1. Real Options Stochastic Processes Prof. Luiz Brandão brandao@iag.puc-rio.br 2009

2. Modeling the Underlying Asset Stochastic Process

3. 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

4. IAG PUC – Rio Brandão Stochastic Process: A variable that evolves over time in a way that is at least partially random. Stochastic Process 95 98 100 103 105 Valor Tempo 95 98 100 103 105 Valor Tempo 50 75 100 125 150 Valor Tempo

5. IAG PUC – Rio Brandão  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

6. IAG PUC – Rio Brandão  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

7. IAG PUC – Rio Brandão  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

8. IAG PUC – Rio Brandão  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: S t+1 = S t + ε t where S t+1 is the value of the variable at t+1 S t is the value of the variable at t ε t is a random variable with probability P(ε t =1 ) = P(ε t =-1) = 0.5 Random Walk

9. IAG PUC – Rio Brandão 9 Random Walk 80 100 120 050100150

10. IAG PUC – Rio Brandão  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 S t+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

11. IAG PUC – Rio Brandão  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

12. IAG PUC – Rio Brandão 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:

13. IAG PUC – Rio Brandão  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)

14. IAG PUC – Rio Brandão Arithmetic Brownian Motion Arithmetic Brownian Motion (with and without drift)

15. IAG PUC – Rio Brandão Arithmetic Brownian Motion

16. IAG PUC – Rio Brandão 16 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.

17. IAG PUC – Rio Brandão  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 whereμ = expected rate of return σ = volatility of the security’s value Geometric Brownian Motion

18. IAG PUC – Rio Brandão  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 Geometric Brownian Motion

19. IAG PUC – Rio Brandão 19  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 Geometric Brownian Motion

20. IAG PUC – Rio Brandão  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 Geometric Brownian Motion

21. IAG PUC – Rio Brandão 21 Simulation Paths of GBM’s Realization

22. IAG PUC – Rio Brandão  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

23. IAG PUC – Rio Brandão  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. Mean Reversion Process

24. IAG PUC – Rio Brandão Simulations of a Mean Reverting Process

25. IAG PUC – Rio Brandão 25 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

26. Simulation with @Risk

27. IAG PUC – Rio Brandão 27 Simulation  Simulation involves the creation of a mathematical model of a real project that incorporates uncertainty, doing calculations and storing results.  The objective of a stimulation is to determine a distribution of results, instead of a single answer.  The simulation process involves: Definition of distribution of each basic variable Perform a series of repetitive calculations to obtain a result Use the obtained values to determine the distribution of the results

28. IAG PUC – Rio Brandão 28 Simulating Distributions  Each distribution is modeled by a probability distribution  The choice of a particular distribution will depend on the random variable we wish to model.  For this we can utilize historic data to model the distribution, make a subjective choice or use prior knowledge for the type of random process.  Most of the time we will adopt a lognormal distribution to model the value of securities.

29. IAG PUC – Rio Brandão 29 Simulating Distributions  Once the desired distribution is chosen, it is necessary to generate random values from this distribution.  There are diverse ways to do this: To develop a software program to perform the simulation. Use the distribution functions of Excel in connection with the RAND function. Use the functions of specialized programs like Crystall Ball and @Risk.  In this course, we will utilize the software @Risk to carry out the necessary simulations.

30. IAG PUC – Rio Brandão 30 Underlying Asset Modeling GBM  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:

31. IAG PUC – Rio Brandão 31  We can also simulate Ln (V) instead of V directly, since:  To simulate the path we have: Underlying Asset Modeling

32. IAG PUC – Rio Brandão 32  This can be modeled in Excel as:  In @Risk the representation is: Underlying Asset Modeling

33. IAG PUC – Rio Brandão 33 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.

34. IAG PUC – Rio Brandão 34 Example: European Call  Underlying asset: Share that doesn’t pay dividends  The share follows GBM  S 0 = $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.

35. Real Options Stochastic Processes Prof. Luiz Brandão brandao@iag.puc-rio.br 2009