Transcript Stochastic Processes.

Real Options
Stochastic Processes
Prof. Luiz Brandão
[email protected]
2009
Modeling the Underlying Asset
Stochastic Process
Modeling Uncertainty

A project’s uncertainty can have more than two outcomes

In practice, the number of outcomes can be infinite
Estado 1
t=1
t=0
Estado 1
Fluxo no
estado 1
Estado 2
Estado 3
Estado 4
Estado 2
Fluxo no
estado 2
Estado 5
Estado n

Fluxo 1
Fluxo 2
Fluxo 3
Fluxo 4
Fluxo 5
............
.
Fluxo n
We are able to obtain a more detailed uncertainty model assuming that a
variable follows a stochastic or a random process.
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Stochastic Process
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Valor
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Tempo
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Tempo
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Valor
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Stochastic Process:
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A variable that evolves over time
in a way that is at least partially
random.
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Tempo
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Stochastic Process

Stochastic processes were initially used in physics to
describe the motion of particles. They can be classified in
the following categories:

Continuous Time Process: The variable can change its
value at any moment in time.

Discrete Time Process: The variable can only change its
value during fixed intervals.

Continuous Variable: The variable can assume any value
within a determined interval.

Discrete Variable: The variable can assume only a few
discrete values.

Stationary Process: The mean and variance are constant
over time.

Non-stationary Process: The expected value of the random
variable can grow without limit and its variance increases over
time.
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Stochastic Process

The majority of real problems are modeled using continuous time
stochastic process with continuous variance.

On the other hand, continuous time processes require the use of
calculus to solve the stochastic differential equations that model
these processes.

Continuous time process can be approximated through discrete
process, which has simpler modeling.

We will study the principal models of continuous process, and
subsequently, the corresponding discrete modeling.
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Markov Process

The Markov Process is a Stochastic process where only the
present value of the variable is relevant to predict the future
evolution of the process.

This means that historic values or even the path through which
the variable arrived at its present value are irrelevant in
determining its future value.

Assume that the price of securities in general, like stock and
commodities, follow a Markov process.

Given this premise, we assume that the current price of a stock
reflects all the historical information as well as expectations
about its future price.

Using this model, it would be impossible to predict the future
value of a stock based on historical price infomation
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Random Walk

Random Walk is one of the most basic stochastic processes.

The name is derived from the path followed by a drunken sailor
walking along the quay. His unsteady steps vary randomly in
direction while its final destiny becomes more uncertain with time.

Random Walk is a Markov process in discrete time that has
independent increments in the form of:
St+1 = St + εt
where St+1
St
εt
is the value of the variable at t+1
is the value of the variable at t
is a random variable with probability
P(εt=1 ) = P(εt=-1) = 0.5
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Random Walk
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Random Walk

Random Walks can include a growth term, or drift, that represents a
long term growth.

Without the drift term, the best estimate of the next value of the
variable St+1 is the present value, if the term of error is normally
distributed with a mean of zero.

With the drift term, or growth, the future values of the variable tend to
grow in a proportional manner to the rate of growth.
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Wiener Process

The Wiener process is a stochastic process that has a
mean of zero and a variance of one per time period.

The Wiener process is a particular case of the Markov
process, and is also known as Brownian Motion.

This process was described for the first time by the botanist
Robert Brown in 1827, and is utilized in physics to describe
the motion of small particles subject to a large number of
small random collisions.

This process has its name in honor of the mathematician
Norbert Wiener, who in 1923 developed the mathematical
theory of the Brownian Motion.
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Wiener Process

The Wiener process has three important characteristics:

It is a continuous time Markov process

Each increment of the process is independent of the previous
increments.

Changes in the process are normally distributed with a variance that
increases lineally with time.
The Wiener process is a continuous version of the Random Walk in
the form:

St 1  St  dz
E  dz   0
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onde dz   dt
var  dz   dt
e   N (0,1)
Arithmetic Brownian Motion (ABM)

The Wiener process is a stationary process, without the drift term. If
we add a long term growth to the Wiener process we obtain a
Movement Arithmetic Brownian (ABM), that has the following
mathematic representation:
St 1  St   dt   dz
dS   dt   dz

dS  N (  dt ,  2 dt )
The evolution of a ABM is a combination of two parts:


A linear growth, with a rate μ
A random growth with a normal distribution and a standard deviation σ

The focus of the ABM is in the change in the value of the variable,
instead of the value of the variable itself.

Since it is also a Random Walk, the ABM also has a normal
distribution.
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Arithmetic Brownian Motion
Arithmetic Brownian Motion (with and without drift)
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Arithmetic Brownian Motion
MAB - Movimento Aritmético Browniano
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ABM Model Limitations
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The ABM is also known as an additive model because the variable grows by
a constant value every period.
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However, modeling securities with ABM presents some problems:


Since the random term is a normally distributed variable, the value of the variable
can occasionally become negative, which cannot happen with the price of
securities.

For a stock that doesn’t pay dividends, the rate of return of the stock decreases
with time as the value of the stock increases for ABM. We know, however, that
investors require a constant rate of return, independent of the price of the stock.

In ABM the standard deviation is constant throughout time, while to better model
securities the standard deviation should be proportional to the value of the
security.
Because of these reasons ABM is not the most appropriate process to
model the prices of stock or securities in general.
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Geometric Brownian Motion

A more appropriate process to model securities is a process
where the return and the proportional volatility of the process are
constant.

This model is known as Geometric Brownian Motion, or GBM, or
multiplying model.
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The evolution of a GBM is a combination of two installments:


A proportional growth, with a rate μ

A random proportional growth with a normal distribution and a
standard deviation σ
The formula in continuous time is
dV  Vdt   Vdz
μ = expected rate of return
where
σ = volatility of the security’s value
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Geometric Brownian Motion



Note that proportional changes in the value of V are
normally distributed, given that dV V   dt   dz
ABM.
is a
In discrete time, dV/V is the return of V.
In continuous time, if
Vt  V0 e v t
V1  V0 e
v
then v is the return of V.

For t = 1 we have

Taking the logarithm we have v  ln V1 V0
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and
ev  V1 V0

Brandão

Geometric Brownian Motion

We can also represent GBM as
dV
  dt   dz
V


1
d ln x  dx
Through differential calculus we know that
x
dx
dV
Therefore if d ln x 
then d ln V 
x
V

Unfortunately we can't directly substitute this in the GBM
equation, because stochastic processes require analysis through
stochastic calculus, or an Ito process..

The correct representation is
d ln V  vdt   dz
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where
1 2
v 
2
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Geometric Brownian Motion

Observe that dV/V is the return of V and has a normal
distribution, because its a ABM.

In continuous time, the return of the price V is given by

Because the returns of V have a normal distribution, V has a
lognormal distribution.

GBM has three characteristics that make it ideal to model the
price of securities:



v  ln V1 V0 
It allows for exponential growth, as in the case of composite
interest.
The returns are normally distributed, which facilitates its
mathematical manipulation.
The value of V cannot become negative, like it occurs with the
price of securities
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Simulation Paths of GBM’s Realization
MGB - Movimento Geométrico Browniano
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Mean Reversion Process
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As we have seen previously, the variable tends to achieve values
very different from its initial value in the GBM.
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Although this can be realistic to model the value of the majority of
securities, there is a group of securities that don’t behave that way.

It is believed that many securities like oil, copper, agricultural
products and other commodities have their price correlated with its
marginal cost of production, while they may suffer random
variations in the short term.

To the extent that the price varies, the producers will increase
production to benefit from the high prices and reduce them to avoid
losses when the prices are low. This will force prices to revert to
their long term equilibrium value.
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Mean Reversion Process

There are many models for mean reverting processes. One of
the most simple is the Ornstein-Uhlenbeck model, which has the
following mathematical expression:
dV   (V  V )dt   dz
where

= reversion speed
V = the long term mean to which V tends to revert

The speed of reversion indicates how quickly the variable reverts
to its long term equilibrium value.
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Simulations of a Mean Reverting
Process
Processo de Reversão à Média
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Final Comments

The ABM, GBM models and the Mean Reverting process are also
known as “models of diffusion,” where the value of the variable
changes in small increments each time.

Processes where the value of the variable changes suddenly are
named ”jump” models.

The ABM is more utilized for physics processes, while the GBM is
widely utilized to model prices of financial securities and real
securities. This will be the principal process that we’ll use in this
course.

Mean reverting process are utilized to model interest rates and
commodity prices
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Simulation with @Risk
Underlying Asset Modeling
GBM
dV  Vdt   Vdz

If the underlying asset follows GBM, we have

To simulate the path followed by V we use a discrete model:
Vt 1  Vt   Vt t  Vt t

  N (0,1)
This can be modeled in Excel as:
Vt 1  Vt   Vt t  Vt NORMSINV ( RAND()) t


With @Risk the representation is:
Vt 1  Vt   Vt t  Vt RiskNormal (0,1) t

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Underlying Asset Modeling

We can also simulate Ln (V) instead of V directly, since:

2 
d ln V    
 dt   dz
2 


2 
ln Vt 1  ln Vt    
 t   t
2 


To simulate the path we have:
Vt 1  Vt e
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 2 
 r  t    t
2 

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Underlying Asset Modeling

This can be modeled in Excel as:
Vt 1  Vt e

 2 
   t  NORMSINV ( RAND ()) t
2 

In @Risk the representation is:
Vt 1  Vt e
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 2

RiskNormal    t , t 
2


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Evaluating Options with Simulation

Options can also be evaluated utilizing the Monte Carlo Simulations.

This is done analyzing each realization of the underlying security’s
path and determining the value of the option at its expiration.

The value of the option is the expected present value of the value of
the option at each realization

The underlying asset and the present value of the value of the
option at expiration are modeled utilizing a risk neutral evaluation.
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Example: European Call

Underlying asset: Share that
doesn’t pay dividends

The share follows GBM

S0 = $100

Volatility =20%

Time to expiration T = 1 year

Exercise price X = $100

Risk free rate is r = 7%

μ = 11%
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
The Monte Carlo
solution with 10,000
iterations is 11.5407

The exact solution
(Black and Scholes) is
$11.5415

Note that the rate of
return μ of the
underlying asset is not
utilized in valuing the
option.
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Real Options
Stochastic Processes
Prof. Luiz Brandão
[email protected]
2009