Stochastic processes
In finance and risk, you will always be running into what are called ‘stochastic processes’. Well, that is just a more complex way of saying that a variable is random. A variable is considered ‘stochastic’ when its value is uncertain. In finance, security returns are usually considered stochastic. In this brief article , we look at some key concepts relating to stochastic variables, including the geometric Brownian motion process (borrowed from particle physics) which is often used to model asset returns.
In finance and risk, you will always be running into what are called ‘stochastic processes’. Well, that is just a more complex way of saying that a variable is random. A variable is considered ‘stochastic’ when its value is uncertain. In finance, security returns are usually considered stochastic. In this brief article , we look at some key concepts relating to stochastic variables, including the geometric Brownian motion process (borrowed from particle physics) which is often used to model asset returns.
Some terms to know:
A Markov process: A Markov process is one where a variable doesn’t follow a trend, ie, the value of a variable depends solely upon its prior value, and is unrelated to the historical trend of its value. Stock prices follow a Markov process.
A Weiner process: A certain type of a Markov process, essentially it means a variable that is a random draw from a standard normal distribution. Same as ‘Brownian motion’ (though not the same as geometric Brownian motion – which comes next in the chain. Often represented as ‘dz’.
Generalized Weiner process: A generalized Weiner process is a normal distribution with a drift and volatility.
dx = a*dt + b*dz, where dx is the change in the variable x, a is a constant drift rate, dt is the time period, b is a constant equal to the volatility of x, and dz is the Weiner process (ie, a random number from a standard normal distribution).
Geometric Brownian motion process: This is the process used to describe the behavior of stock prices. This is slightly different from the generalized Weiner process in that the drift rate a is a not a constant number, but a percentage rate that reflects the expected return on the stock, so that the actual dollar number of the return changes as the stock price itself changes. Takes the form dS/S=μdt + σdz. With this process as the assumption, the expected stock price at time T is given by ST = S0eμT
Behavior of future prices:
Now let me highlight an important thing about ST = S0eμT. This is the same as saying that for a starting price of $1, the price one year hence is eμ. We know that if x is normally distributed with a mean and variance of μ and σ, then exp(x) is lognormally distributed with an expected value of exp(μ + σ2/2). If the process for a stock price is given by dS/S=μdt + σdz, which implies dS/S has a mean of μ and a variance of σ. So why isn’t the future price equal to exp(μ + σ2/2))? Why did the σ2/2 term drop out? That is because dS/S is not the same as ln(S1/S0). dS/S represents discrete returns calculated as (S1 – S0)/S0. For very short periods of time, these two terms (ie ln(S1/S0) and (S1 – S0)/S0 are near identical, but not truly so. I plan to write another article (okay – finally written after considerable delay here: http://www.riskprep.com/all-tutorials/35-exam-1/129-volatility-returns-and-stock-prices) with some examples to illustrate the difference, but for the moment suffice it to say that if the mean of dS/S (calculated as the mean of (S1 – S0)/S0) is μ with a volatility of σ, then the mean of ln(S1/S0) is μ – σ2/2, and its variance is σ. This means the expected value of S1 will be exp(μ – σ2/2 + σ2/2), or exp(μ).
Now a little bit more about returns and prices:
The Weiner process describes how the value of a variable is determined at a point in time from its mean and standard deviation. It is based upon a physical phenomenon – the generalized Brownian motion process – according to which molecules randomly move about as they are suspended in a medium.
The Weiner process, when applied to an asset’s price, simply says this: If we know the price of an asset today to be S0, then its price S1 at a future point in time will be [S0 + expected returns during the time period S0 to S1 + an error that would depend upon the volatility of the returns on this asset]. Another way of saying this is that the price of the asset would change from its present level of S0 by an amount equal to expected returns plus a random number that is determined by the variability of the expected returns. If the returns are not variable at all, ie they are risk free, then the change in price would be exactly equal to expected returns. As the variability of returns increases, the change in price also becomes increasingly unpredictable. According to the Weiner process, the random number is equal to a drawing from the normal distribution.
Therefore: Change in price = Expected return + Number drawn randomly from a standard normal distribution,
Or Δp = μ + ϕ(0, σ), where Δp is the change in price, μ is the expected return (as a percentage of the stock price), ϕ is a drawing from the normal distribution and σ is the standard deviation of the expected return.
Since μ and σ are defined in terms of time (ie, the returns for one day are different from the returns for one year, and likewise for standard deviation), we often express this equation as follows, where Δt reflects an adjustment for time.
Δp = μ Δt + ϕ(0, σ √Δt)
Which is the same as saying: Δp = ϕ(μ Δt, σ √Δt)
Also, if P is the current price, then the future price P1 is given by:
P1 = μ PΔt + P.ϕ(0, σ √Δt)
The future price is determined by the current price (P), expected return (μ Δt) and standard deviation (σ √Δt).
Example: Consider a security that has an expected return of 24% annually with a volatility of 16%. How can we simulate its price at the end of one month? If the security follows the Weiner process, then the change in its price will be equal to expected returns, which would be 2% for the month, plus a random drawing from the normal distribution with a mean of 2% (=24%/12) and volatility of 4.6% (=16%/SQRT(12)). This can be expressed as:
Change in price = ϕ(μ Δt, σ √Δt)= ϕ (2%, 4.6%)
Modeling in Excel: Note that it is easy to produce a random drawing from a normal distribution in Excel using the formula “=NORMINV(RAND(),expected return,volatility)”.