For Use by Permission Only
1/ Partial Differential Equation
8. According to the risk-free argument the random term is
null, eliminating the random term dX; therefore write:
9. Plug the new-found value of ∆ into the deterministic term:
10. Which gives the Black Scholes Merton Equation:
28 March 2016 10
2 2 2
2
risk discount
free termdiffusion
drift
BSME
2
V S V V
rS rV
t S S


  
  
   
   
  
: 0
0
SdX
S dX
V V
S S




  
    
  
2 2 2
2
0
2
0
V S V V
r V S S S
t S S
V V
S S

 

   
    
  



 
 

11. Proprietary & Confidential
For Use by Permission Only
To go from the BSM “Equation” to the BSM “Formulae”
for option pricing, solve the BSM PDE:
11. Formulate a function U which is the forward value @T of
the present value @t of the derivative V(S,t):
12. Derive V by t and by S and express V as a function of U:
13. Replace V by its expression of U in the LHS of the BSME :
28 March 2016 11
1/ Partial Differential Equation
   
     
 
"numeraire" @T "numeraire" @t simplified
notation w/o T
, , , ,r T t r T t
TU S t e V S t V S t e U S t  
  
  
   
 
 
     
       22
2 2
,
,
, ,
, ,
r T t r T t
r T t
r T t r T t
U S tV
re U S t e
t t
V S t e U S t
U S t U S tV V
e e
S S S S
   
 
   
 
 
  
  
         
   
 
   
2 2 2, ,
2
2 2 2
2
BSME
2
Backward Kolmogorov Equation
BSME 0 BKE
identical to BSME w/o discount2
r T t
V S t e U S t
r T t r T t
rV
U S U U
re U e rS rV
t S S
U S U U
rS
t S S


 

   

   
     
   
  
    
  


12. Proprietary & Confidential
For Use by Permission Only
14. First change of variable, on time t:
15. With ̃ replacing t, the BKE becomes the FKE (or Forward
Kolmogorov Equation, aka Fokker Planck Equation):
16. Second change of variable, on the stock S:
28 March 2016 12
1/ Partial Differential Equation
 
chain rule
limitsvariable
. . .
;
0
parameter
t T
t tt T T t

 
 


    
             
  
 

 

2 2 2
2
Notice the sign of
BSME 0 FKE(1)
the time derivation2
T t U S U U
rS
S S
 

    
     
 



chain rule
first order
2 2
variable
2 2 2 2second order
. . 1 .
ln
. 1 . 1 . 1 . 1 . .
S S S
S S
S S S S S S S

 

    
    
  
    
   
           
                  

13. Proprietary & Confidential
For Use by Permission Only
17. With ς replacing S, the FKE(1) becomes FKE(2):
18. Third change of variables:
28 March 2016 13
1/ Partial Differential Equation
2
2
2 2 2ln
2 2
2 2 2
2
1 1 U
BSME 0
2
Notice the change to theU
BSME 0 FKE(2)
drift (& diffusion) terms2 2
S
T t
U UU
t S
S
U S U U
rS
S S
U U
r



  
 
 

 
    

    
      
    
   
       
   
 
 

 
 
2
2
2 2 2 2
2 2
. . . . .
2
, . . . .2
. . . .
x
r
x x
x r x
x x
x x x
 
   

    
   
  
  
        
      
       
             
      
                
      

  
 
 

14. Proprietary & Confidential
For Use by Permission Only
19. Replace the old with the new variables into FKE(2) to get W:
20. Therefore finally:
with:
28 March 2016 14
 
 
   
2
2
2
2 2 2 2, ,t
2
U
,
2
0
2 2 2
FKE(2)
W x U S
UU
x r
W W W W
r r
x x x



   
  
  


  
 
    
  
      
            
      




 
 
 
   
2
2 2 2 2 2, ,
2 2
ln
2(classic) Black Scholes Merton Equation Heat Equation
BKE FKE
0
2 2
rt
V S t e W x
x S r
T t
V S V V W W
rS rV
t S S x




 



 
    
 
 
    
     
      
1/ Partial Differential Equation
   
2
2
, , derivative's value @t
correspondence of variables
r
x r
V S t e W x
S e t T






 
   
 
  


     

15. Proprietary & Confidential
For Use by Permission Only
To value the derivative V(S,t) is equivalent to solve the Heat
Equation. Therefore with the fundamental solution, seek the
pdf of the Gaussian RP W for the alternate set of variables x
and τ.
21. May solve the heat equation by similarity reduction, i.e.
transform the PDE (two variables) by an ODE (single
variable) by the appropriate change of variable
22. For that new variable z, assume α, β and y three parameters
tbd and write:
23. Derive the variable z once wrt τ and once wrt x:
28 March 2016 15
1/ Partial Differential Equation
 
   
3
,
, ,y / , ]0, ]
W x f z
x T x y
z


 
  

 

       


 
1
1
z x y
x y
z
z
x




 



 
  
  
 


16. Proprietary & Confidential
For Use by Permission Only
24. Derive the function W once wrt τ and twice wrt x:
28 March 2016 16
   
 
 
   
    
 
 
 
 
 
1
1
1
1
1
2
2
2
'
,
'
'
1
'
''
''
f zW z
f z
z
x y
f z f z
W x f z
zf z f z
x y
z f zW z
x z xz x y
f z
z
f zW z
x
x z x
z
f z
x
f z
 
 





 
 
 
 
 
 
 

 
  
 

 











  
 
  
  
    
 
            
     
     
 
    
  














1/ Partial Differential Equation

17. Proprietary & Confidential
For Use by Permission Only
25. Go back to the Heat equation and substitute the above
.
:
26. By similarity reduction, seek to reduce dimensionality – it
is sufficient to pick β=1/2 to eliminate τ, hence:
27. And then:
28 March 2016 17
    
 
      
     
2
2
2 2
2
2
1 1 2
2
2
2
2
1 2
0
2
' ' '' 0
2
''
' '' 0
2
W W
x
W W
x
W
zf z f z zf z f z f z
W
f z
x
f z zf z f z
   
 




      



  
  
 
 
 

 
  
  
 
       
 

 
 
   

           
1
2 22
1 2
' '' 0 ' '' 0
2 2 2
z
f z zf z f z f z f z f z

 
   


      
   
 
1
,2
,
W x f z
x y x y x y
z z W x f
  


 
  
 
   
      
 
1/ Partial Differential Equation

18. Proprietary & Confidential
For Use by Permission Only
28. The values of α and β are selected so that, at any time,
ταf(z) is pdf if the Normal distribution (since W is the
fundamental solution of the heat equation), and therefore:
28 March 2016 18
1/ Partial Differential Equation
 
   
 
 
1
2
1
2
must be constant vav
alwaystherefore this term can
independentonly also be equal to 1
from
10,
2
0, , 1
0, , 1
0, , 1
1
1
0
2
x y
z
dx dz
T
T
x y
T f dx
T f z dz


 



 
 
 

 









  
  
 
   
 
   
 
  






 
1
2
and is the pdf of the
1
normal distribut
α is necessarily
equal to 1/
o
2
i n
f
f z dz
  
  

19. Proprietary & Confidential
For Use by Permission Only
29. From there, f is defined by the following ODE:
28 March 2016 19
           
    
   
 
 
   
2 * by -2
2
product
rule
2
anti-
derivative
vav z
2
boundary
conditions
2
0
' 0
1
' '' 0 ' '' 0
2 2 2
'' 0
' Constant
' 0
z
z
f z
f z
z
f z f z f z f z zf z f z
d
zf z f z
dz
zf z f z
zf z f z








 
       
  
  
  
1/ Partial Differential Equation

20. Proprietary & Confidential
For Use by Permission Only
30. Continuing the resolution of the ODE:
31. And the constant C2 is such that:
28 March 2016 20
   
 
 
2
2
2 2
2
2
2
2
12
' 0
1
ln 1 C1 constant
2
2* 2
z
C
df z
zf z f z zf z
dz
df z
dz
f
df
zdz
f
z
f C
f C e C e
 




    

 

 

   
   
 
 
2
2
2
2
2
2
1
1 2* 1 2
z
f z dz C e dz C
e d






     

 

1/ Partial Differential Equation

21. Proprietary & Confidential
For Use by Permission Only
32. By simple change of variable:
33. Therefore, we get the formulation of f:
From here W is the Gaussian distribution of mean y and variance
.
28 March 2016 21
2
2
2
2
2 2
2
2
2
1
12 2 2
2
2
2
1
2
2
z
e dz
C e d
Cz
dz d
C



   
  






 

  
  

 





 
   
 
 22
2 2
1
,
2 2
2 2
1 1
,
2 2
yW x f z x yz
yx y
z
f z e W x e


  


  
  


  
1/ Partial Differential Equation

22. Proprietary & Confidential
For Use by Permission Only
34. At any time between t=0 (inception, τ=T) and t=T (expiration,
τ=0), write for the pdf and CDF of that Gaussian RP:
35. Therefore at the τ-zero-plus limit, when the option expires, the
three conditions above show that the pdf Wy tends to a y-Dirac
delta function:
28 March 2016 22
 
   
 
 
   
0
2
0
Gaussian pdf
, 0 1, 0, ,
, 2, ,
0, , , 1 3
y
y
y
y
x y W xx T
x y W xW x N y
T W x dx



  
 





       

   

   



   
 
   
 
 
 
 
2
2
2
2 0
1
, 0, , , lim ,
2
0
where, by definition, y-Dirac is st:
1
x y
y y y
y
y y
y
x T W x e W x x
x y x
x x y y
x dx
 

   
 

 

 

      
   


    



1/ Partial Differential Equation

23. Proprietary & Confidential
For Use by Permission Only
36. According to the sifting (aka sampling) property of Dirac-
delta, for a continuous function g(x):
37. Therefore write (note that x and y are interchangeable):
38. For t<T, Wy above is the Gaussian pdf of mean y and variance
. At the limit τ=0+ (i.e. t=T -), with g the payoff of the
derivative V, the integral’s limit gives the expected value of
the derivative V at expiration, aka that payoff, and:
28 March 2016 23
1/ Partial Differential Equation
     
g -continous
y-Dirac delta y
y
g x x dx g y


 



   
 
 
 
   
     
2
2
0
2
2
2
2 , 0
Gaussian pdf N ,
or N x,
1
, , lim ,
2 x x
x y
x y
y x x
W y y
y
W x W y e W y g y dy g x

 
  
 
 
  
 


 

 
   



0
2 0
00
ln
2
T Tx r S S e
 



  
 

 
       
 




24. Proprietary & Confidential
For Use by Permission Only
39. At the boundary for τ=0+:
40. Prior that, for τ>0, integrate a Green function (Gaussian
transition pdf) multiplied by the derivative payoff (boundary
condition) over all possible values of the underlying stock
(state variable) to get the derivative value (solution) @t<T:
28 March 2016 24
1/ Partial Differential Equation
   
 
   
       
0 0
0
2 0 0
0
0
aka payoff @t=T
aka derivative's value @t=T
aka boundary condition @t=T
ln
x ln
,0 , , g2
, ,
, ,
t
x
T T
t T x
y T T
r T t
t t
t y
T t S
S e e S
x r
W x U S T V S T e
V S t e U S t
U S t W x


 

  

 

 
  
 
    
       
         

 
 

 







     
       
forward
derivative's (Green solution) Gaussian pdf derivative's payoff @Tvalue t T
value@t (Green function) (boundary condition)
0, , , , ,
0,
r T t y
t t y xt T U S t e V S t W x W y g e dy
t
  

     
  
    
 
   
 
 
2
2
2
2
2
, ,
2,
ln
2
y xr
r y
t y
t
e
V S t e W x e g e dy
T
x r S T t

  

  

 

   
 


  


 
        
 


25. Proprietary & Confidential
For Use by Permission Only
41. Now return to observable variables for the pricing of the
derivative between inception and expiration, first write for
payoff:
42. Then:
28 March 2016 25
   ln y
y
d
dy d
y g e dy G
e

 
    
  
 
 
 
 
 
 
 
 
 
 
 
 
 
2
2
2
2
2
*
2
2
2
*
2
2
ln ln
2
2
2
ln
2
2
2
risk free rate
present valuation
exp
,
2
e
2
1
e
2
t
t
x yr
y
t
S r T t
r T t
T t
S
r T t
r T t T t
e
V S t e g e dy
e d
G
T t
d
e G
T t

 




 




 
  
         
   

  
          
  


 


 







present value of the expected payoff
ected value of the payoff wrt
lognormal transition probability density function


1/ Partial Differential Equation

26. Proprietary & Confidential
For Use by Permission Only
28 March 2016 26
43. Pricing a Euro-Call; for K the strike and Υ the value of the
underlying @T, the Call payoff is:
44. Hence write for V(S,t) @t<T, with :
   G K

   
 
 
 
 
 
 
 
 
 
 
 
 
 
2
2
2
*
2
2
2
2
ln
2
2
2
ln
2
... 0
2
2for a Call
ln
2
2ln
, e
2
e
2
e
2
t
t
K
y
t
S
r T t
r T t
T t
t
S
r T t
r T t
T t
K
e S r
d
dy r T t
K y K
e d
V S t K
T t
e d
K
T t
e
T t










  
           

  
            
 
   
  

  

  

 
  






 
 
 
2
2
2
ln
T t y
T t y
K
e K dy
 
   
 
 

1/ Partial Differential Equation

27. Proprietary & Confidential
For Use by Permission Only
45. Split the integral in two parts:
46. For S, t and T known, introduce the variable uε ( 1) s.t.:
47. And the integral boundaries s.t.:
28 March 2016 27
 
 
 
 
 
 
 
 
 
2
2
2
2
2
2
ln
2
2
2 ln
ln
2
2
2 ln
1 e
2
, 1 2
2 e
2
t
t
S r T t y
r T t
T t y
K
t
S r T t y
r T t
T t
K
e
e dy
T t
V S t
Ke
dy
T t






  
        
   
 
  
        
   
 


 
 
  


 
 


 
2
ln
2
tS r T t y
dy
u du dy T tdu
T t T t
  



 
 
    
       
 
   
2 2
ln ln
2 2
ln
t tS S
r T t r T t
K K
y K u u
T t T t
y u


 

 

   
        
       
 
    
1/ Partial Differential Equation

28. Proprietary & Confidential
For Use by Permission Only
48. Rewrite the second integral, with the variable u-:
Which is the result for the second integral.
28 March 2016 28
 
 
 
 
 
 
   
 
2
2
2
2
2
2
2
ln
2
2
2 ln
ln
2
2
2
2
Gaussian CDF to =d
2 e
2
e
2
1
e
2
t
y
S r T t y
r T t
T t
K
K u
r T t udy T tdu
uu
u u dur T t r T t
u
Ke
dy
T t
Ke
T tdu
T t
e K du e KN d











 


  
        
   
 

    
 
   




 

 




1/ Partial Differential Equation

29. Proprietary & Confidential
For Use by Permission Only
49. Now rewrite the first integral:
50. With that first integral, we notice with the numerator of
the exponent that (see tedious but straightforward algebra
next):
28 March 2016 29
 
 
 
 
 
 
 
 
 
      
 
2
2
2
2
2
2
2
2
2
2
ln
2
2
2 ln
ln
2
2
2 ln
ln 2
2
2
2 ln
1 e
2
1
e
2
1
e
2
t
t
t
S r T t y
r T t
T t y
K
S r T t y
y r T t
T t
K
S r T t y T t y r T t
T t
K
e
e dy
T t
dy
T t
dy
T t










  
        
   
 
  
        
      
  
            
  
 









      
   
2
2
2
2
2
2
ln 2
2
ln 2 ln
2
t
t t
S r T t y T t y r T t
S r T t y T t S




  
          
  
  
         
  
1/ Partial Differential Equation

30. Proprietary & Confidential
For Use by Permission Only
51. Below, brief demonstration of the result above; the
intuition is to change the square of a difference in d2 (=u-) to
the square of a sum in d1 (=u+) , while brushing lightly over
the terms that are identical between d1 to d2:
28 March 2016 30
      
       
   
2
2
2
2
22 2
keep together
remarkable square identity with diffrence = n
2 2
2
2
l
ln
2
ln
2
2
2 ln
2
t
t
S y r T
t
t
t
S r T t y
S y
T t y r T
r T t S y r T t
t


 

  
          
  
      
  
   
          
   
  


 
   
           
       
2 2 2
4 2
2 2
22 2
22 2
22 22 2
4
2 22
2
0 0
ln 2 l
2 2
n
4 2
ln 2 ln
4
2
2
2 22t t
t t
S y r
y
r T t S y r T t
S y r T t S
r r
T t r T t
y T t r T t
r yr T t r


 

 


  

 
   
           
   
 
   
  
     
 
   
    


 
        
       
   
2
2
remarkable square ident
2
2
ity
22
with s
22
2
um = ln
2
2
2
2
2 l
ln 2 ln
2
n
2
2
2 2
t y r
t
S
t
T t
t
T t
S y S y
S y T t
r
y T t r
r t
t
t T
T
T

 




  
          
   
 
  
 
 
 
 
     
 
  
 
 


 
        
 
   
20 2
22
ln
2
2
2 2
2
2 2
22
ln 2 l
2
2 l
n
2n
tT t S
t
t tS r T t y
r T t yr T t S
T t
ty t TT
S






  
  
   
   
     

 
 
 

 

1/ Partial Differential Equation

31. Proprietary & Confidential
For Use by Permission Only
52. Back to the first integral:
28 March 2016 31
 
      
 
      
   
 
 
2
2
2
2
2
2
2
22
2
2
ln 2
2
2
2 ln
ln 2
2
ln
ln 2 ln 2
2
2
1
1 e
2
1
e
2
t
t
t
t t
S r T t y T t y r T t
T t
K
S r T t y T t y r T t
S r T t y
S r T t y T t S
dy
T t
T t










  
            
  
 
  
           
  
  
                       
  





 
 
 
 
 
 
 
 
2
2
2
2
2
2
2
2
2
2 ln
2
ln
ln
2
ln
2
2 ln
ln
2
2
2 ln
1
e
2
e
2
t
t
t
t
T t S
T t
K
S r T t y
S
T t
K
S r T t y
T tt
K
dy
dy
T t
S
dy
T t








  

 
  
        
    
  
        
  
 







1/ Partial Differential Equation

32. Proprietary & Confidential
For Use by Permission Only
53. From there now write, with the variable u+:
54. Therefore the BSM formulae for an Euro-Call (same
approach for an Euro-Put):
28 March 2016 32
 
 
 
 
 
2
2
2
2
2
1
1
ln
2
2
2 ln
ln
2
2 u
u
2
1
Gaussian CDF to u =d
1 e
2
e
2
1
e
2
t
y
S r T t y
T tt
K
K u
udy T tdu
t
u
u u d
t t
S
dy
T t
S
T tdu
T t
S du S N d














  
        
  
 

  
 
 




 

 




       
 1 2, 1 2 , r T t
t t tV S t V S t S N d e KN d BSMF 
     
1/ Partial Differential Equation

33. Proprietary & Confidential
For Use by Permission Only
55. By solving the PDE, with the final condition aka the Euro-
Option Payoff @T, for the Black and Scholes Formulae,
get for Euro-Calls and Euro-Puts:
28 March 2016 33
   
     
 
 
 
1 2
@ ( )
2
1
2
@
2 1
,t
1
ln1 2
,
ln
2
r T t
t t
OptionValue
t T
t
T T
OptionPayoff
tT
V S S N d Ke N d
Call S
r T tPut K
d
V S T S K T t
S
r T t
K
d d T t
T t
   
 

 



 


  


                   
      
          
      







1/ Partial Differential Equation

34. Proprietary & Confidential
For Use by Permission Only
56. Remarks:
• At the infinitesimal level, the BSME is a linear parabolic
differential equation
– First order in time
– Second order in underlying asset
• At the integrated level, solving BSME requires boundary
conditions; one may use:
– One time condition (e.g. payoff at T)
– Two asset boundary condition (e.g. V for S=0 and S=∞)
• The drift μ of the underlying GBM of S is not a factor in the
pricing of the derivative V – only r is. Hence “risk neutral
pricing” - a direct consequence of the dynamic hedging in
the risk neutral portfolio.
• Another reading of the dynamic hedging argument is that –
in a complete market – the value of the derivative can be
replicated with cash and ∆ stocks, where ∆ is the sensitivity
of the derivative to the underlying asset.
28 March 2016 34
1/ Partial Differential Equation

35. Proprietary & Confidential
For Use by Permission Only
57. Variations of the BSME:
28 March 2016 35
1/ Partial Differential Equation
Stocks S with dividends s
2
Currency spot C with domestic
rate (rd) and foreign rate (rf).
C expressed with foreign
numeraire and domestic base
2
Commodity Q with cost of
carry q 2

36. Proprietary & Confidential
For Use by Permission Only
28 March 2016 36
58. Variation of the BSMF (ε=±1, Euro Call and Put)
Stocks S with dividends s
Currency spot C, domestic
rate (rd) and foreign rate (rf)
Commodity Q with cost of
carry q
   
   
 
   
2
,t
1
ln
2
s T t r T t
t t
t
V S S e N d Ke N d
S
d r s T t
KT t
    


   
 

 
   
             
1/ Partial Differential Equation
   
   
 
   
2
,t
1
ln
2
f dr T t r T t
t t
t
d f
V S C e N d Ke N d
C
d r r T t
KT t
    


   
 

 
   
             
   
   
 
   
2
,t
1
ln
2
q T t r T t
t t
t
V S Q e N d Ke N d
Q
d r q T t
KT t
    


  
 

 
   
             

37. Proprietary & Confidential
For Use by Permission Only
2/ Binomial Tree
28 March 2016 37

38. Proprietary & Confidential
For Use by Permission Only
2/ Binomial Tree and Time Limit
1. At t, the Stock price is St ; move in time from t to t+δt.
2. At t+δt, the Stock price may move up to uS or down to vS,
with 0<v<1<u; probability to move up is p, to move down is
1-p (hence “binomial”):
3. Chose u, v and p (not unique) by using the drift μ and the
diffusion σ of S (note: that choice will later lead to coincide at
the continuous limit δt→0 with the GBM dS=μSdt+σSdX):
28 March 2016 38
1 for t small enough:
1 2 2
0<p<1
1 1
0<1-p<11
2 2
t
p
u t
v t t
p
  
  
   


    
   
       
t t+δt→
S
S+=uS
S-=vS
p
1-p

39. Proprietary & Confidential
For Use by Permission Only
4. Look at the Option price at V(t+δt); no assumption yet on
Option value V(t). Option value at t+δt is either V+ if
S(t+δt)=uS(t), or V- if S(t+δt)=vS(t) (note: with S going up, a Call
V+ goes up, a Put V+ goes down):
5. At t, construct a portfolio Π(t), long 1xOption and short a
fixed value ΔS of the Stock S; go from t to t+δt:
2/ Binomial Tree and Time Limit
28 March 2016 39
t t+δt
→
V
V+
V-
p
1-p
t t+δt
→
Π
Π+
Π-
p
1-p

40. Proprietary & Confidential
For Use by Permission Only
28 March 2016
2/ Binomial Tree and Time Limit
6. Write the portfolio Π(t) and Π(t+δt) :
7. Chose ∆ so that Π+(t+δt) =Π-(t+δt), i.e. the value of the
portfolio at t+δt is indifferent of whether the stock goes up
or down between t and t+δt (risk neutral argument):
8. General remark: S(t) is such that S+(t+δt)>S(t)>S-(t+δt) , and:
– for a Call, V+(t+δt)>V-(t+δt)therefore Δ>0
– for a Put ,V+(t+δt)<V-(t+δt), therefore Δ<0 40
     
     
     
t t t t t V t t S t t
t V t S t
t t V t t S t t
   
  
 
  
  
      
    
      
           
   
   
   
   
t t t t V t t S t t V t t S t t
V t t V t t
S t t S t t
V t t V t t
u v S t
     
 
 
 
     
 
 
 
              
  
  
  
  
  


41. Proprietary & Confidential
For Use by Permission Only
2/ Binomial Tree and Time Limit
9. The choice of Δ is such that the return of Π is risk-free (again,
the risk neutral argument); therefore:
10. For instance, in the S+ up case Π+(t+δt) on the RHS (or equally
Π-=Π+ in the down case) and substituting in Π(t) on the LHS
11. And replacing the value and S+(t+δt)=uS(t), and solving for V(t):
28 March 2016 41
   
 
 
value of portfolio at t+ t
with up move of the underlying S
risk free rate portfolio
growth over t value at t value of portfolio at t+ t
with down move of the
1
t t
r t t
t t

 







  
  
  

 
     
real world expectation of the value of portfolio at t+ t
underlying S
1p t t p t t

  

      


        
            
1 or resp. =
1 or resp. w/
r t t t t t t
r t V t S t V t t S t t S
  
  
 
  
      
        
          
   
   
1
1
r t V t S t V t t uS t
V t t uS t
V t S t
r t
 




      
  
   


42. Proprietary & Confidential
For Use by Permission Only
2/ Binomial Tree and Time Limit
12. Replace ∆ in the previous equation with its prior found
value:
13. To lighten up the notation, now remove the explicit
reference to the time steps:
28 March 2016 42
 
   
   
 
 
   
   
 
1
V t t V t t
V t t uS t
V t t V t t u v S t
V t S t
u v S t r t
 

 


 

 

  
 
   
 
 


 
 
   
   
1
1
1
1
1
V V
V uS
u v SV V
V S
u v S r t
V V V V
V u
u v r t u v
V V uV vV
u v r t u v



 

 
   

   



 
 
  
       
 
 
  

43. Proprietary & Confidential
For Use by Permission Only
2/ Binomial Tree and Time Limit
14. Replace u and v with the original values:
28 March 2016 43
 
     
 
   
 
      
 
   
 
present
valuation
1
1
1 11
12 2
1 1 11
1 2
1
1 2
1 1
1 2 2
q
V V uV vV
V t
u v r t u v
t V t VV V
r tt t
r t V V t V t V
r t t
r t t V r t t V
r t t
r t
r t

   
   
    
  
     
  

 
   
 
 
   
 

 
 
  
  
 

      
 
 
 
   
 
 
 
 
     
 

 
 
   
1
risk neutral expectation of the value of the
derivative V at t+ t (no drift μ, replaced by r)
1 1
2 2 1 1
q
qV t tr t
V V
r t q V t t


  
 

 

 
     
                
 
 


44. Proprietary & Confidential
For Use by Permission Only
2/ Binomial Tree and Time Limit
15. Where is Black and Scholes? Look at V(S,t), and perform a
Taylor expansion between t and t+δt, denoting ε=±1:
16. Side remark, the expression of a familiar term:
28 March 2016 44
 
   
   
 
uS or vS,
depending on ε=±1
2Taylor
2
2
2 2 2
2
2
2
2
2
2
2
2
1
1 ,
1 1
2 2
,
2
,
2
,
2
V t S t t
V V V
V
V
S t S t S
S t S
V V S V
V S t S t t t
S t S
V V S
V S t S t t
V
t S t o t
t Sdt
t
S t
o
   
     

   



 

 

 
 
  
 
 
 
  
   
  
  
   
  
  
  





 


 2
V
t
S
o t 


0
2
2 2
t
V
S t
V V VS
StS tS

 
   
 


    


45. Proprietary & Confidential
For Use by Permission Only
2/ Binomial Tree and Time Limit
17. Combine risk neutral Option value with Option value Taylor
expansion:
28 March 2016 45
 
 
     
     
2 2 2
21
2 2 2
2
1 1 1
,
1 2 2 2 2
, ,
2
1
1 , ,
2 2 2
S S t
V
r t r t
V S t V V
r t
V V S V
V S S t t V S t S t t t o t
S t S
r t V V S V
r t V S t V S t S t t t
S t S
  


 
  

      
 
    


 



    
                  
  
       
   
    
            
 
   
       
2 2 2
2
2 2 2
2
δt 0
1
,
2 2 2
1 , , 2
2 2
V
r t V V S V
V S t S t t t o t
S t S
V S V r t V
r t V S t V S t t t S t o t
t S S
V
t
 
    

 
     




 
 
 
  
 
 
    
             
 
  
      
  

 



2 2 2
2
BSME
2
S V V
rS rV
S S
 
  
 

46. Proprietary & Confidential
For Use by Permission Only
18. Practical remarks, when constructing trees:
– The underlying S is computed forward in time, from S=S0
@t=0
– The derivative V is computed backward in time, from the
payoff, aka the final boundary condition
– The portfolio Π is computed forward in time, from t=0
19. The construction of the risk neutral portfolio transitions the
“Real World” probability p to the “Risk Neutral” probability q.
this is called a “change of measure” (see martingale chapter).
Also see Cox Ross Rubinstein next for other possible
formulations of p and q.
28 March 2016 46
2/ Binomial Tree and Time Limit
 
asset numeraire
drift drift
-measure, Real World -measure, Risk Neutral
1 1
2 2 2 2
t r t
p q
  
 
    

 


47. Proprietary & Confidential
For Use by Permission Only
2.1/ Binomial Tree with
Cox-Ross-Rubinstein
28 March 2016 47

48. Proprietary & Confidential
For Use by Permission Only
2.1/ Cox-Ross-Rubinstein
20. For CRR, go back to the beginning of the tree presentation,
and take the following assumptions on 1/ , and the
discount factor D – with RFR r and Stock Vol σ. Make the
following Taylor approximations (arbitrary for u and v):
21. Similar steps as the previous tree example (in 2/) result in the
LHS below, which writes equivalently as the RHS below:
28 March 2016 48
 
 
 
2
2
1 discount factor on backward t
up probability for S on forward t
1
rounded @ o t2
down probability for S on forward t
1
rounded @ o t2
r t
D e r t o t
u t t
v t t

  

  


  


    
   
   
     
 
 
1 1 1V V t
V V V t t
V V uV vV
V D u v V v V u V
u v u v D D D  

   
 
  
     
          
     

49. Proprietary & Confidential
For Use by Permission Only
22. By δt-order Taylor expansion, write:
23. For the up or down stock movements from t to t+δt write:
24. Lightening up notations as below and with Taylor:
28 March 2016 49
 
 
 
 
2
2 δtsmall
2
2
2
1
1 1 1
2
1 2
2 1
2
21
2
o t
o t
o tTaylor
D r t o t r t
v t r tD
D
u t t u v t
u t r tu v t
Dv t t



   
  

    

   

  
 
                 
        
             
     
2.1/ Cox-Ross-Rubinstein
 
 
 
 2
2
2 δtsmall
2 2
1 12
o t o t
Taylor
t t
S t t S t S S t t
t t t t
 
 
 

      
  
      
        
       
 
 
 
 
21
2
2
1
,
2
,
S t t t S
V V V
V S t V V V S t S o t
S t S
V S S t t V
 

  
  
  
   
 

   
   
      
  
   

50. Proprietary & Confidential
For Use by Permission Only
25. Take the following expression:
and plug the Taylor expansion on RHS below (ε=±1) in the
above :
Which gives on the next page…
28 March 2016 50
 
 
 
 
 
2
2
2 2 2
2
2
2
2 22
2 2
2
2 2 2 2
2
1
2
1
2 2
2 2
S S t t o t
S S t o t
V V V
V V S t S o t
S t S
V V V
V V t t S t S t o t
S t S
V V S V S V
V V S t t o t
S t S S


  

    

   

   

      
 
   
  
 
  
    
  
   
       
   
    
       
    
2.1/ Cox-Ross-Rubinstein
 
1 1 1
u v V v V u V
D D D
 
   
       
   

51. Proprietary & Confidential
For Use by Permission Only
26. The three terms below:
28 March 2016 51
 
 
 
 
 
 
1
2
2 2 2 2
2
1
1 2 01
1 2
2 2
1
o t
o t
o t
u v
D
v
D
V
r t t V
u v V
D
V
V S t
S
t r t
V S V S Vv V tD t S S
u V
D



  
 

  
 


 

 


 
 

  
                                     





  
     
 

 
 
 
2
2 2 2 2
2
1
1
1
2
2 2
o t
o t
u
D
V
V
V S t
S
t r t
V S V S V
t
t S S


 

  
 


 









 





                                   




2.1/ Cox-Ross-Rubinstein

52. Proprietary & Confidential
For Use by Permission Only
27. And further, with ε=±1:
28 March 2016 52
2.1/ Cox-Ross-Rubinstein
3
2
2 2 2 2 2
2
2
2
2 2 2 2 2
3
2 2
2
2 2 2
2
2 2 2
o t
V V S V S V
t r t V S t t
S t S S
V
V t r V S t
S
V V S V S V
r S t
S t S S
V t

  
       

    
  
   
 
 
 
          
                     
   
        
       
                

3
2
2
2
3 3 3 2 2
3
2
2
2 3 2 2
3
2 2
2
2
2 2 2
2 2
1 1 2 2
o t
V
r V S t
S
V S V V S V S V
r S t
S S t S S
V V V S V
V t r V S t r S t
S t S S
V
V t


  
  
  
 
       
  
 
 
 
   
       
        
                 
       
                  

    
3 2 2
3
2
2
2
V S V
r S t
t S S

 
  
  
   

53. Proprietary & Confidential
For Use by Permission Only
28. Therefore, get the BSME as below:
28 March 2016 53
 
  
 
 
 
3 3
2 2
3
2
3 2 2
3
2
2
2 2 2
2
2 2 2
1 1 1
0 1 1
1 2 2 2
2
1
2
2
o t o t
o t
o t
o t
u v V v V u V
D D D
V V S V
r t t V V t r S t
t S S
V V S V
r t V V rS t
t S S
V V S V
r tV rS
t S
 




       

 


   
   
   
 
 
 
 
   
            
   
   
      
   
   
      
   
  
   
  2
2 2 20
2
2
t
t
S
V V S V
rS rV BSME
t S S



 
 
 
  
    
  

54. Proprietary & Confidential
For Use by Permission Only
29. Also note the following results:
30. By writing q as below, interpret q as a probability where the
asset drift μ in the p formulation is replaced by RFR r in q:
28 March 2016 54
   
 
 
   
1/
for δt small enough
0<q<1, a probability
1 1
1
1/ 1/
D v
q
u v
V V uV vV
V D
u v u v
qV t tvD uD
V V V t D
u v u v q V t t
D v u D
D V V
u v u v


   


 
 

 
  
   

  
             
     
      
     
 
 
 
2
-measure, Real World-measure, Risk Neutral
21/ 1 1
2 2 4 2 22
o t
t r t o t
D v r t
q q t p
u v t o t


   
  

   
  
    
                 
  
 

 
2.1/ Cox-Ross-Rubinstein

55. Proprietary & Confidential
For Use by Permission Only
31. Therefore, V(t) is the risk free present value of the risk neutral
expectation of V(t+δt) – i.e. the RHS independent of μ:
32.Derivative (backward): in a discrete formulation V is
expressed with :
28 March 2016 55
   
 
risk free risk neutral probabilityvalue of the value of the derivatdiscount of up move ofderivative @t over t the underlying Swith value of the
between t and t+ tunderlying of S @t
, ,V S t D q V S S t t


    

   
ive @t+ t risk neutralprobability value of the derivative @t+ t
with up move of the underlying S of down move of with
between t and t+ t the underlying S
between t and t+ t
1 ,q V S S t t
 


     
 
   
down move of the underlying S
between t and t+ t
1
1/ 1
2 2
o t o t
r t D v r t
D e q S S t
u v

 




  



  
  
  
  
  
  
   

  
           

 1
for any back time step j the previous time step's valuefor any node level i
from N-1 to 0 of the derivative, expectedat back time step j
from following time stefrom 0 to j
, , ,N jj i V i j    
   
   
one step risk neutral following time step risk neutralrisk free probability and up node value probabilitdiscount to comep's of the derivative
back downvalue of the derivative
1, 1 1-D q V i j q   
 
 
following time step
y and down node value
to come of the derivative
back up
, 1V i j
 
 
 
 
 
 
 
 
 
2.1/ Cox-Ross-Rubinstein
 1 /j jt t t T t N    

56. Proprietary & Confidential
For Use by Permission Only
33.Underlying (forward): From origin 0 to any time index j
(i.e. at time j*δt=j*(T-t)/N, going forward in time), write by
hypothesis for the underlying:
34. At the boundary (final time step N), write the payoff value V
for time step N (taking the Call case for the demonstration):
35. The discrete formulation is:
36. From N, roll V back in time to the previous time step N-1:
28 March 2016 56
  0
any time step j any node level i
at time step j
, , , j i j i
N jj i S i S u v 
     
 
    0, ,N ,N i N i
Ni V i S i K S u v K
 
           
        1(j 1), ,N 1 , 1,N 1- ,NNN i V i D qV i q V i       
        1, , , 1, 1 1- , 1N jj i V i j D qV i j q V i j         
2.1/ Cox-Ross-Rubinstein

57. Proprietary & Confidential
For Use by Permission Only
37. From N-1, roll back V to the previous time step N-2:
… a pattern which suggest an induction.
28 March 2016 57
 
 
   
      
 
        
 
 
   
   
2
1,N 1
,N 1
2
2
2
1,N 1
(j 2), ,N 2 ,
1- ,N 1
2,N 1- 1,N
1- 1,N 1- ,N
2,N
2 1- 1,N
1- ,N
N
V i
V i
qV i
N i V i D
q V i
q D qV i q V i
D
q D qV i q V i
q V i
D q q V i
q V i

  
 
  
          
   
 
 
   
 
 

  



   
2
22
2
0
1 ,N
kk k
k
D C q q V i k


 
 
 
 
 
 
  
2.1/ Cox-Ross-Rubinstein

58. Proprietary & Confidential
For Use by Permission Only
38. For the Induction, assume that for time step M-h:
39. From there, prove that (induction):
40. First, change the indexation in the discrete formulation:
41. Above is the starting point of the next development:
28 March 2016 58
 
     
0
: , ,
,N 1 ,N
N h
h
h kh k k
h
k
j N h assumed i
V i h D C q q V i k



   
   

    
      
1
1
11
1
0
1 : , ,
,N 1 1 ,N
N h
h
h kh k k
h
k
j N h to prove i
V i h D C q q V i k
 

 


    
    

        
        
1
for any back time step j value of thefor any node level i
derivative @ i and jand at back time step j
1 1
, , , 1, 1 1- , 1
, , ,N 1 1,N h 1-
N j
N N h
j i V i j D qV i j q V i j
h i V i h D qV i q V

  
        
          
 
 
    ,N hi 
2.1/ Cox-Ross-Rubinstein

59. Proprietary & Confidential
For Use by Permission Only
28 March 2016 59
        
   
 
     
 
 
0 0
1,N h ,N h
1 1
,N (h 1) 1,N 1- ,N
induction
1 1,N 1- 1 ,N
assumption
1
h h
h k h kh k k h k k
h h
k k
V i V i
hh k k
h
V i D qV i h q V i h
D q D C q q V i k q D C q q V i k
D C q q
 
 
    
 
      
 
 
        
 
 
 
 
 
 
     
       
 
1
0 0
1
1 11 1
1
1
0
1
1
factorisation
1,N 1 ,N
of D, q, 1-q
index boundaries
1 ,N 1 ,N
change
1
on the
,
H
N 1
L S
h h
k h kk k
h
k k
h h
h k h kh k k k k
h
h h
k
k
h
h
k
k
h
h
V i k C q q V i k
D C q q
C q V i h
V i k C q
C q
q V i k
D
  
 

    
 
 

 
      
 
 
       





 
 
 
   
       
1 10
1
1
1
1
0
1 isolate 0 on RHS
and h+1 on LH
1 ,N ,
S
1
,N
N
k k h
k k
h
h k
h
h
h
h k
hk
h
k
k
k
hC q q V i k C q V i
V
D
q i k
  

 
   
 


 
 
 
  
 
 
 
 




 




 
     
   
   
1
1
0
1
1
10
1
1
1
1
1
1 0
111 1
1
1
1,N
1N
N
1 , ,
,
N
k
h
h
h
h
h h
h kh k k
h
h kk k k
h h
k
h h
h
C
h
C
h
C
k
D C q q V
C
C C q q V i k i k
V
h
q i
C q V i




 



  









 
 
 
 
   
    
  
 
 
 
 

 




 




60. Proprietary & Confidential
For Use by Permission Only
42. Induction’s conclusion:
• For h=1:
• And for h=2 (optional):
• And from h to h+1, it was demonstrated that:
• Conclusion: for an Euro-Call (resp. []- payoff for an Euro-Put):
28 March 2016 60
        1(j 1), ,N 1 , 1,N 1- ,NNN i V i D qV i q V i       
     
2
22
2 2
0
(j 2), ,N 2 1 ,N
kk k
N
k
N i V i D C q q V i k



       
     
     
0
1
11
1
0
,N 1 ,N
,
,N (h 1) 1 ,N
h
h kh k k
h
k
N h h
h kh k k
h
k
V i h D C q q V i k
i
V i D C q q V i k


 
 


 
    
  
  
       
  



 *
risk freefor any back time step for any node i at a value of the derivative discount backfrom the expiration given time step N-h at node i, time step N-h from ma
, ,N h
N N hh i V i h D      
  
   
0
binomial risk neutral boundary value of the
probability mass function derivative at time step Nturity to
and at node i+ktime step N-h
1 ,N
h
h kk k
h
k
C q q V i k


   
2.1/ Cox-Ross-Rubinstein

61. Proprietary & Confidential
For Use by Permission Only
43. Now write at the present (h=N) value V(k,N) of a derivative
with defined Payoff at the boundary for any k level :
44. With the following interpretation:
the formulation above gives the value of the derivative as the
RFR present value of the future RN expected payoff cash flows,
aka Fundamental Asset Pricing Formulae, aka Feynman-Kac.
28 March 2016 61
   0
0
1 ,N
N
N kN k k
N
k
V D C q q V k


 
 
    
0
present valuationvalue of thee binomial risk neutralwith risk free ratederivative now probability mass function, iefrom expiration to now
, 1(N period discount)
1
N kk k
N
N kN k k
N
f B N q k C q q
V D C q q


  
   
 
 
risk free present value of the derivative's risk neutral ex
0
k-payoff
at time step N
of the derivative
derivative's risk neutral expected payoff, ie ,N
,N
N
k
V k
V k

  


 


pected payoff
2.1/ Cox-Ross-Rubinstein

62. Proprietary & Confidential
For Use by Permission Only
45. Now develop the expression as follow:
28 March 2016 62
   
   
 
     
       
0
0
0
0
0
0 ,N
0
0 ,N ,N
0
1 ,N
1 ,N Euro-Call option case
1
1
1 1
N
N kN k k
N
k
N
N kN k k
N
k
N
N kN k k k N k
N
k
N
N kN k k k N k
N S k K
k
N
N k N kN k k k N k k k
N NS k K S k
k
V D C q q V k
D C q q S k K
D C q q S u v K
D C q q S u v K
D C q q S u v K C q q




 

 


 
 

 
      
    
  
   





1
1 1
          
0
0 ,N ,N
0 0
1 2
1 1
N
K
k
N N
N kk N kk N k k
N NS k K S k K
k k
S C Dqu D q v D K C q q

 
 
 
 
 
 
 
 

 
 
1 1
2.1/ Cox-Ross-Rubinstein

63. Proprietary & Confidential
For Use by Permission Only
46. The second term is:
since the expectation of the indicator function is the
probability of the indicator condition. Here, the probability is
q, under the risk neutral -measure.
47. For the first term, notice that:
48. Therefore the first term is under the new -measure:
28 March 2016 63
    
       
 
 
  
,N
0 0,N
0underlying binomial risk neutral
S, @t=0 probability mass function, ie
,D 1
1 1 ,N
S k K
N kkk
N
N
N kkk
N S k K
k
f B N qu k C Dqu D q v
S C Dqu D q v S S k K


 






 
  



 




 1
1
 
   
 
1/ 1/
1 1 1
1 1 another probability measure,
u Duv Duv vD v D v u v
Dqu D q v Du Dv
u v u v u v u v
D q v Dqu
     
        
    
     
  
    
 
 ,N
,N
callrisk free 0
binomial risk neutraloptionpresent
probability mass function, iestrikevaluation
, 1
risk neutral expectation,
2 1
S k K
N kk k
N
N
N kN k k
N S k K
k
f B N q k C q q
D K C q q





 
 



 



 
 

  1
1   ,NN
D K S k K


2.1/ Cox-Ross-Rubinstein

64. Proprietary & Confidential
For Use by Permission Only
49. From there write:
50. From the previous expressions of D, q and u, infer the
probability measure implied in o( ) approximation by these
results:
Form above, note that the probabilities in the respective
measures and the impact of an increase in the asset volatility
28 March 2016 64
          
  
0 0 ,N ,N
0 0
1 2
0 0
value of value of
call option underlying
@
binomial cu
t=0 S,
mulativ
t=0
e
@
1 1
,N
N N
N kk N kk N k k
N NS k K S k K
k k
V S C Dqu D q v D K C q q
V S S k K
 
 
 
 
  
  
 
 
 

1 1
   
Eurodiscount at
Callthe risk free rate
striketo t=
density cumulative density
with the measure with the me0 asure
,NN
D K S k K 
 

   
2.1/ Cox-Ross-Rubinstein
Small δt Dqu D(1‐q)v q (1‐q) p, (1‐p)
Taylor approx.
1
2 2 2
1
2 2 2
1
2 2 2
1
2 2 2
1
2 2
Measures ‐measure ‐measure ‐measure

65. Proprietary & Confidential
For Use by Permission Only
2.1/ Cox-Ross-Rubinstein
51. And from the CLT on the Binomial at the limit for → ∞,
convergence to the normal distribution, from which we get
the BSMF:
51. The -measure is the risk neutral measure under the
currency numeraire, and N(d2) the probability of exercise
of the option under the -measure.
52. The -measure is the risk neutral measure under the asset
numeraire, and N(d1) the probability of exercise of the
option under the -measure.
28 March 2016 65
   
 0 0 1 2
BSMF for the
N
Call Option
r T t
V S d e KN d 
  

66. Proprietary & Confidential
For Use by Permission Only
3/ Risk Premium and
Equivalent Martingale Measures
28 March 2016 66

67. Proprietary & Confidential
For Use by Permission Only
3/ Risk Premium & Equivalent -Martingale
1. Assume a stock following a GBM:
2. Assume a risk free asset (RFA) such as money market
account with constant RFR r:
3. Discount the asset process to t=0 (removing the drift r of
the RFA, or time value of money, from the Stock’s drift),
and normalize by the initial stock value at t=0. Get the
normalized Underlying Security Excess Return (USER) υt:
28 March 2016 67
 
2
2
00, ,
tt X
t
t t
t
dS
t T dt dX S S e
S

 
 
 
   
 
     


  1 1
drift only
no diffusion
0, , 1
t
rt
t
t
t
dA rAdt
t T A e
d dA rA dt
A
 


     
   
 

 
0 0
is the normalized Underlying1
0, ,
Security Excess Return (USER)
rt t t
t
t
S S
t T e
S S A

 
    

68. Proprietary & Confidential
For Use by Permission Only
4. Definitions and Goals:
• A martingale is a driftless Random Process Mt such that:
• Equivalent measures: are two measures which are
absolutely continuous v.a.v. each other - or equivalently, for
which no impossible event under one measure may be possible
under the other measure
• Adapted (Measurable) Process: the Random Process Mt is
said to be adapted to the filtration (or measurable wrt t), iff
Mt is known given the information set t
• Goal: seek a -equivalent measure , under which υ is a
martingale - in other terms, seek an equivalent martingale
measure for the USER, υ. We use:
1. Radon-Nikodym Derivative
2. Doléans Exponential and
3. Girsanov’ s theorem. 68
   
2 1 1 2
denoting thereafter:
0, , 0, ,
t
t t t tt s t t
M
t T s T t
M MM M
       
                    

  
3/ Risk Premium & Equivalent -Martingale

69. Proprietary & Confidential
For Use by Permission Only
28 March 2016 69
5. The USER follows the RFR adjusted GBM, removing the time value
of money (apply Itô):
– Note: if μ=r, then υ is a -martingale, and reciprocally
– If μ-r>0 (the natural case - investors expect positive excess
returns) then υ is a -submartingale i.e. ℙ
υ υ
– If μ-r<0 then υ is a -supermartingale i.e. ℙ
υ υ
– Assume μ r going forward (i.e. υ is no -martingale)
6. Radon-Nikodym states: if  and  are equivalent measures,
then there exists a RP Λ(t) such that (see RHS below):
 
2 2
2 2
0
Ito on ln( )
t tt X r t X
rt rtt
t
t
t
t
S
e e e e
S
d
r dt dX
 
   



 

   
               
  
   
 

 
 
 
such that for any
there exists a random set A of events
process over , , we have
, , , ,
A
dA A
 
        



  


3/ Risk Premium & Equivalent -Martingale

70. Proprietary & Confidential
For Use by Permission Only
7. The Novikov condition on a RP θt and horizon T is such that:
8. Introduce the Doléans Exponential ε of a RP θt, defined as
below:
9. With the Novikov condition, the Doléans exponential is a
martingale – actually an exponential martingale:
28 March 2016 70
   
2
0 0
1
2
0
0, ,
t t
s s st ds dX
s st T dX e
 
 
    
 
2
0
2
0
1
2
1
2
which can be stated
T
s
T
s
ds
ds
e
e d

 

 
  
 
       
  




     
2
0
1
2
0 0
martingale condition
Novikov condition
0, ,
T
s ds T t
t s s s se t T dX dX

   
             
 



 
3/ Risk Premium & Equivalent -Martingale

71. Proprietary & Confidential
For Use by Permission Only
10. The Girsanov theorem states that:
11. And in that case, the following corollary:
• Radon-Nikodym states the existence of the relationship in
the measures. Girsanov expresses the value and the
explicit correspondence between the two measures – given
θ. In practice, one needs to find an appropriate θ.28 March 2016 71
 
   
 
0
then
if there exists a
such that its Doélans'random process θ
expectation under is one
for any
random process X
, ,
, ,
,
/ 1
/
,
T
s sdX
X
   

    
  
 



 










P  
 
 
 
0
the Doléans exponential is the
Radon Nikodym Derivative
there exists a such-equiv thata
measure at any ti
t
0
n
m
le
e
, 0, ,
t
s s
T
t s s
d
dX and
d
t T
dX

 
 


 









 
 




  
and the Doleans exponential is a martingal
0
e
t
s sdX

 

 
 
 
 
 
 
 
  
       
 



   0
0, , : driftless ABM on , , a martinga, le
t
t t s tt T X X ds X     
 


3/ Risk Premium & Equivalent -Martingale

72. Proprietary & Confidential
For Use by Permission Only
12. From the -GBM υ in (5) and the -ABM X in (11) write:
13. For υ to be a -martingale, it is sufficient to cancel its drift
under . That sufficient condition also gives an explicit
formulation for θ:
28 March 2016 72
 
   
 
0
t t t t
t
t t s t t t
t
t t
t
t
t t
t
d r dt dX
X X ds dX dX dt
d
r dt dX dt
d
r dt dX
   
 

  


  

  


     
    
    

 



 
market price of risk
aka risk premium
aka the Sharpe ratio
t t t t
r
d dX

   


    
3/ Risk Premium & Equivalent -Martingale

73. Proprietary & Confidential
For Use by Permission Only
14. Therefore with RND (the Novikov condition is also valid):
15. By changing measure from  to :
28 March 2016 73
     
2
0 0
2
0
Doléans exponential
1
2
1
2
0, ,
t t
t
t
t
s
RND
r
r r
dt dX
r r
t X
d r
t T t t dX
d
e
e

 

 
 
 
 



 
 
 
  
  
 
    
    
   
 
      
 
 







  2
2
0
expression of X with
22
0
is a -martingale
(no drift, diffusion only)
0, rt t
t
t
t
t t
ABM
S
e
r t XSt X
t t t t t
r
X t X
t T
d dX e S S e
  



   


 
     
 

 


  

    











3/ Risk Premium & Equivalent -Martingale

74. Proprietary & Confidential
For Use by Permission Only
16. Remarks: from  to , and vice-versa:
17. The Market Price of Risk, aka Risk Premium, aka Sharpe
Ratio:
– Specifies the Doléans exponential as the Radon Nikodym
Derivative, and
– Enables the measure transition from Real-World to Risk-
Neutral, and vice-versa from Risk-Neutral to Real-World
with its inverse.
28 March 2016 74
2 2
0 0
2 2
RND, Doléans RND, Doléans
Real Risk
World Neutral
t t
t t t t
t t
t t s t t s
t X t X
X X ds X X ds
d d
e e
d d
r
 
 
 



  
    
   

 

 
 

 


 



3/ Risk Premium & Equivalent -Martingale

75. Proprietary & Confidential
For Use by Permission Only
18. Replicate our derivative Vt with (1) stock S and (2) risk-
free asset At=ert through a self financing portfolio Wt s.t.:
19. The Excess Returns of the Replicating Portfolio ω (RPER)
and that of the Derivative ν (DER) are:
28 March 2016 75
 

qty qty priceprice
Stock RFA FRAStock
Ito integral Riemann integral
0 0 0
change in value change in value
between 0 and t
portfolio composition @t
0, ,
S A
t t t t t
t t
S A
t t t t t
W S A
t T
W W dS dA
    
 
     
  
 

in S or A
or between S and A - only
no consumption or contribution





 




  0 0
0 0
1
normalized RPER
0, ,
1
normalized DER
rt t t
t
t
rt t t
t
t
W W
e
W W A
t T
V V
e
V V A





  

  
   

3/ Risk Premium & Equivalent -Martingale

76. Proprietary & Confidential
For Use by Permission Only
20. W is the replicating and self financing portfolio, and through
the Itô product rule, simplified here with constant r:
28 March 2016 76
1 1
1 1
0
0
1 1
, replication self financing
condition condition
1
t t
t t
t t
t
t t t t t t
t
dA rA dt
dA rA dt
S A S A
t t t t t t t t t t
W dW
W
d d W d dA W A dW
W A
rA dt S A A dS rA dt
 
 
 


 
 
 
    
 
  
 
         
 
  
 
 
 
   
0
1 1
1 1
1
1
0
delta hedging
0 0
0 0
dynamic hedging driftless martingale
t
t
t
t
t
t
S S
t t t t t t
S
t t t t t
S
S A
S S
t t t t t
d
dX
S S
t t t t t t t
rA dt S A dS
dA S A dS
d A S d S
S S
d d d dX
W W






   
 
 




 
 
 

    
  
   
     



  
3/ Risk Premium & Equivalent -Martingale

77. Proprietary & Confidential
For Use by Permission Only
21. Consider the following Risk-Neutral results:
• -Martingale: ωt is a -martingale and Wt is given below:
• -Expectation: further with the Itô integral:
• -Conditional Expectation: by definition of a martingale :
28 March 2016 77
 
   
0 0
0 0
0 0
Ito integral
expectation of Ito integral = 0
1 1
0, , 1 constant RN Expectation
t t
S S
t t t t t t t
t
S S
dX dX
W W
t T
 
  

 
               
     
 
    




  

   0, , Martingale propertyT t t T tt T          
 
 
 
0 0
0 0
0 0
0 0
Ito integral
1
1
0
0 00 0
0 deterministic stochastic
RPER is
0, ,d
Martingale
1
t
t
t
rt
t
t t tS S
t t t t t t t t
W
W At t
S rt rt S
t t t t t t t t
A e
S S
t T dX d dX
W W
S
dX W e W e S dX
W



   

   



       

       
 
 
 
 

  
replicating
portfolio


3/ Risk Premium & Equivalent -Martingale

78. Proprietary & Confidential
For Use by Permission Only
22. Completeness: the replication condition gives the identity
between the replicating portfolio and the derivative, as well
as their respective excess returns, including boundaries, over
the time period:
23. At t=T, VT=payoff=derivative value=replicating port value WT
28 March 2016 78
 
 
 
0 00 0
1
0, ,
payoff
t
t t t t t T
T T T T
W V
t T W V
W V
  
  
 
   
 
       
    




     
 
 
 
 
 
0
0
is a martingale
replication condition
1
0, ,
V simplifies
0, ,
derivative's option payoff @T
discount tovalue @t =
t T t
t
t t
t
t
t
t t t T t T
t t T
t t T
V
t T
V A
T
t t t
T
t T
V
V A
A
 

 

   
 
 



  
    
 
 
 
    
 

 





 



origin 0

Feynman
-Kac
3/ Risk Premium & Equivalent -Martingale

79. Proprietary & Confidential
For Use by Permission Only
QUANT SHEET:
79
2
2
00
0
0
0
1
W is replicating portfolio
V is Euro-derivative
1
1
filtration
& adaptation
tr t Xt
t t s t
t
t
t
rt
t
t
t
t
r
t
t
t
t
t
t
t
X X ds S S e
dS
dt dX d
S
A e
S
S A
W
W A
V
V A






 



 
   
 

 

   

  



























2
2
underlying security excess return
0 0
0
0 0
dynamichedging replicating portfolio excess return
(complete) r
1
tt X
t t t t
t
S S
t t t t t t t
t t
dX e
S S
d d dX
W W


   

   
 
 
  
     






 
   
 
 
eplication condition
0
Radom Nikodym Derivative
aka Doléans Exponential
RPER Expectation, constant,
1
no drift under
RPER Conditional Expectation
diffusion o
t
s s
t
t T t
d
t dX
d
 

 











   
 
 









nly, martingale
Derivative Value @t - (Feynman-Kac)
Warning: Euro-style only
T
t t t
T
V
V A
A























 

  
   
  



‐martingales
3/ Risk Premium & Equivalent -Martingale

80. Proprietary & Confidential
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24. Now, price derivative V:
1. Equivalent Martingale Measures and
Probability Calculations
2. Equivalent Martingale Measures and Change
of Numeraire
28 March 2016 80
3/ Risk Premium & Equivalent -Martingale

81. Proprietary & Confidential
For Use by Permission Only
3.1/ Risk Premium
Equivalent Martingale Measures
and Probability Calculations
28 March 2016 81

82. Proprietary & Confidential
For Use by Permission Only
25. From Feynman-Kac in (23):
25. From (1) the GBM and (15) change of measure:
26. Therefore, Yt;T follows a normal distribution with:
28 March 2016 82
 
   
   
2
2
2ln
2
;
the ln is a normal Random
Proces with boundary @T
0, ,
ln ,
2
t t
T
T tt
r
X X t
t t
t t
t t
r T t X X
T t
T
t T
t
dS dS
t T dt dX rdt dX
S S
S S e
S
Y N r T t T t
S




  



 

 
     
 
 
  
      

 
 
     
  

 







 
 ;
2
;
2
2
T
t
t
TY r T t
T tY


  
       

    



   
 0, ,
rt
tA e
r T tT
t t t t t T
T
V
t T V A V e V
A

  
     
 
 
 
3.1/ EMM & Probability Calculations

83. Proprietary & Confidential
For Use by Permission Only
27. From the definition of expectation of Yt;T, where g is the
payoff function:
28. The SNRV Z~N(0,1) is such that:
29. Hence:
30. In the case of a Euro-Call:
28 March 2016 83
   
   
   0, , r T t r T t y
t t T tt T V e V e g S e p y dy
   

    


 
 
2
2
2
2
Y r T t
Z Y r T t Z T t
T t




 
   
         
  
 
 
2
2
2
2
=pdf
of SNRV=T-payoff RV @t
2
z
r T t z T t
r T t
t t
e
V e g S e dz



  
         

 
 
 
 



     
 
2
2
2
2
2
z
r T t z T t
r T t
T T t t
e
g S S K V e S e K dz




  
          

 
     
 
 

3.1/ EMM & Probability Calculations

84. Proprietary & Confidential
For Use by Permission Only
31. Notice that:
32. Therefore:
28 March 2016 84
 
 
 
2
2
2
2
0
0 ln 0
2
ln
2
r T t Z T t
t
t
t
S
S e K r T t T tZ
K
S
r T t
K
Z z
T t






 
     
 
 
         
 
 
    
    

 
 
 
 
2
2
2
20
0
2
2
... 0
2
2
2
2
z
z
r T t z T t
r T t
t t
z
r T t z T t
r T t
tz
e
V e S e K dz
e
e S e K dz









  
         

            

 
  
 
 
 
  
 
 


3.1/ EMM & Probability Calculations

85. Proprietary & Confidential
For Use by Permission Only
33. Then:
28 March 2016 85
 
 
 
 
 
 
 
2 2
2
0 0
2 2 2
0 0
22
0 0
2
0
2 2
2
2 2 2
1
2 2
1
2
2 2
2 2
2 2
2
z z
r T t z T t
r T t
t tz z
z z
T t z T t
r T t
t z z
z
z T t
r T t
t z z
z T t
z
t
e e
V e S e dz K dz
e e
S dz e K dz
e e
S dz e K dz
e
S dz






 
 
 

   
         
 
     
  
 
   
  
 
  

 
 
  
 
 
 
 

 
 
 

  
 
 
 
2
0
0,1,1
00
2
2
NN T t
z
zr T t
P Z zP Z z
e
e K dz


 

 


 
 
3.1/ EMM & Probability Calculations

86. Proprietary & Confidential
For Use by Permission Only
34. From there:
28 March 2016 86
 
   
 
 
 
 
 
 
 
 
 
,1 0,1
0 0
2
2
1
1
2
0,1
0 2
ln
2
ln
2
ln
2
N T t N
t
t
t t
t
N
P Z z P Z z T t
S
r T t
K
N Z T t
T t
S
r T t
K
N Z N d
T
Z
t
V S N d e
S
r T t
K
P Z z N
T t
N d









 
 
 
 
    
  
    
     
 
  
 
 

 
 

     
     
 
  
 
  
  
    
  







 
 2
r T t
N dK
BSMF
 

3.1/ EMM & Probability Calculations

87. Proprietary & Confidential
For Use by Permission Only
3.2/ Risk Premium,
Equivalent Martingale Measures
with Change of Numeraire
28 March 2016 87

88. Proprietary & Confidential
For Use by Permission Only
35. Start from Feynman-Kac in (23); to lighten the notation, look
at the Euro-Call (for Euro-Put: opposite payoff or C-P parity):
28 March 2016 88
3.2/ EMM & Change of Numeraire
 
 
 
   
2
2
0
A deterministic
2
0
2
0 0
0, , Feynman-Kac
Euro Call case
V
tt T
T
T
t t t
T
T
t t
T T T
r T t X X
t
t t
T
r T Xt
rT
V
t T V A
A
S K
A
A S K
A
S e K
A
V e S e K







 
     
 
 
       
 
    
 
 
  
   
  
     
   

  

 
  




 




   
2
2
0
2
1
@ 0
T
T
T
r T X
rT
S K S K
rT
t
e S e e K



 
    




 
     
  
 
       
 

 


 1 1

89. Proprietary & Confidential
For Use by Permission Only
36. For the second term, quite simple (the expectation of the
indicator function is the probability):
28 March 2016 89
     
2
probability of theexpectation of
indicator conditionthe indicator
2
0
2
0
2
ln
2
T
T
T TS K S K
rT rT rT
r T X
rT
rT
S K
K
K
e K e K e K
e K S e
S
e K r



  

 
     

 
        
 
 
 
 
  
   
  




 





 1 1
 
 
2
2
0
2
ln
2
0
2
,1
T
rT T
T
T
d
rT
T X
S
r T
X X
e K N
T T
K
T
X
e
T
KN d






 
 
 
   
    
       
 
 
  

 
    


 

 


3.2/ EMM & Change of Numeraire

90. Proprietary & Confidential
For Use by Permission Only
37. Form the first tem:
38. Notice on the RHS the Doléans Exponential, hence a new RND
and its corresponding new measure :
39. Change to new measure , with θs=σ:
28 March 2016 90
 
2 2
0 02 2
0
T T
tTT X dt dX T
t
d
e e dX
d
 
 
 
      
 
 

  2
2
expression of X with
22
0 0
0,
=
t
t
t t
ABM
r t Xt X
t t
X X t
t T
S A S e S e
 


 
    
 
  

  




 


   
2
2
2 2
0 01 T
T
T
TS
r T X T X
T
S
r
K K
e S e S e
  
 
 
       
   
    
     


 
 1 1
3.2/ EMM & Change of Numeraire

91. Proprietary & Confidential
For Use by Permission Only
40. Go back to the first term calculation:
28 March 2016 91
     
 
2
2r deterministic
(constant)
2 2
0 0 0
, aka RND
expectation
0
to change of mea
1 T T T
T
T
T
r T X T X
rT
d
S K S K S K
S K
d
d
e S e S e S
d
d
S d
d
  
 
       


 



  
               
 


  

 
 

 




  1 1 1
1      
2
0 0 0
-probability of the-probability expectation of the
indicator conditionindicator functionsure with RND
2
0 0
T T TS K
d
d
r
S K
S d S S
S S e
S K



 




 
    



 


 
 

1 1
 
 
1
2
0
0
2
0
0 1
0 0
ln
2
ln
2
1
,1
T
T
T X
T
T T
T
d
S
K
K
K
S
S r T X
S
r T
X X
S T
S
N X
T
N
T
d
T









                     
   
    
        
 
 
  

   
 


 



 


3.2/ EMM & Change of Numeraire

92. Proprietary & Confidential
For Use by Permission Only
41. Conclusion:
42. Corollaries:
– N is the normal distribution, which is pervasive throughout
the demonstration
– N(d2) is the -probability of exercising the option at
maturity, under the risk neutral measure with the currency
numeraire
– N(d1) is the -probability of exercising the option at
maturity, under the risk neutral measure with the asset
numeraire.
28 March 2016 92
   10 20
rT
V S N e Nd K d
 
3.2/ EMM & Change of Numeraire

93. Proprietary & Confidential
For Use by Permission Only
4/ Dual Variables Parameters with
Forward Kolmogorov Equation (aka
Fokker-Planck Equation) and with
Transition Probability Density Function
28 March 2016 93

94. Proprietary & Confidential
For Use by Permission Only
4/ Dual Variables Parameters with FKE
1. Fundamental Asset Pricing Formulae of Feynman-Kac for
Euro-Style:
2. Switch around T (expiration) and K (Strike), for given St
and t. For the Call Option case (i.e. with ε=+1):
28 March 2016 94
 
Feynman-Kac, aka Fondamental Asset Pricing Formulae
risk free risk neutral processriskneutral
discout payoff @Texpectation@t
forPV@t
,t (T)
T
t
rdt
t tV S e Payoff

  
     
  
   

 
 

 
with derterministic
any measure works with FK,
provided measure identical
for expectation and process
= 1 for Euro-Call
or Euro-Put Options
T
t
rdt
t Te S K


 

       
 




   
 
Euro-Call option payoff case
r
,t r T t
t t TV S e S K
    
 


   
   
risk freeparameters variablesvariable parameters transitionprobabilitycalloptionpresent
densityfunctionpayoffvaluation
S ,t ; K,T , ; , , = , ; ,r T t
t t T T tV V K T S t V K T e S K p S T S t
 
  
    
   
   
   
Risk NeutralExpectation@t
0
(tpdf)
TdS





95. Proprietary & Confidential
For Use by Permission Only
4/ Dual Variables Parameters with FKE
3. For the Call case write, at given St and t:
4. Derive V w.r.t. K, once & twice:
28 March 2016 95
   
   
   
   
0
0
...dS 0
, ,
, ,
K
T
r T t
T T T
r T t
T T TK
V K T e S K p S T dS
V K T e S K p S T dS
  

 
 

  


   
     
 
 
   
 
 
 
( ) ( ) 0 ( ) (.) 0
Leibniz
integral
ru
0
2
le
2
,
= , ,
= ,
,
,
,
T
r T t
T T T T KK
S K p S and K K p
r T t
T TK
r T t
T K
r T t
V K T
e p S T dS S K p S T
K
e p S T dS
V K T
e p S T
K
e p K T
  
    

 
 
 
 
 
        
 
 


    




