Real Options Stochastic Processes Prof. Luiz Brandão [email protected] 2009.

Real OptionsStochastic Processes

Prof. Luiz Brandão

[email protected]

2009

Modeling the Underlying Asset

Stochastic Process

IAG PUC – Rio Brandão

A project’s uncertainty can have more than two outcomes

In practice, the number of outcomes can be infinite

We are able to obtain a more detailed uncertainty model assuming that a variable follows a stochastic or a random process.

Modeling Uncertainty

Estado 1

Estado n

Estado 5

Estado 4

Estado 3

Estado 2

Fluxo 1

Fluxo n

Fluxo 5

Fluxo 4

Fluxo 3

Fluxo 2

.............

Fluxo noestado 2

t = 0

Fluxo noestado 1

Estado 1

Estado 2

t = 1


Stochastic Process:

A variable that evolves over time in a way that is at least partially random.

Stochastic Process

95

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105Valor

Tempo95

98

100

103

105Valor

Tempo

50

75

100

125

150Valor

Tempo


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.

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.

Stochastic 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

Markov Process


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 is the value of the variable at t+1

St is the value of the variable at t

εt is a random variable with probability

P(εt=1 ) = P(εt=-1) = 0.5

Random Walk


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Random Walk

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100

120

0 50 100 150


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.

Random Walk


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.

Wiener Process


Wiener Process

1 onde e (0,1)t tS S dz dz dt N

0 varE dz dz dt

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:


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:

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.

Arithmetic Brownian Motion (ABM)

1

2( , )

t tS S dt dz

dS dt dz dS N dt dt


Arithmetic Brownian Motion

(10)

0

10

20

Arithmetic Brownian Motion (with and without drift)


Arithmetic Brownian Motion

0

5

10

15

0 50 100 150 200 250 300


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ABM Model Limitations The ABM is also known as an additive model because the variable grows by

a constant value every period.

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.


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.

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

where μ = expected rate of return

σ = volatility of the security’s value

Geometric Brownian Motion

dV Vdt Vdz


Note that proportional changes in the value of V are normally distributed, given that is a ABM.

In discrete time, dV/V is the return of V.

In continuous time, if then v is the return of V.

For t = 1 we have and

Taking the logarithm we have

dV V dt dz

0v t

tV V e

1 0vV V e

1 0ve V V

1 0lnv V V



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We can also represent GBM as

Through differential calculus we know that

Therefore if then

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

where

1lnd x dx

x

lnd V vdt dz 21

2v

dVdt dz

V


lndx

d xx

lndV

d VV


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:

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

1 0lnv V V



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Simulation Paths of GBM’s Realization

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100

0 50 100 150 200 250 300


As we have seen previously, the variable tends to achieve values very different from its initial value in the GBM.

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.

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:

where = reversion speed

= 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.

( )dV V V dt dz

V

Mean Reversion Process


Simulations of a Mean Reverting Process

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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

Simulation with @Risk


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Underlying Asset ModelingGBM

If the underlying asset follows GBM, we have

To simulate the path followed by V we use a discrete model:

This can be modeled in Excel as:

With @Risk the representation is:

dV Vdt Vdz

1 (0,1)t t t tV V V t V t N

1 ( ())t t t tV V V t V NORMSINV RAND t

1 (0,1)t t t tV V V t V RiskNormal t


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We can also simulate Ln (V) instead of V directly, since:

To simulate the path we have:

2

2

1

ln2

ln ln2t t

d V dt dz

V V t t

2

2

1

r t t

t tV V e

Underlying Asset Modeling


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This can be modeled in Excel as:

In @Risk the representation is:

2

( ())2

1 et NORMSINV RAND t

t tV V

2

,2

1 eRiskNormal t t

t tV V

Underlying Asset Modeling


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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%

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.

Real OptionsStochastic Processes

Prof. Luiz Brandão

[email protected]

2009

Real Options Stochastic Processes Prof. Luiz Brandão [email protected] 2009.

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