Transcript Black-Scholes Formula

第四章
Brown运动和Ito公式
Chapter 5
European Option Pricing
-----Black-Scholes Formula
Chapter 5
European Option Pricing
-----Black-Scholes Formula
Introduction

In this chapter,
 we will describe the price movement of an underlying
asset by a continuous model --- geometrical Brownian
motion.
 we will set up a mathematical model for the option
pricing (Black-Scholes PDE) and find the pricing formula
(Black-Scholes formula).
 We will discuss how to manage risky assets using the
Black-Scholes formula and hedging technique.
History
 In
1900, Louis Bachelier published his
doctoral thesis ``Thèorie de la Spèculation",
- milestone of the modern financial theory. In
his thesis, Bachelier made the first attempt
to model the stock price movement as a
random walk. Option pricing problem was
also addressed in his thesis.
HistoryIn 1964, Paul Samuelson, a Nobel Economics
Prize winner, modified Bachelier's model,
using return instead of stock price in the
original model.
 Let St be the stock price, then dSt / St is its
return. The SDE proposed by P. Samuelson is:
dSt
 rdt   dWt
St
 This correction eliminates the unrealistic
negative value of stock price in the original
model.

History- P.
Samuelson studied the call option pricing
problem (Ć. Sprenkle (1965) and J. Baness (1964) also studied it
at the same time). The result is given in the
following. (V,etc
as
before)
 T
 T
V e
d1 
c
 Se
S
N (d1 )  KN (d 2 )  ,
ln  S / K    S  1/ 2 2  T
,
 T
1
d 2  d1   T , N ( x ) 
2
x
e

 2 / 2
d
History--In 1973, Fischer Black and Myron Scholes
gave the following call
 option pricing formula  rT

V  SN (d1 ) - Ke
N (d 2 )
 S & c
 Comparing to Samuelson’s one,
are no
longer present. Instead, the risk-free interest
r enters the formula.
History---
The novelty of this formula is that it is independent
of the risk preference of individual investors. It
puts all investors in a risk-neutral world where the
expected return equals the risk-free interest rate.
The 1997 Nobel economics prize was awarded to
M. Scholes and R. Merton (F. Black had died) for
this brilliant formula and a series of contributions
to the option pricing theory based on this formula.
Basic Assumptions
 (a)
The underlying asset price follows the
geometrical Brownian motion:
dSt
  dt   dWt
St
μ
– expected return rate (constant)
 σ- volatility (constant)
dWt

- standard Brownian motion
Basic Assumptions  (b)
Risk-free interest rate r is a constant
 (c) Underlying asset pays no dividend
 (d) No transaction cost and no tax
 (e) The market is arbitrage-free
A Problem
 Let
V=V(S,t) denote the option price. At
maturity (t=T),

( S  K ) , call
V (S , T )  

( K  S ) , put
where K is the strike price.
 What is the option's value during its lifetime
(0< t<T)?
Δ-Hedging Technique
 Construct

a portfolio
  V  S
(Δ denotes shares of the underlying asset),
choose Δ such that Π is risk-free in (t,t+dt).
Δ-Hedging Technique portfolio Π starts at time t, and Δ remains
unchanged in (t,t+dt), then the requirement
Π be risk-free means the return of the
portfolio at t+dt should be
 If
 t  dt   t
 rdt
t
i.e. dVt  dSt  d  t dt  r (Vt  St )dt
Δ-Hedging Technique - Since
Vt  (St , T ),
where the stochastic process
SDE*, hence by Ito formula
St
satisfies
 V 1 2 2  2V
V 
V
dVt  
  S
 S
dWt
 dt   S
2
S
S 
S
 t 2
 V 1 2 2  2V

V
  S
 S
 S  dt

2
So
S
S
 t 2

V


  S
 S  dWt  r (V  S )dt
S


Δ-Hedging Technique -- Since
the right hand side of the equation is
risk-free, the coefficient
dWtof the random term
on the left hand side must be zero.
Therefore, we choose

V

S
Δ-Hedging Technique ---Black-Scholes Equation
 Substituting
it, we get the following PDE:
V 1 2 2  2V
V
  S
 rS
 rV  0
2
t 2
S
S

 This
is the Black-Scholes Equation that
describes the option price movement.
Remark
 The
line segment {S=0, 0< t< T} is also a
boundary of the domain . However, since
the equation is degenerated at S=0,
according to the PDE theory, there is no
need to specify the boundary value at S=0.
Black-Scholes Equation
in Cauchy Form


2
2
 V 1

  V
2  V

 
r 
 rV  0,

2
x 
2  S
  2

(, )  (0, T ]



x


 e  K  , call
V  0  
x 

 K  e  , put


Well-posed Problem
 By
the PDE theory, above Cauchy problem
is well-posed. Thus the original problem is
also well-posed.
Remark

The expected return μ of the asset, a parameter in
the underlying asset model , does not appear in
the Black-Scholes equation . Instead, the risk-free
interest rate r appears it. As we have seen in the
discrete model, by the Δ---hedging technique, the
Black-Scholes equation puts the investors in a
risk-neutral world where pricing is independent of
the risk preference of individual investors. Thus
the option price arrived at by solving the BlackScholes equation is a risk-neutral price.
An Interesting Question
1) Starting from the discrete option price
obtained by the BTM, by interpolation, we can
define a function V ( S , t ) on the domain
Σ={0< S<∞,0<t<T};
 2) If there exists a function V(S,t), such that

lim V ( S , t )  V ( S , t ), (( S , t )  ),
t 0
3) if V(S,t) 2nd derivatives are continuous in Σ,
What differential equation does V(S,t) satisfy?

Answer of the Question
 V(S,t)
satisfies the Black-Scholes equation
in Σ. i.e., if the option price from the BTM
converges to a sufficiently smooth limit
function as Δ 0, then the limit function is a
solution to the Black-Scholes equation.
Black-Scholes Formula (call)

V ( S , t )  SN (d1 )  Ke
 r (T  t )
N (d 2 )
ln( S / K )  (r   / 2)(T  t )
d1 
,
 T t
2
d 2  d1   T  t ,
1
N ( x) 
2

x

e
 2 / 2
d
Black-Scholes Formula (put)

V ( S , t )  Ke
 r (T  t )
N (d 2 )  SN (d1 )
ln( S / K )  (r   / 2)(T  t )
d1 
,
 T t
2
d 2  d1   T  t ,
1
N ( x) 
2

x

e
 2 / 2
d
Generalized Black-Scholes Model (I) ----Dividend-Paying Options
 Modify
the basic assumptions as follows
(a^)The underlying asset price movement
satisfies the stochastic differential equation
dSt
  (t )dt   (t )dWt
St
(b^) Risk-free interest rate r=r(t)
(c^) The underlying asset pays continuous
dividends at rate q(t)
(d) and (e) remain unchanged
Δ-hedging
the Δ-hedging technique to set up a
continuous model of the option pricing, and
find valuation formulas.


V


S
 Construct a portfolio
 Choose Δ, so that Π is risk-neutral in
[t,t+dt].
 the expected return is
 Use
t dt  t  rt dt
Δ-hedging  Taking
into account thedividends, the
t  dt
portfolio's value at
is
t  dt  Vt  dt  t St qt dt  t St  dt

Therefore, we have
dVt  t dSt  rt t dt  t St qt dt
B-S Equation with Dividend
Ito formula, and choose t  V / S
we have
 Apply
 V  2 (t ) 2  2V

S

2

t
2

S



V

 dt  r (t ) V  S
S


V

dt
 dt  q(t )S
S

Thus, B-S Equation with dividend is
V  2 (t ) 2  2V
V

S
  r (t )  q(t )  S
 r (t )V  0
2
t
2
S
S
Solve B-S Equation with dividend

Set

choose α&βto eliminate 0 & 1st terms:
u  2 (t ) 2  2u
u  Ve
t

2
 (t )
y
 (t )
, y  Se
y 2
u
  r (t )  q (t )   '(t )  y
  r (t )   '(t )  u  0
y
Solve B-S Equation with dividend α&β be the solutions to the following
initial value problems of ODE:
d
 d
 Let
 r (t )  q (t )  0,
 r (t )  0,

dt
 dt
 (T )   (T )  0
 The
solutions of the ODE are
T
T
 (t )    r ( )  q( )d ,  (t )   r ( )d
t
t
Solve B-S Equation with dividend - Thus
t
T
under the
2transformation,
^
2 and take
    ( )d, T    ()d
0
0
the original problem
is reduced
to
2
 u

u
2
 t  y y 2  0


u ^  Ve  (t )
 (y  K)
^
t

T
t T

Apply B-S Formula
 Let
σ=1, r=0, T=Tˆ, t=τ,
u ( y, )  yN (d )  KN (d )
^
1
d 
^
1
ln y / K  1/ 2(T   )
^
T 
^
d  d  T 
^
2
^
2
^
1
^
European Option Pricing
(call, with dividend)
 Back
to the original
variables,
we have
  (t )
 (t )
^
^


V (S , t )  e
Se
N
(
d
)

KN
(
d
)
1
2


T

 Se t

q ( ) d
T

N (d )  Ke t

^
1
r ( ) d
N (d 2^ )
T
d1^ 
ln S / K    r ( )  q( )   2 ( ) / 2  d
t


T
t
d d 
^
2
^
1

T
t
 ( )d
2
 2 ( )d
European Option Pricing
(put, with dividend)
T

V ( S , t )  Ke t

d 
^
1
r ( ) d
ln S / K  
T
t

d d 
^
1

T
t
q ( ) d
N (d )
 r ( )  q( )   2 ( ) / 2  d
T
t
^
2
T

N (d )  Se t
^
2

 2 ( )d
 ( )d
2
^
1
Theorem 5.1
c(S,t) - price of a European call option
 p(S,t) - price of a European put option,
with the same strike price K and expiration
date T.
 Then the call---put parity is given by

T
c(S , t )  Ke t

r ( ) d
T
 p(S , t )  Se t

q ( ) d
where r=r(t) is the risk-free interest rate, q=q(t) is
the dividend rate, and σ= σ(t) is the volatility.
Proof of Theorem 5.1
Consider the difference between a call and a put:
W(S,t)=c(S,t)-p(S,t).
 At t=T,



W (S , T )  (S  K )  ( K  S )  S  K

W is the solution of the following problem
 W  2 (t ) 2  2W
W

S
  r (t )  q(t )  S
 r (t )W  0

2
2
S
S
 t
W
 t T  S  K
Proof of Theorem 5.1 Let
W be of the form W=a(t)S-b(t)K, then
a '(t ) S  b '(t ) K   r (t )  q (t )  Sa (t )
 r (t )  a(t ) S  b(t ) K   0
 Choose a(t),b(t) such that
a '(t )   r (t )  q (t )  a (t )  r (t )a (t )  0
b '(t )  r (t )b(t )  0
a (T )  b(T )  0
Proof of Theorem 5.1- The
solution is
T

t
a(t )  e

 Then
q ( ) d
T

t
, b(t )  e

r ( ) d
we get the call---put parity, the
theorem is proved.
Predetermined Date Dividend
If in place of the continuous dividend paying
assumption (c^), we assume
 (c~) the underlying asset pays dividend Q on a
predetermined date t=t_1 (0<t_1<T)
 (if the asset is a stock, then Q is the dividend per
share).
 After the dividend payday t=t_1, there will be a
change in stock price:
S(t_1-0)=S(t_1+0)+Q.

Predetermined Date Dividend However, the option price must be continuous at
t=t_1:
V(S(t_1-0),t_1-0)=V(S(t_1+0),t_1+0).
 Therefore, S and V must satisfy the boundary
condition at t=t_1:
V(S,t_1-0)=V(S-Q,t_1+0)
 In order to set up the option pricing model (take
call option as example), consider two periods
[0,t_1], [t_1,T] separately.

Predetermined Date Dividend -0≤ S<∞, t_1 ≤ t ≤ T, V=V(S,t) satisfies the
boundary-terminal value problem
 In

V  2 (t ) 2  2V
V

S
 r (t ) S
 r (t )V  0
2
t
2
S
S
V
t T
 Obtain
 S  K 

V=V(S,t) on t_1
Predetermined Date Dividend --
in 0 ≤ S<∞,0 ≤ t ≤ t_1,V=V(S,t) satisfies
V  2 (t ) 2  2V
V

S
 r (t ) S
t
2
S
V t t  V  S  Q, t1  0 
2
S
 r (t )V  0
1
 By
solving above problems, we can
determine the premium V(S_0,0) to be paid
at the initial date t=0 (S_0 is the stock price
at that time).
Remark1
 Note
that there is a subtle difference
between the dividend-paying assumptions
(c^) and (c~) when we model the option
price of dividend-paying assets.
Remark1
In the case of assumption (c^), we used the
dividend rate q=q(t), which is related to the
return of the stock. Thus in [t_1,t_2], the
dividend payment alone
will cause the stock
t
price St  St exp  t q(t )dt By this model, if
the dividend is paid at t=t_1 with the intensity
t1 t
d_Q,
lim 
q(t )dt  dQ
1
1
t 0 t1


2
1

 dQ
then at t=t_1 the stock price St1  St1 e
Thus we can derive from the corresponding
option pricing formula.
Remark1-
In the case of assumption (c~), we used the
dividend Q, which is related to the stock price
itself. So at the payday t=t_1, the stock price
St1  St1  Q
Thus we have at t=t_1 the boundary condition for
the option price.
 We should be aware of this difference when
solving real problems.

Remark 2
For commodity options, the storage fee, which
depends on the amount of the commodity, should
also be taken into account.
 Therefore, when applying the Δ-hedging
technique, for the portfolio:

  V  S , d   dV  dS  q dt
^

where Δq^dt denotes the storage fee for Δ amount
of commodity and period dt.
Remark 2
Similar to the
 derivation
V / S we did before, choose
such that Π is risk-free in (t,t+dt). Then we get the
terminal-boundary problem for the option price
V=V(S,t) V  2  2V
V
t
V


t T
S2
S
2
 S  K 

2
 (rS  q ^ )
S
 rV  0
This equation does not have a closed form
solution in general. Numerical approach is
required.
Remark 2-the storage fee for Δ items of commodity
and period dt is in the form of Δq^Sdt,
proportional to the current price of the
commodity, then
V  2 2  2V
V
^
 If
t
 And

2
S
S
2
 (r  q ) S
S
 rV  0
the option price is given by the BlackScholes formula.
Generalized B-S Model (II) ------Binary
Options
 There
are two basic forms of binary option
(take stock option as example):
Cash-or-nothing call
 Cash-or-nothing
call (CONC):
In Case: t=T: stock price < strike price, the
option =0;
 In Case: t=T: stock price > strike price, the
holder gets $1 in cash.
Asset-or nothing call
 Asset-or
nothing call (AONC):
In Case: t=T: stock price < strike price, the
option =0;
In Case: t=T: stock price < strike price, the
option pays the stock price.
Modeling
 If
the basic assumptions hold, then the
binary option can
be
modeled as
2
2

V  2  V
V

S
 (r  q) S
 rV  0
2
t
2
S
S
 H ( S  K ), (CONC )
V t T  
 SH ( S  k ), ( AONC )
Relation of CONC,AONC & VC
 Consider
a vanilla call option, a CONC and
a AONC with the same strike price K and
the same expiration date T. Their prices are
denoted by V, V_C and V_A, respectively.
On the expiration date t=T, these prices
satisfy

V ( S , T )  VA ( S , T )  KVC (S , T )
Relation of CONC,AONC & VC 
And V(S,t), V_A(S,t) and V_C(S,T) each
satisfies the same Black-Scholes equation. In
view of the linearity of the terminal-boundary
problem, therefore in Σ{0≤ S<∞,0 ≤ t ≤ T},
V (S , t )  VA (S , t )  KVC (S , t )

i.e throughout the option's lifetime, a vanilla
call is a combination of an AONC in long
position and K times of CONC in short position.
Theorem 5.2

VA ( S , t , r , q)  SVC ( S , t , q, r )
^
where r  2q  r  
^
2
Proof of Theorem 5.2
 Let
V_A(S,t)=Su(S,t). It is easy to verify that
u(S,t) satisfies:2
u  2  2u
u
2

S
 (r  q   ) S
 qu  0
2
t
 Define
2
S
S
r ^  2q  r   2
, then the above
equation can be
written
as
2
2
u  2  u
u
^

S
 (q  r ) S
 qu  0
2
t
2
S
S
Proof of Theorem 5.2 
In Σ, compare the terminal-boundary problem for
u(S,t), and the terminal-boundary problem for
V_C(S,t). If the constants r, q are replaced by q
and r^, then u(S,t) and V_C(S,t) satisfy the same
terminal-boundary value problem. By the
uniqueness of the solution, we claim
u ( x, t )  VC ( S , t ; q, r )
^

Thus the Theorem is proved.
Solve pricing of CONC & AONC
 Once
the CONC price V_C(S,t;r,q) is found,
the AONC price V_A(S,t;r,q) can be
determined by Theorem 5.2.
 In order to solve the CONC problem, make
transformation
  T  t , x  ln( S / K )
H ( S  K )  H (e  1)  H ( x)
x
Solve pricing of CONC & AONC  Then
the Ori Prob. is reduced to a Cauchy
problem
2
2
2
 V   V
 V

 (r  q  )
 rV

2
2 x
2 x
 
V ( x, 0)  H ( x)


Analogous to the derivation of the BlackScholes formula, we have
Solve pricing of CONC & AONC-V ( x, )  e
 r
 x  (r  q   / 2) 
N

 


2
 Back
to the original variables (S,t), and by
Theorem 5.2, we have
2

ln(
S
/
K
)

(
r

q


/ 2)(T  t ) 
 r (T  t )
VC ( S , t ; r , q )  e
N

 T t


2

ln(
S
/
K
)

(
r

q


/ 2)(T  t ) 
 q (T  t )
VA ( S , t ; r , q )  Se
N

 T t


Generalized B-S Model (III) -----Compound Options
A compound option is an option on another option.
There are many varieties of compound options.
Here we explain the simplest forms of compound
options.
 A compound option gives its owner the right to buy
(sell) after a certain days (i.e. t=T_1) at a certain
price K^ a call (put) option with the expiration date
t=T_2 (T_2>T_1) and the strike price K. There are
following forms of compound options:

Compound Options
 1.
At t=T_1 buy a call option on a call
option;
 2. At t=T_1 buy a call option on a put
option;
 3. At t=T_1 sell a put option on a call
option;
 4. At t=T_1 sell a put option on a put
option;
Compound Options  Three
risky assets are involved: the
underlying asset (stock), the underlying
option (stock option) and the compound
option.
 First, in domain Σ_2{0≤ S<∞,0 ≤ t ≤ T_2},
define the underlying option price, which can
be given by the Black-Scholes formula,
denoted as V(S,t).
Compound Options -on Σ_1{0≤ S<∞,0 ≤ t ≤ T_1}, set up a
PDE problem for the compound option
V_{co}(S,t). For this we again make use of
the Δ-hedging technique to obtain the BlackScholes equation for V_{co}(S,t).
 Then
Compound Options -- At
is
t=T_1, the corresponding terminal
value

 For
V ( S , T )  K ^  , call
1

Vco ( S , T1 )  

^
 K  V ( S , T1 )  , put

the case when r,q,σ are all constants,
we can obtain a pricing formula for this form
of the compound option. Take a call on a
call at t=T_1 as example.
Compound Options --- At
t=0,
 qT2
Vco ( S , 0)  Se M (a1 , b1; T1 / T2 )
 Ke rT2 M (a2 , b2 ; T1 / T2 )  e rT1 K ^ N (a2 ),

where
ln( S / S * )  (r  q   2 / 2)T1
a1 
, a2  a1   T1
 T1
ln( S / K )  (r  q   2 / 2)T2
b1 
, b2  b1   T2
 T2
Compound Options ----
S^* is the root of the following equation:
*
2

ln(
S
/
K
)

(
r

q


/ 2) T2  T1  
^
*  q T2 T1 
K S e
N




T

T
2
1


*
2

ln(
S
/
K
)

(
r

q


/ 2) T2  T1  
 r T2 T1 
 Ke
N



 T2  T1


and M(a,b;ρ) is the bivariate normal distribution
function:
M(a,b;ρ)=Prob{X≤a,Y≤b},
where X~ N(0,1), Y~ N(0,1) are standard normal
distribution, Cov(X,Y)=ρ(-1<\rho<1).
Chooser Options (as you like it)
 Chooser
option can be regarded as a
special form of compound option. The
option holder is given this right:
 at t=T_1 he can choose to let the option
be a call option at strike price K_1,
expiration date T_2 or let the option be a
put option at strike price K_2, expiration
date T^_2,(T_2,T^_2}>T_1>0).
Chooser Options  Here
four risky assets are involved:
the underlying asset (stock),
the underlying call option (stock option; strike
price K_1, expiration date T_2),
the underlying put option (stock option; strike
price K_2, expiration dateT^_2)
the chooser option.
Chooser Options - Denote
VC ( S , t )  the underlying call option price
VP ( S , t )  the underlying put option price
both are solutions given by the BlackScholes formula.
Chooser Options -- In
order to find the chooser option price
V_{ch}(S,t) onΣ_1{0≤ S<∞,0 ≤ t ≤ T_1}, we
need to solve the following terminalboundary value problem:
2
2
 V  2  V
V


S
 (r  q ) S
 rV  0,

2
2
S
S
 t
V
 max VC ( S , T1 ), VP ( S , T1 )
t

T

1
Chooser Options --- If
the underlying call option V_C and put
option V_P have the same strike price K and
expiration date T_2, then by the call-put
parity:
 r T2 T1 
 qT2 T1 
VP (S , T1 )  VC (S , T1 )  Ke
 Se

thus
V
t T1
 max VC ( S , T1 ),VP ( S , T1 )
 VC ( S , T1 )  e
 q T2 T1 
( Ke
 r  q T2 T1 
 S )
Chooser Options ---- Then
by the superposition principle of the
linear equations, we have ^
Vch ( S , t )  VC ( S , t )  V ( S , t )
where
V (S , t )  e
^
 q (T2 T1 )
 put option(T1 , Ke
 ( r  q )(T2 T1 )
)
Numerical Methods (I) -----Finite Difference Method
 With
the computation power we enjoy
nowadays, numerical methods are often
preferred, although for European option
pricing closed-form solutions do exist.
Especially for complex option pricing
problems, such as compound options and
chooser options, numerical methods are
particularly advantageous.
BTM vs Finite Difference Method
 The
binomial tree method (BTM) is the most
commonly used numerical method in option
pricing.
Questions:
a. How to solve the Black-Scholes equation
by finite difference method (FDM)?
 b. What is the relation between FDM and
BTM?
 c. How to prove the convergence of BTM,
which is a stochastic algorithm, in the
framework of the numerical solutions of
partial differential equations?

Introduction to
Finite Difference Method
 Finite
difference method is a discretization
approach to the boundary value problems
for partial differential equations by replacing
the derivatives with differences.
Types of Approaches
There are several approaches to set up finite
difference equations corresponding to the partial
differential equations.
 Regarding the equation solving techniques, there
are two basic types:

 1. the explicit finite difference scheme, whose solving
process is explicit and solution can be obtained by direct
computation;
 2. the implicit finite difference scheme, whose solution
can only be obtained by solving a system of algebraic
equations.
Definition 5.1
Lu  0
 Suppose
is a finite difference
equation obtained from discretization of the
PDE Lu=0.
 If for any sufficiently smooth function ω(x,t)
there is
lim | L  L  | 0,
t , x 0
 then

the finite difference scheme
is said to be consistent with Lu=0.
Lu
Lax's Equivalence Theorem
 Given
a properly posed initial-boundary
value problem and a finite difference
scheme to it that satisfies the consistency
condition, then stability is the necessary and
sufficient condition for convergence.
Implicit Finite Difference Scheme vs
Explicit One

According to the numerical analysis theory of PDE, for the
IB problem, the implicit FDS is unconditionally stable,
whereas the explicit FDS is stable if a^2Δt/Δx^2=α≤1/2,
and unstable if α>1/2. This result indicates, although the
explicit FDS is relatively simple in algorithm, but in order to
obtain a reliable result, the time interval must satisfy the
condition Δt≤(1/2a^2)Δx^2. In contrast, although the implicit
FDS requires solving a large system of linear equations in
each step, the scheme is unconditionally stable, thus has
no constraint on time interval Δt. That means if the
computation accuracy is guaranteed, Δt can be large, and
the result is still reliable.
Explicit FDS of the B-S Equation
 We
have shown the Black-Scholes equation
can be reduced to a backward parabolic
equation with constant coefficients under the
transformation x=ln S:
 V  2  2V
 2 V

 (r  q  )
 rV  0,

2
2 x
2 x
 t
V ( x, T )  (e x  K ) 

Theorem 5.4
Set   ( t ) / x ,
2

if   1 & 1 
 then
2
1

2
|rq

2
2
| x  0,
the FD scheme of B-S is stable.
Numerical Methods (II) -----BTM & FDM
 BTM
is essentially a stochastic algorithm.
However, if S is regarded as a variable, and
option price V=V(S,t) is regarded as a
function of S,t, then BTM is an explicit
discrete algorithm for option pricing. If the
higher orders of Δt can be neglected, we will
be able to show that it is indeed a special
form of the explicit FDS of the BlackScholes equation.
Theorem 5.5
 If
ud=1, ignoring higher order terms
 of Δt, then for European option, pricing the
BTM and the explicit FDS of the BlackScholes equation
 (ω=σ^2Δt/(ln u)^2=1) are equivalent.
Theorem 5.6
(Convergence of BTM for Eoption)
 If
 then
r q 
1


2
t  0
as Δt→ 0, there must be
lim V ( S , t )  V ( S , t ),
t 0
where V_Δ(S,t)$ is the linear extrapolation
of V_m^n.
Properties of European Option
Price
 European
option price depends on 7 factors
(take stock option as example): S (stock
price), K (strike price), r (risk-free interest
rate), q (dividend rate), T (expiration date) , t
(time), σ (volatility).
Dependence on S
c
 q (T  t )
^
e
N (d 1 )  0
S
p
 e  q (T t ) [1  N (d 1^ )]  0
S
 That
is, as S increases, call option price
goes up, and put option price goes down.
Dependence on K
c
 r (T t )
^
 e
N (d 2 )  0
K
p
 r (T  t )
^
e
[1  N (d 2 )]  0
K

 For
different strike prices, call option price
decreases with K, and put option price
increases with K.
Dependence on S & K Financially
 When
the stock price goes up or the strike
price goes down, the call option holders are
more likely to gain more profits in the future,
thus the call option price goes up; In
contrast, the put option holders have smaller
chance to gain profits in the future, thus the
put option price goes down.
Dependence on r
c
 r (T  t )
^
 K (T  t )e
N (d 2 )  0
r
p
 r (T  t )
^
  K (T  t )e
[1  N (d 2 )]  0
r
 If
the risk-free interest rate goes up, then the
call option price goes up, but the put option
price goes down.
Dependence on r Financially

The risk-free interest rate raise has two effects: for
stock price, in a risk-neutral world, the expected
return E(dS/S)=(r-q)dt will go up; For cash flow,
the cash K received at the future time (t=T) would
have a lower value Ke^{-r(T-t)} at the present time
t. Therefore, for put option holders, who will sell
stocks for cash at the maturity t=T, thus the above
two effects result in a decrease of the put option
price. For call option holders, the effects are just
the opposite, and the option price will go up.
Dependence on q
c
 q (T  t )
^
  S (T  t )e
N (d 1 )  0
q
p
 q (T  t )
^
 S (T  t )e
[1  N (d 1 )]  0
q
 If
the dividend rate increases, then the call
option price goes down, and the put option
price goes up.
Dependence on q Financially
 The
dividend rate directly affects the stock
price. In a risk-neutral world, as the dividend
rate increases, the expected return of the
stock
E(dS/S)=(r-q)dt decreases, thus the call
option price decreases, but the put option
price increases.
Dependence on σ
c p
 q (T t )
^

e
N '(d 1 ) T  t  0
 
a stock has a high volatility σ, its
option price (both call and put) goes up.
 when
Dependence on σFinancially


An increase of the volatility σ means an increase of the
stock price fluctuation, i.e., increased investment risk. For
the underlying asset itself (the stock), since E(σdW_t)=0,
the risks (gain or loss) are symmetric. But this is not true
for an individual option holder.
Example (call): The holder benefits from stock price
increases, but has only limited downside risk in the event
of stock price decreases, because the holder's loss is at
most the option's premium. Therefore the stock price
change has an asymmetric impact on the call option value.
Therefore the call option price increases as the volatility
increases. Same reasoning can be applied to the put
option.
Dependence on t&T
c
 q (T  t )
^
 r (T  t )
^
 qSe
N (d 1 )  rKe
N (d 2 )
t
 e q (T t ) SN (d 1^ )
c


(sign is not fixed)
T
2 T t
p
 r (T  t )
^
 q (T  t )
^
 rKe
[1  N (d 2 )]  qSe
[1  N (d1 )]
t
 q (T  t )
^
e
S [1  N (d 1 )]
p


(sign is not fixed)
T
2 T t
Dependence on t&T Financially
No matter how long the option's lifetime T is, a
European option has only one exercise. A long
expiration does not mean more gaining
opportunity. So, European options do not become
more valuable as time to expiration increases.
 As for t, larger t, smaller T-t, means closer to the
exercise day. Therefore, for European options we
cannot predict whether the option price will go
down or go up as the exercise day comes closer.

Dependence on t&T Financially
However, there is an exception. In the case q=0,
 SN (d ^ )
1
c
 r (T  t )
^
 rKe
N (d 2 ) 
0
t
2 T t

i.e. with the expiration day coming closer, the call
option on a non-dividend-paying stock will go
down.
Table of European Option Price
Changes
call
put
S
+
-
K
-
+
r
+
-
q
-
+
σ
+
+
T
?
?
t
?
?
Risk Management—Δ(Sigma)
V

S

Δ
is the partial derivative of the option or its
portfolio price V with respect to the
underlying asset price S. The seller of the
option or its portfolio should buy Δ shares of
the underlying asset to hedge the risk
inherited in selling the option or portfolio.
Risk Management—Γ(Gamma)
  V

 2
S S
2


Since Δ is a function of S & t, one must
constantly adjust Δ to achieve the goal of the
hedging. In practice, this is not feasible
because of the transaction fee. Therefore in
real operation one must choose the frequency
of Δ wisely. This is reflected in the magnitude
of Γ. A small Γ means Δ changes slowly, and
there is no need to adjust in haste; Conversely,
if Γ is large, then Δ is sensitive to change in S,
there will be a risk if Δ is not adjusted in time.
Risk Management—Θ(Theta)
Θ
is the rate of change in the option or
portfolio price over time. The Black-Scholes
equation gives the relation between Δ, Γ and
Θ:


2
2
S   (r  q) S   rV
2
Risk Management—V (Vega)


¶V
V=
¶s
V is the partial derivative of the option or its
portfolio price with respect to the volatility of
the underlying asset.
 The underlying asset volatility σ is the least
known parameter in the Black-Scholes
formula. It is practically impossible to give a
precise value of σ Instead, we consider the
sensitivity of the corresponding option price
over σ This is the meaning of V
Risk Management—ρ(rho)
ρ
is the partial derivative of the option or
portfolio price with respect to the risk-free
interest rate.
How to Manage Risk?
 For
European options, we have obtained the
expressions of these Greeks in the previous
section. Now we will explain how to use
these parameters (especially Δ and Γ) in
risk management.
A Specific Example

Suppose a financial institution has sold a stock
option OTC, and faces a risk due to the option
price change. Therefore it wants to take a hedging
strategy to manage the risk. Ideally, a hedging
strategy should guarantee an approximate
balance of the expense and income, i.e., the
money spent on hedging approximately equals the
income from selling the option premiums.
How does the hedging strategy work?

At t=0 the seller buys Δ_0 shares of stock at S_0 per
share, and borrows Δ_0 S_0 from the bank. At t=t_1, to
adjust the hedging share to Δ_1 (S_1 is the stock price at
t=t_1), the seller needs to buy Δ_1-Δ_0 shares at S_1 per
share if Δ_1>Δ_0 and sell Δ_0- Δ_1 shares at S_1 per
share if Δ_1<Δ_0; and borrow (save) the money needed
(gained) for (from) buying (selling) the stocks, and at t=t_1
pay the interest Δ_0 S_0rΔt to the bank for the money
borrowed at t=t_0. In general, at t=t_n, the seller owns Δ_n
shares of stock, and has paid hedging cost D_n:
How does the hedging strategy work?

On the option expiration day t=T, the seller owns
Δ_N shares of stock, i.e.
 if S_T>K (i.e. the option is in the money), the seller
owns one share of stock,
 if S_{T}<K (i.e. the option is out of the money), the
seller owns no share of stock.
 If the option is in the money, the option holder will
exercise the contract to buy one share of stock S from
the seller with cash K;
 if the option is out of the money, the option holder will
certainly choose not to exercise the contract.
How does the hedging strategy
work?-The above hedging strategy successfully hedges
the risk in selling the option.
 In this deal the sellor's actual profit is
profit=V_0e^{rT}-D_T,
where V_0 is the option premium. If there is a
transaction fee for each hedging strategy
adjustment, then the seller's profit is
profit=V_0e^{rT}-D_T-Σ_{i=0}^{N-1}e_i,
where e_i is the fee for the i-th adjustment.

Remark

In practical operation, hedging adjustment
interval Δt is not a constant, and depends
2


V . If Γ is large, adjustment
on  
 2
S S
is made more frequently; If Γ is small,
adjustment can be made less frequently.
Summary 1

Introduced a continuous model for the underlying
asset price movement---the stochastic differential
equation. Based on this model, using the Δhedging technique and the Ito formula, we derived
the Black-Scholes equation for the option price,
by solving the terminal value problem of the BlackScholes equation, we obtained a fair price for the
European option, independent of each individual
investor's risk preference---the Black-Scholes
formula.
Summary 2
 As
derivatives of an underlying asset, a
variety of options can be set up in a various
terminal-boundary problem for the BlackScholes equation. To price these various
options is to solve the Black-Scholes
equation under various terminal-boundary
conditions.
Summary 3
 BTM
is the most important discrete method
of option pricing. When neglecting the
higher orders of Δt, BTM is equivalent to an
explicit finite difference scheme of the
Black-Scholes equation. By the numerical
solution theory of partial differential
equation, we have proved the convergence
of the BTM.
Summary 4
 The
option seller can manage the risk in
selling the option by taking a hedging
strategy. Since the amount of hedging
shares Δ= Δ(S,t) changes constantly,
the seller needs to adjust Δ at
appropriate frequency according to the
magnitude of Γ(S,t), to achieve the goal
of hedging.