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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. IAG PUC – Rio Brandão Stochastic Process Valor Valor 105 103 103 100 100 98 98 105 Tempo 95 Tempo 95 Valor 150 125 Stochastic Process: 100 A variable that evolves over time in a way that is at least partially random. 75 Tempo 50 IAG PUC – Rio Brandão 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. IAG PUC – Rio Brandão 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. IAG PUC – Rio Brandão 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 IAG PUC – Rio Brandão 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 IAG PUC – Rio Brandão Random Walk 120 100 80 0 IAG PUC – Rio 50 Brandão 100 150 9 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. IAG PUC – Rio Brandão 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. 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: St 1 St dz E dz 0 IAG PUC – Rio Brandão 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. IAG PUC – Rio Brandão Arithmetic Brownian Motion Arithmetic Brownian Motion (with and without drift) 20 10 0 (10) IAG PUC – Rio Brandão Arithmetic Brownian Motion MAB - Movimento Aritmético Browniano 15 10 5 0 0 IAG PUC – Rio 50 Brandão 100 150 200 250 300 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. IAG PUC – Rio Brandão 16 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. 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 IAG PUC – Rio Brandão 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 IAG PUC – Rio 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 IAG PUC – Rio Brandão where 1 2 v 2 19 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 IAG PUC – Rio Brandão Simulation Paths of GBM’s Realization MGB - Movimento Geométrico Browniano 100 75 50 25 0 0 IAG PUC – Rio Brandão 50 100 150 200 250 300 21 Mean Reversion Process 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. IAG PUC – Rio Brandão 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. IAG PUC – Rio Brandão Simulations of a Mean Reverting Process Processo de Reversão à Média 12 9 6 3 0 IAG PUC – Rio Brandão 50 100 150 200 250 300 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 IAG PUC – Rio Brandão 25 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 IAG PUC – Rio Brandão 30 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 IAG PUC – Rio Brandão 2 r t t 2 31 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 IAG PUC – Rio Brandão 2 RiskNormal t , t 2 32 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. IAG PUC – Rio Brandão 33 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% IAG PUC – Rio Brandão 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. 34 Real Options Stochastic Processes Prof. Luiz Brandão [email protected] 2009