By: Brian Scott

Topics Defining a Stochastic Process Geometric Brownian Motion G.B.M. With Jump Diffusion G.B.M with jump diffusion when volatility is stochastic and beyond… Monte Carlo, Applicability, and examples

So, What is Stochastic? A stochastic process, or sometimes called random process, is a process with some indeterminacy in the future evolution of the variables being examined (i.e. Stock Prices, Oil Prices, Returns of the Finance Sector, etc…) Don’t Fret though because, we can describe the parameters and variables by probability distributions which allows for a fun new way to solve math problems with random variables!!! Stochastic Calculus!

However, that is well beyond the scope of this MIF meeting, what I will do is give you a visual interpretation of stochastic process and how they are used though So what is the Problem??? We Want to analyze how stock prices progress over time, which Is a stochastic process… To do this we’ll start simple(non-stochastic) and get progressively more complex

Lets start as simple as it gets What We know… – Price of the stock today – Some Approx. of μ Return (μ = mean/average) Ok so lets model that…

What does that look like? Just as terrible as you expected

Time to get a little more realistic What else do we know???? Volatility! Lets take the last equation and add some volatility to it…

Time for something even more realistic Lets step into the world where the variance of the daily returns isn’t fixed but rather a random sample from a Normal Distribution

Now things get interesting

Geometric Brownian Motion Change in the Stock Price Drift Coefficient Random Shock

Accounting for natural phenomena's

What are jumps??? Speculation/ Self Fulfilling Prophecies with market or individual stock conditions Earnings Reports (Beating or Missing) drastically Some completely unrelated catastrophe i.e. a terrorist blowing up a building, or a meteor hitting earth (harder to model…)

So How de we capture this phenomena??? Don’t Worry Good ole’ Poisson Distributions from COB 191 to the Rescue!!! Used to describe discrete known events ( i.e. earnings reports!!!) Lets see how we can use this insight!

Geometric Brownian motion with “Jump Diffusion” Some assumptions… Jumps can only occur once in a time interval ln(J) ~ N

G.B.M. with Jump Diffusion notes We can Add in a myriad of jump factors for different forecasted phenomena’s We don’t have to let jumps be fixed, with some alterations in algebra using the fact that ln(J)~N you can add stochastic Jump sizes!!! Or we can get dynamic, if we want the jump size to be randomly between -15% and 9% (i.e. earnings, or an FDA drug approval)

Completely Random Jump Size

With a little math we can let the random jump be bounded between -15 % and 9% w.r.t. Maximum likelihood Estimation

Where do we go from here? Well… one major assumption of all the model thus far, is that we assumed σ constant An quick empirical look at volatility will clearly show that this could not be farther from the truth!

Lets get crafty… Since Volatility is not actually constant lets let volatility become stochastic as well!! Notice that volatility follows a bursty pattern that stays around an average

Ornstein Uhlenbeck Processes Rock! Well it just so happens there is a stochastic process that can model this! It is a class of stochastic differential equations known as Mean-Reverting Function of Ornstein Uhlenbeck Models

Examples of Stochastic Volatility using mean reversion

Putting it all together we now have Geometric Brownian Motion With Jump Diffusion, when Volatility is Mean Reverting Stochastic where W and Z are independent Brownian motions, ρ= the correlation between price and volatility shocks, with ρ < 1. m= long run mean of volatility dQt= jump term α = rate of mean reversion Β = volatility of the volatility μ = drift of the stock

Going beyond… Notice that things follow their moving averages… You can correlate random variables of stochastic volatility so that its reverting mean is a correlated stochastic process and not stagnent

Geometric Brownian Motion With Stochastic Jump Diffusion, when Volatility is Mean Reverting Stochastic process, to a correlated stochastic mean where W, Z, and X are independent Brownian motions, ρ= the correlation between price and volatility shocks, with ρ < 1. m= long run mean of volatility dQt= jump term α = rate of mean reversion Β = volatility of the volatility μ = drift of the stock Where m = f ( ρ σma,σ,, X )

Monte Carlo Simulation.. A Monte Carlo method is a computational algorithm that relies on repeated random sampling to compute results. Monte Carlo methods are often used when simulating financial systems/situations.computationalalgorithmrandom Because of their reliance on repeated computation and random or pseudo-random numbers, Monte Carlo methods are most suited to calculation by a computer. randompseudo-randomcomputer Monte Carlo methods tend to be used when it is infeasible or impossible to compute an exact result with a deterministic algorithmdeterministic algorithm

Applicability As you can probably see, it is easy to create and run these projections hundreds of times using a program like Crystal Ball! – Calculate Certain Parameters and Correlations – Take into account upcoming events (i.e. earnings) – Make some predictions based on historical data, and upcoming events for the market/company about jump sizes – Run 100,000’s or times and analyze results – Then Run testing for sensitivity to changing decision variables

And you thought you’d never understand stochastic processes… Any Questions??? Going Further – Test with historical data the relative errors of all methods – Get more computing power than showker to run the models

The End Special Thanks to… My mom ( she always believed in me!!) and… Showker computer lab for running out of virtual memory every time I try running crystal ball