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Chp.9 Option Pricing When
Underlying Stock Returns are
Discontinuous
• In this chapter, an option pricing formula is
derived for the more general case where the
underlying stock returns are generated by a
mixture of both continuous and jump
processes.
9.1 Introduction
• The critical assumptions in the BlackScholes derivation is that trading takes
place continuously in time and that the price
dynamics of the stock have a continuous
sample path with provability one.
• What will happened if there is a jump?
9.2 The Stock-Price and OptionPrice Dynamics
• The total change of stock price is divided
into two parts:
– normal vibrations(振动), modeled by a
standard geometric Brownian motion;
– abnormal vibrations, usually due to firm
specific information, modeled jump process;
• so the stock price sample path: Wiener
process + Poisson-driven process
• The Poisson-driven process is described as
follows:
prob the event does not occur in the time interval  t , t  h   1   h  o  h 
prob the event occurs once in the time interval  t , t  h    h  o  h 
prob the event occrus more than once in the time interval  t , t  h   o  h 
• Y: the random variable description of the
drawing from a distribution to determine the
impact of the information on the stock price.
Then, neglecting the continuous part
•
S t  h   S t Y
• Then the stock-price
returns can be
described as
dS
    k  dt   dZ  dq
S
if the Poisson event does not occur
   k  dt   dZ

   k  dt   dZ  Y  1 if the Poisson event occurs
• If
 ,  , k ,
are constants, then


S t 

2
 exp   
  k  t   Z t  Y  n 
S
2



Y  n    j 1Y j
n
• According to Ito’s lemma the function of
stock price and time
dW
 W   kW
W
 dt
  W dZ  dqW
1 2 2
 S FSS  S , t      k  SFS  S , t   Ft   F  SY , t   F  S , t 
W  2
F  S,t 
FS  S , t   S
W 
F  S,t 
YW  F  SY , t  / F  S , t 
• （following the three assets model）
• Consider a portfolio strategy which holds
the stock, the option, and the riskless asset,
if P is the value of the return dynamics on
the portfolio can written as
dP
  p   k p  dt   p dZ  dq p
P
• p  w1   r   w2  w  r   r  w1  w2 w  1  w1  w2  r
 p  w1  w2 w
Yp  1  w1 Y  1 
w2  F  SY , t   F  S , t 
F S,t 
 F  SY , t  
 w1 Y  1  w2 
 1
 F S,t  
• 1.  0, dq  0 no-jumps => standard B-S
  r w  r


w
1 2 2
 S FSS  rSFS  rF  Ft  0
2
• 2. jumps can not be hedge
dP*
*
*
*




k
dt

dq


p
p
p
P*
  *p   k *p  dt


*
*
*
  p   k p  dt  Yp  1
if the Poisson event does not occur
if the Poisson event occurs
Yp*  1 
w2*  F  SY , t   F  S , t   FS  S , t  SY  S  
F  S,t 
 F  SY , t  
w 
 1  w1* Y  1
 F  S,t 

*
2
• because of the concave of option price to
stock price  F  SY , t   F  S , t   F  S , t  SY  S 
• is always positive
Y  1 will not be 0。
S
*
p
• Economic implications：
• （1）following B-S hedging : long stock  w  0 
and short option, 平时收益高于预期，跳
跃时损失很大；reverse B-S hedging: short
 w  0
stock and long option, 平时收益低于预期，
跳跃时收益很大；
• （2）无跳跃时期权卖方获利，有跳跃时
买方获利。
*
2
*
2
9.3 An Option Pricing Formula
• Pricing technique 1:
• If one knew the required expected return on
the option   g  S , 
w
1 2 2
 S FSS     k  SFS  g  S ,  F   F  SY ,   F  S , 
2
F  0,   0
F  0,   max  0, S  E 
• Pricing technique 2: Assumed that the CAPM
was a valid description of equilibrium security
returns.
• Stock-price dynamics were described two
components:
– Continuous part---new information;
– Jump part---important new information, usually firm
(or even industry) specific such as discovery of an
important new oil or the loss of a court suit,
“nonsystematic” risk.
• 跳跃部分属于非系统风险，不产生风险
*
溢价，根据 CAPM  p  r
  r w  r


w
1 2 2
 S FSS   r   k  SFS  F  rF   F  SY ,   F  S , 
2
• Even though the jumps represent “pure”
nonsystematic risk, the jump component
does affect the equilibrium option price.
That is, one cannot “act as if” the jump
component was not there and compute the
correct option price (nonsystematic risk has
s none zero price?)
• Define W to be the B-S option pricing
formula for the no-jump case.
W  S , ; E ,  2 , r   S   d1   E exp  r    d 2 
  y 
1
 s2 
 exp   2  ds
y
 2 
1/ 2
• Define Xn the random variable to have the
same distribution as the product of n i.i.d.
(identically distributed to Y) random
variables
exp     
F  S ,   
 n W  SX n exp   k  , ; E,  2 , r 
n!
n 0

n


• There is not a closed-form solution, but it
does admit to reasonable computational
approximation
• There are two special cases where can be
vastly simplified.
• Example 1: There is a positive probability
of immediate ruin, i.e. if the Poisson event
occurs, then the stock price goes to 0. that is
Y=0 with probability one.
F  S ,   exp   W  S exp    , ; E ,  2 , r   W  S , ; E ,  2 , r   
• inedntical with the standard B-S solution
but with a larger “interest rate,”. As was
shown in Merton (1973a, Ch.8), the option
price is an increasing function of the interest
rate, and therefore an option on a stock that
has a positive probability of complete ruin
is more valuable than an option on a stock
that dose not.
• Example 2: Y has a log-normal distribution
• Define
f n  S ,   W  S , ; E , vn2 , rn 
• then
exp       
F  S ,   
f n ( S , )
n!
n 0

n
• Clearly, f (S , )is the value of the option,
conditional on knowing that exactly n
Poisson jumps will occur during the life of
the option. The actual value of the option,,
is just the weighted sum of each of these
prices where each weighted equals the
probability.
n
• (9.16) was deduced from the twin
assumptions that CAMP is valid and the
jump component of a security’s return is
uncorrelated with the market. One can
hardly claim strong empirical evidence to
support these assumptions.
• Another technique to derive (9.16): the Ross
model for security pricing.
• 有m个跳跃过程互相独立的股票，对应有
m个共同形式（股票＋期权）的套利组合
•
dPj*
Pj
*
  *j   j k *j  dt  dq*j
j  1, 2,
m
• 用m个套利组合与无风险资产组成一个组
合：
dH
   H  H kH  dt  dqH
H
 H  x j   r   r , H kH   x j  j k , dqH   x j dq*j
m
j 1
m
*
j
j 1
m
*
j
j 1
• 随着m的增加，组合接近无风险（推导），
所以有
1 m
*
u


j  j  r  0
m j 1
 *j ，此时成立。
r
• 所以有
• 两种推导过程基于同样的原理：跳跃过
程可以分散。
• 为消除系统风险必须卖出的股票数量。
N  W / S    d1 
exp       
*
N 
  d  n  
n!
n 0
n

d n 
log  S / E    rn   2 / 2   n 2 / 2
   n 
2
2 1/ 2
9.4 A Possible Answer to an
Empirical Puzzle
• 如果投资者用B-S公式来为期权股价，它
与理论价格的差距在哪？
• Assume Y is log-normal
• The investor’s estimate is the true
unconditional variance
v2  h    2   2  v2
• 真是的应该是一个条件方差（考虑了跳
跃过程）
 2  n 2 / h
• The problem becomes, if the investor uses
as his estimate of the variance rate in the
standard B-S formula, then how will his
appraisal of the option’s value compare with
the “true”?
2
2
Tn     n
T   T    2   2   v 2
W  S , ; E , u 2 , r   E exp  r W  S , ;1,1, 0 
• Inspection of (9.18) shows that, from (9.27),
fn can be rewritten as
Fe  S ,   E exp  r  W  X , T 
F  S ,   E exp  r   W  X , T 
F  S ,   E exp  r W  X , T 
• so the answer will depend on whether
 W  X , T   W  X , T   0
•
2
     2 / 4 
a

 W /  



2
W /  
2   
2
2
0
a  0  S  E exp  r  , Fe  F , m arg inall  in  the  money

 2
1  S EorS E , Fe  F , deep  in  the  money
a

deep  out  of  the  money

