Phantom Works
Mathematics & Computing Technology
Real Options and Mean-Reverting
Prices
Gregory F. Robel
Mathematics & Engineering Analysis
July 14, 2001
gregory.f.robel@boeing.com
Acknowledgements
Phantom Works
Mathematics & Computing Technology
I would like to thank several colleagues for useful
discussions, including Dr. Stuart Anderson, Dr. Mike Epton
and Dr. Roman Fresnedo of The Boeing Company; and
Dr. Dan Calistrate and Professor Gordon Sick of
The University of Calgary.
The usual disclaimer applies.
2
Outline
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•
•
•
•
•
•
•
•
Standard capital budgeting paradigm
Real option extension of standard paradigm
McDonald-Siegel model
Survey and critique of some proposed models of meanreverting prices
Specification of a “new,” more appropriate model
Analytical solution of some option problems for the “new”
model, and some comparative statics
Remarks on numerical approaches for more complex
problems
Conclusion
3
Shareholder Value and Capital Budgeting
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• A firm’s investment decisions are the most important
determinant of shareholder value (Modigliani-Miller theorem).
• The standard Net Present Value rule can be expressed as
follows:
-- Estimate the value S0 which the project’s future net
cash inflows would have, if they were traded in the
financial markets.
-- Estimate the cost I of launching the project.
• If S0 > I , then the project has a positive net present value:
NPV = S0 - I > 0. Launching the project creates shareholder
value, since in so doing, the firm is acquiring an asset for less
than the shareholders could themselves acquire a
comparable asset in the financial markets.
4
Basis for NPV Rule
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• The Fisher Separation Theorem states that if markets are
perfect and complete, then maximizing the market value of
the firm’s equity simultaneously enables all shareholders to
maximize their expected lifetime utility of consumption.
• Estimating the market value of a hypothetical set of cash
flows requires the existence of a portfolio of traded securities
which tracks these cash flows sufficiently well. (This is a
weak form of the complete market assumption.)
• This estimation usually involves an asset pricing model such
as the Capital Asset Pricing Model (CAPM) or the Arbitrage
Pricing Theory (APT), to estimate the expected rate of return
which the market would demand for these cash flows.
-- Empirical support less than overwhelming, especially
for CAPM.
• In principle, each project has its own cost of capital.
5
Critique of NPV Rule
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• Correct, as far as it goes.
• Implicit assumptions:
-- Project has a directly associated stream of future net
cash inflows
-- Decision must be made now or never
-- Management strategy fixed in advance
-- Similar to evaluating stocks or bonds
6
Real Options
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• Many projects create future opportunities, which may be a
significant, or even the sole, source of value.
• These opportunities are analogous to financial options.
• The term real options has been coined to refer to a firm’s
discretionary opportunities involving real assets.
• Discounted cash flow methods fail for the evaluation of
options.
• Financial economists have recognized the analogy between
real and financial options.
• The same assumptions which justify the use of discounted
cash flow methods for evaluating a project also justify the
use of an option pricing model for evaluating options
associated with the project.
Reference: [CoA]
7
Types of Options
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• In general, an option is a financial contract which gives its
holder the right, but not the obligation, to buy or sell another,
underlying asset, at a specified price, on or until a specified
expiration date.
• An option to buy is a call option.
• An option to sell is a put option.
• An option with an infinite expiration date is said to be
perpetual.
• An option which can be exercised only on its expiration date
is said to be European.
• An option which can be exercised at any time up until and
including its expiration date is said to be American.
8
Exponential Growth under Uncertainty
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• Suppose that some random quantity (e.g., a stock price)
St > 0 evolves continuously in time, in such a way that:
1. On the average, St grows at a constant exponential
rate  .
2. At any instant, there may be random errors between
the expected value and the observed value.
3. The past history of these (relative) errors is of no
use in predicting their future values.
4. Each possible path of St is a continuous function of
time, almost surely.
9
Geometric Brownian Motion, I
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• Then it can be shown1 that St is governed by the following
expression, which is called a stochastic differential equation
(SDE) :
d St   St d t   St d Bt ,
where
 is the expected rate of growth of St ,
 determines the rate of growth of the
variance of St ,
and
Bt is a particular random process
called a standard Brownian motion
or “random walk.”
• One refers to St as a Geometric Brownian Motion, with drift 
and volatility  .
1Assuming that the variance of the error between the logarithm of S and its
t
expected value is proportional to time.
10
Two Possible Paths of Geometric Brownian Motion
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• S0 = 42, e = 1.15,  = 0.20.
11
Geometric Brownian Motion, II
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• Explicit solution:
St
 S0 e




 



2
2


 t   Bt 



• Conclusion: St > 0 almost surely, and the logarithm of St at
any time t > 0 is normally distributed.
• Expected value and variance of St :
E St

 S0 e  t
Var  St

 S02 e 2 t
12
e
2t
 1

.
McDonald - Siegel Model, I
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• Suppose that a firm is considering whether to launch a new
project. Assume that its estimate Vt of the market value of
the completed project evolves as a geometric Brownian
motion, so that
dVt
  Vt d t   Vt d Bt .
for some  > 0 .
• Suppose that the fixed costs associated with launching the
project are known, irreversible, and equal to I .
• Then one can ask two questions:
-- How much is this opportunity worth?
-- At what point is it optimal to launch the project?
References: [DP], [McDS]
13
McDonald - Siegel Model, II
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• Suppose that the required rate of return on the project (its
cost of capital) is m .
• For the problem to make economic sense, one must
have m >  .
-- If m =  , then the value of the option is the same
as that of the underlying asset, and one would never
rationally exercise it.
-- If m   , then the value of the underlying asset and
the option are both infinite.
• Let d = m -  > 0 , and let r > 0 be the riskless
interest rate.
14
McDonald - Siegel Model, III
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• Then there exists a critical value V* such that the value
C ( V ) of the opportunity is given by
CV

 A V p , if 0  V  V * ;

 
 V  I , if V *  V .

• The project should be launched at the first time t  0 for
which Vt  V* .
15
McDonald - Siegel Model, IV
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• The parameter p , the coefficient A , and the optimal
exercise threshold V* are given by
1
p 

2
A 
V* 

r  d 
2
p  1
pp I p1
 r  d
  2

p  1
p
I
p 1
, and
 I .
16

1 
 
2
2

2r
2
 1,
McDonald - Siegel Model, V
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• For example, suppose that
m = 0.13 ,
 = 0.08 , and
 = 0.20 .
• Suppose that the required investment I = 1 , and that the
riskless interest rate r = 0.05.
• Then one finds that A = 0.2253 , p = 2.1583 , and V* = 1.8633 .
Hence, the value of the option to launch the project is
C V

 0.2253 V 2.1583 , if 0  V  1.8633 ;

 
 V  1 , otherwise .

17
McDonald - Siegel Model, VI
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F( V ) = A V p
Payoff = max( V - 1 , 0 )
V* = 1.8633
18
McDonald - Siegel Model, VII
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Optimal Rule
NPV Rule
19
McDonald - Siegel Model, VIII
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• If d  r , then
• If d  r , then
lim V *  I .
  0
lim V *  I .
  0
• McDonald and Siegel also considered the case where the
exercise price I is stochastic, and where the value of the
underlying asset might suddenly jump to zero.
• Returning to the case where the exercise price I is
nonstochastic, other authors (e.g., Dixit and Pindyck) have
noted that the same methodology can be applied more
generally, to problems where the value of the underlying
asset can be written as some function of some state variable
which follows a geometric Brownian motion.
20
Motivation for Mean-Reverting Prices
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• The “workhorse” stochastic process for much of the early
work on real options is geometric Brownian motion.
• This is a natural model for exponential growth (or decay)
under uncertainty.
• Consider a real option where the underlying state variable is
the price of some input or output.
• One might argue that the price may be subject to random
shocks, but that in the long run, competitive pressures will
ensure that it tends to revert to some sustainable level.
21
Candidate “Mean-Reverting” Price Processes, I
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• Mean-Reverting Ornstein-Uhlenbeck process:
d Xt
   X  X t  dt
  d Bt
• Explicit solution:
Xt


X
X
0
 X  e  t
 

t
0
e   t  s  d Bs
• Conclusion: Xt is normally distributed, with
E X t

Var  X t 

X

X0
 X  e  t
2
1  e  2  t

2

• The process does indeed revert to a mean, but can take
negative values.
References: [KP], [Si]
22
Candidate “Mean-Reverting” Price Processes, II
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• Exponential of a Mean-Reverting Ornstein-Uhlenbeck
process:
Xt
dYt
 e Y , where
   Y  Yt  d t   d Bt
t
• From Ito’s lemma, one then finds that
d Xt

1 2
    Y 

2




  log  X t   X t dt


  X t d Bt
• Conclusion: Xt is lognormally distributed, and the logarithm
of Xt reverts to a mean.
Reference: [Si]
23
The Homogeneity Condition
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• Question: If the price of one widget reverts to some mean
value, shouldn’t the price of two widgets revert to twice that
same mean value, “in the same manner”?
• More generally (and more precisely), it seems reasonable to
require the following: Suppose that the price Xt follows the
stochastic process
dXt

f  X t , X  d t  g  X t , X  d Bt .
Then for any  > 0 , Yt   Xt should follow the stochastic
process
dYt  f  Yt , Y  d t  g  Yt , Y  d Bt ,
where Y   X .
• In a word, the drift and diffusion of the process should be
homogeneous functions of degree one, of the pair  X t , X  .
• The previous model violates this condition.
24
Candidate “Mean-Reverting” Price Processes, III
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• Stochastic logistic or Pearl-Verhulst equation:
   X  X t  X t dt
d Xt
  X t d Bt
• Explicit solution:
Xt
•
•
•
•

1
X0

2
exp   X 
 t   Bt
2 

t

2
   exp   X 
s 
0
2 





 Bs  d s

Conclusion: Xt > 0 almost surely, for all t > 0 .
What are E[ Xt ] and Var[ Xt ] ?
Does this process really revert to a mean?
This model violates the homogeneity condition.
References: [DP], [KP], [Pin]
25
Candidate “Mean-Reverting” Price Processes, IV
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• Inhomogeneous geometric Brownian motion (IGBM) :
d Xt
   X  X t  d t   X t d Bt
   X t d t   X t d Bt   X d t
• Explicit solution:
Xt
 
2
 exp    
t
2 
 

  Bt   X 0

 X
t
0



2
exp   
s


B
d
s


s 
2




• Conclusion: Xt > 0 almost surely, for all t > 0 .
(Remark: If Xt follows an inhomogeneous geometric
Brownian motion, and if Yt = 1 / Xt , then it follows from Ito’s
lemma that Yt observes a stochastic logistic equation, as on
the previous slide.)
• This model satisfies the homogeneity condition.
References: [B], [KP], [Pil], [Si]
26
First and Second Moments of Inhomogeneous
Geometric Brownian Motion, I
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• Let Xt be an inhomogeneous geometric Brownian motion,
and let
m1  t   E[ X t ] ,
2
m2  t   E[ X t ] .
• Then one can show that
d m1
dt
d m2
dt
  m1  t    X , and

 2 
  2  m 2  t   2  X m1  t  .
Reference: [KP]
27
First and Second Moments of Inhomogeneous
Geometric Brownian Motion, II
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• One finds that
E[ X t ] 
X

X
0
 X  e  t .
• Hence, the expected value of Xt tends to
• Moreover,
 
X , as t   .
log  2 
,
t1 / 2
where t1 / 2 is the “half-life” of the deviation between E[ X t ] and X .
28
First and Second Moments of Inhomogeneous
Geometric Brownian Motion, III
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• One also finds that

2
  X t  X X  X0

 X2
1  e   2    t

2
 2  2
 X
  2    t
1

e
2
 2  



2
2
 e


 t
E[ X t ] 
2
1 ,
if 2   2  0 ;
  X  X  X 0  t e   2    t ,
if    2  0 ;
2

 X  X  X 0   t
e
 e   2    t , if  2   2     2   0 .
2
 

2
29

Density Function of Inhomogeneous Geometric
Brownian Motion
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• Suppose that
d Xt
   X  X t  d t   X t d Bt , X 0 given .
• Let p( x , t ) be the probability density function of Xt .
• Then one can show that p( x , t ) is a solution of the following
initial value problem for Kolmogorov’s forward equation (also
known as the Fokker-Planck equation):
p
t
1 2

2 2
 X  x p  ,   x   , 0  t ;




x
p

2  x2
x
p x , 0   d  x  X 0  ,    x   .
Reference: [W]
30
Mean-Reverting and Lognormal Distributions, I
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Maximum error = 0.0347
X0
 0.2 , X
 1 ,   0.0347  t1/ 2  20  , 
31
 1, t  1
Mean-Reverting and Lognormal Distributions, II
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Maximum error = 0.0142
X0
 5, X
 1 ,   0.0347  t1/ 2  20  , 
32
 1, t  1
Valuation of an Asset Contingent on an Inhomogenous
Geometric Brownian Motion, I
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• Suppose that there exists a traded security Mt such that risk
uncorrelated with changes in Mt is not priced. Suppose that
there are no cash payouts associated with Mt , and that
dMt
  M M t dt
  M M t d Z t , where
Zt is a standard Brownian motion, and where M > 0 and
M > 0 are constant.
• Suppose that Xt follows the inhomogeneous geometric
Brownian motion
d X t    X  X t  d t   X t d Bt .
• Suppose that there is a constant instantaneous risk-free
interest rate r > 0 .
• Let r dt be the instantaneous correlation between dBt and
dZt , for some constant | r |  1 .
• Let    M  r
be the market price of risk.
M
33
Valuation of an Asset Contingent on an Inhomogenous
Geometric Brownian Motion, II
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• Consider an asset of market value V( x , t ) , which is
completely determined by x = Xt and by time t  0 .
• Suppose that this asset generates a cash flow of c( x , t ) dt
during the time interval ( t , t + dt ) .
• Then, under certain “perfect market” assumptions, one can
show that V( x , t ) satisfies the partial differential equation
(PDE)
1 2 2  2V
 x
2
x 2

  X
 x   r  x 
for 0 < x <  , 0 < t .
34
V
x

V
t
 rV
  c x , t 
Valuation of an Asset Contingent on an Inhomogenous
Geometric Brownian Motion, III
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• Moreover, suppose that X̂ t follows the inhomogeneous
geometric Brownian motion
d Xˆ t
where


 ˆ Xˆ  Xˆ t d t
  Xˆ t d Bt , Xˆ 0 given ;
ˆ    r   and Xˆ


  r 
X .
• Then for any time t0 and any value of the state variable X0 ,
V( X0 , t0 ) may be evaluated as follows:
V  X 0 , t0




t0
 
E c Xˆ s , t


Xˆ t  X 0 e  r s d s .
0
• One refers to X̂ t as the risk-neutralized version of X t .
References: [Co], [Sh], [Si]
35
Derivation of (PDE), I
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• Consider a portfolio P = V - h M .
• Then, using Ito’s lemma and the SDEs for V and for M ,
d P  dV
 h dM
V

dt
t

V
dX
X

1  2V
dX
2 X 2
V
V
   X  X  dt

dt 
t
X
 h  M M dt   M M d Z 
 V
 
 t
2
 h  M M dt
  X dB
V
1 2 2  2V
 X  X 

 X
X
2
X 2
V
 hM M dZ   X
dB .
X
36


 M M dZ
1  2V 2 2
 X dt
2 X 2

 h M M  dt


Derivation of (PDE), II
Phantom Works
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• Choose h such that
 h M M dZ
  X
V
dB
X
is uncorrelated with dZ :

E  h M M  dZ

 h 
• Then
P  V

2
  X

V
dB dZ   0
X

r  X V
 M M X
r  X V
M
 M M X
is uncorrelated with M .
37
 V

r  V
X
 M X
Derivation of (PDE), III
Phantom Works
Mathematics & Computing Technology
• Therefore, for this choice of h , the expected return on
portfolio P must equal the riskless rate r .
• Portfolio P ‘s (instantaneous) expected return, moreover,
must equal its expected (instantaneous) capital gain plus its
(instantaneous) cash flow.
• Therefore


r  V

r P dt  r  V 
X  d t 
 M X


 V

 t
V
 X  X 
X

1 2 2  2V
 X
2
X 2
M r 
V 

X
 dt  c  X , t  dt .
M
X 
• The PDE follows after rearranging, and substituting X = x
and
M  r
 
.
M
38
Valuation of a Perpetuity Contingent upon an
Inhomogeneous Geometric Brownian Motion
Phantom Works
Mathematics & Computing Technology
• Suppose that
d Xt
   X  X t  d t   X t d Bt , X 0 given ;
and consider a perpetual cash flow of Xt dt , beginning at
t =   0.
• Then the current ( t = 0 ) value of this perpetuity is
 


E Xˆ s

r


Xˆ 0  X 0 e  r s d s 
Xe
r    r  


 Xˆ   X

X
 X0 
  r 
 
r   
39
  r
e

r 
0
 
ˆ
 Xˆ e  s e  r s d s
   r   
.
Valuation of an Annuity Contingent upon an
Inhomogeneous Geometric Brownian Motion
Phantom Works
Mathematics & Computing Technology
• Suppose that
d Xt
   X  X t  d t   X t d Bt , X 0 given ;
and consider an annuity with instantaneous cash flow Xt dt ,
beginning at t = 0 and ending at t =   0 .
• Then the current ( t = 0 ) value of this annuity is (as given by
Bhattacharya (1978))
r
Xe
 1  e r 
r    r  


X
 X0 
  r 
 
r   
Reference: [B]
40
  r
e

r 
   r   
1  e 
 r    r 

.
Valuation of a Perpetual American Call Option on an
Underlying Asset which is a Function of an IGBM, I
Phantom Works
Mathematics & Computing Technology
• Suppose that some asset value or commodity price V = V( x )
is completely determined by x = Xt , where Xt evolves in
accordance with
d Xt
   X  X t  d t   X t d Bt , X 0 given ;
and consider a perpetual American option to purchase V for
an exercise price I .
• While it is optimal to hold the option, its value F = F( x ) is
governed by the ordinary differential equation (ODE)
1 2 2 d 2F
dF




 x


X

x

r


x
 rF  0 , 0  x   .
2
2
dx
dx
• We will impose the natural boundary condition that F( x ) is
bounded as x  0+ .
41
Valuation of a Perpetual American Call Option on an
Underlying Asset which is a Function of an IGBM, II
Phantom Works
Mathematics & Computing Technology
• For convenience, rewrite the ODE as
2
dF
2 d F


x


x

b
 F  0 ,
2
dx
dx
 2   r  
2 X
2r
,
b

,
and


.
where  
2
2
2



• Note that  > 0 and b > 0 . Moreover, we will assume that
 <  , which is equivalent to r    - (  + r ) .
• Up to a constant multiplier, there is exactly one solution to
the ODE which is bounded as x  0+ . It is
F1  x  
 b x  U  p, 2 p
1
p
 2   , b x 1  ,
where p is the positive solution of the quadratic equation
p2

1
 p  
 0 ,
and where U( a, b, z ) is Tricomi’s confluent hypergeometric
function.
42
Valuation of a Perpetual American Call Option on an
Underlying Asset which is a Function of an IGBM, III
Phantom Works
Mathematics & Computing Technology
• Tricomi’s confluent hypergeometric function U( a, b, z ) is
defined by
U  a, b, z  
 1  b 
M  a, b, z  
 1  a  b 
 b  1 
M 1  a  b , 2  b , z  ,
b 1
 a  z
where ( . ) is Euler’s gamma function, and where M( a, b, z )
is Kummer’s confluent hypergeometric function, defined by
M  a, b, z  


k 0
 a  k   b  z k
.
 a   b  k  k !
• The function U( a, b, z ) can be computed numerically using a
rational function approximation algorithm devised by Luke
(1977).
References: [L], [SpO]
43
Valuation of a Perpetual American Call Option on an
Underlying Asset which is a Function of an IGBM, IV
Phantom Works
Mathematics & Computing Technology
• The particular solution F1( x ) of the ODE satisfies the initial
conditions
F1  0   1 ,
d F1
dx
0


b
 0 .
• Moreover, using the facts that  > 0 , b > 0 , and  <  ,
and some classical identities for confluent hypergeometric
functions, one can show that
d F1
dx
x
 0 and
d 2 F1
dx
2
x
44
 0 , for 0  x   .
Valuation of a Perpetual American Call Option on an
Underlying Asset which is a Function of an IGBM, V
Phantom Works
Mathematics & Computing Technology
• The value of the American call option is
F x  
A F1  x  ,
for some constant A > 0 .
• The constant A must be chosen so that F( x ) solves the
free boundary problem
1 2 2 d 2F
 x
2
d x2
F x

  X
 x   r  x 
 bounded as
x  0 ;
F x*   V  x*   I ;
dF
 x*  
dx
dV
 x*  .
dx
45
dF
dx
 rF
 0 , 0  x  x* ;
Example: Valuation of a Perpetual American Option on
a Mean-Reverting Perpetuity, I
Phantom Works
Mathematics & Computing Technology
• Consider a perpetual American call option on a meanreverting perpetuity, which begins payment immediately after
the option has been exercised (so that  = 0 , on page 19 ).
• Then the free boundary problem for this option is
1 2 2 d 2F
 x
2
d x2
F x


 X
 x   r  x

dF
dx
 rF
 0 , 0  x  x* ;
bounded as x  0  ;
F x*  
dF
 x*  
dx
X
  r
 I ;
r    r
x* 
X

r    r 
1
r    r
.
46
Example: Valuation of a Perpetual American Option on
a Mean-Reverting Perpetuity, II
Phantom Works
Mathematics & Computing Technology
• To be specific, suppose that
X
 1 ,   0.14  t1/ 2  5  ,   0.2 , r  1 ,   0.4 , and r  0.05 .
• Suppose that I = 14.0438 . (This is the value the perpetuity
would have if the currently observed value of Xt were 1 .)
• Then one finds that A = 0.0937 , x* = 1.2402 . The option
value is plotted below, as a function of x .
47
Example: Valuation of a Perpetual American Option on
a Mean-Reverting Perpetuity, III
Phantom Works
Mathematics & Computing Technology
• Suppose that we fix
X
 1 ,   0.14  t1/ 2  5  ,   0.2 , r  1 ,   0.4 , and I  14.034 .
r = 0.04
r = 0.05
r = 0.06
48
x = 1
Example: Valuation of a Perpetual American Option on
a Mean-Reverting Perpetuity, IV
Phantom Works
Mathematics & Computing Technology
• Suppose that we fix
X
 1 ,   0.14  t1/ 2  5  ,   0.2 ,   0.4 , r  0.05 , and I  14.034 .
r = 0
r = 0.5
r = 1
49
x = 1
Example: Valuation of a Perpetual American Option on
a Mean-Reverting Perpetuity, V
Phantom Works
Mathematics & Computing Technology
• Suppose that we fix
X
 1 ,   0.14  t1/ 2  5  , r  1 ,   0.4 , r  0.05 , and I  14.034 .
 = 0.1
 = 0.2
 = 0.3
50
x = 1
Example: Valuation of a Perpetual American Option on
a Mean-Reverting Perpetuity, VI
Phantom Works
Mathematics & Computing Technology
• Suppose that we fix
X
 1 ,   0.2 , r  1 ,   0.4 , r  0.05 , and I  14.034 .
t1/2 = 2.5
t1/2 = 5
t1/2 = 10
51
x = 1
Binomial Tree Approach, I
Phantom Works
Mathematics & Computing Technology
• For more complex problems, one can use a binomial
approximation to the risk-neutralized mean-reverting process.
• Calistrate (2000) suggests the following approach:
--Let t = T / n , where T is the time horizon and n is
the number of time steps.
--Let u  e 
t
and d

1
.
u
--The value of the state variable at time i , after j
“up” steps, is
Xˆ  i , j  
52
X0 u j di j
.
Binomial Tree Approach, II
Phantom Works
Mathematics & Computing Technology
--The probability of an “up” step at time i , after j
“up” steps, is
 1

t
p i , j   P   1 
m *  i , j    ,


 2
where
 X  Xˆ  i , j 
2
m * i , j  
 r  
,
ˆ
2
X i, j 


and where
 0 , if x  0 ;

P x    x , if 0  x  1 ;
 1 , if 1  x .

• The analytical models presented earlier can be used to
validate a binomial tree.
Reference: [Ca]
53
Conclusion
Phantom Works
Mathematics & Computing Technology
• Of the models we have considered for a mean-reverting price
process, inhomogeneous geometric Brownian motion is the
most appropriate.
--It is guaranteed to be positive
--One can compute its moments
--It really does revert to a mean
--It has an appealing homogeneity property
--The available evidence suggests that it is (at least
approximately) lognormal
• Bhattacharya (1978) showed how to value certain cash flow
streams which depend on such a process.
• We have shown how to value certain options on such cash
flow streams.
54
References, I
Phantom Works
Mathematics & Computing Technology
[B]
[BØ]
[Ca]
[Co]
[CoA]
[DP]
S. Bhattacharya, “Project Valuation with Mean-Reverting
Cash Flow Streams,” Journal of Finance 33, December
1978, pp. 1317-1331.
K. Brekke and B. Øksendal, “The High Contact Principle
as a Sufficiency Condition for Optimal Stopping.”
Stochastic Models and Option Values, D. Lund and
B. Øksendal, editors, Elsevier Science Publishers, 1991,
pp. 187-208.
D. Calistrate, “Setting Lattice Parameters for Derivatives
Valuation”, private communication, 2000.
G. Constantinides, “Market Risk Adjustment in Project
Valuation,” Journal of Finance 33, May 1978, pp. 603-616.
T. Copeland and V. Antikarov, Real Options: A
Practitioner’s Guide, Texere, 2001.
A. Dixit and R. Pindyck, Investment under Uncertainty,
Princeton University Press, 1994.
55
References, II
Phantom Works
Mathematics & Computing Technology
[KP]
P. Kloeden and E. Platen, Numerical Solution of Stochastic
Differential Equations, Springer-Verlag, 1992.
[L]
Y. Luke, “Algorithms for Rational Approximations for a
Confluent Hypergeometric Function,” Utilitas Math. 11
(1977), pp. 123-151.
[McDS] R. McDonald and D. Siegel, “The Value of Waiting to
Invest,” Quarterly J. Econ. 101 (1986), pp. 707-728.
[Pil]
D. Pilipovic, Energy Risk, McGraw-Hill, 1998.
[Pin]
R. Pindyck, “Irreversibility, Uncertainty, and Investment,”
J. Econ. Lit. 19 (1991), pp. 1110-1148.
[R]
B.L.S. Prakasa Rao, Statistical Inference for Diffusion Type
Processes (Kendall’s Library of Statistics 8), Arnold / Oxford
University Press, 1999.
[Sh]
D. Shimko, Finance in Continuous Time--A Primer,
Kolb, 1992.
56
References, III
Phantom Works
Mathematics & Computing Technology
[Si]
[SO]
[SpO]
[T]
[W]
G. Sick, “Real Options.” Finance (Handbooks in Operations
Research and Management Science, Volume 9), R. Jarrow,
V. Maksimovic, and W. Ziemba, editors, pp. 631-691,
North-Holland, 1995.
I. Shoji and T. Ozaki, “Comparative Study of Estimation
Methods for Continuous Time Stochastic Processes,”
J. Time Series Analysis 18 (1997), pp. 485-506.
J. Spanier and K. Oldham, An Atlas of Functions,
Hemisphere, 1987.
L. Trigeorgis, Real Options, MIT Press, 1996.
P. Wilmott, Paul Wilmott on Quantitative Finance, Wiley,
2000.
57