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By: Brian Scott
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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
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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!
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• 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
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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…
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•What does that look like?
•Just as terrible as you expected
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Time to get a little more realistic
• What else do we know????• Volatility!• Lets take the last equation and add some
volatility to it…
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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
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Now things get interesting
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Geometric Brownian Motion
Change in the Stock Price
Drift Coefficient
Random Shock
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Accounting for natural phenomena's
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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…)
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• 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!
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Geometric Brownian motion with “Jump Diffusion”
•Some assumptions…•Jumps can only occur once in a time interval • ln(J) ~ N
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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)
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Completely Random Jump Size
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With a little math we can let the random jump be bounded between -15 % and 9% w.r.t.
Maximum likelihood Estimation
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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!
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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
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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
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Examples of Stochastic Volatility using mean reversion
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Putting it all together we now haveGeometric 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 termα = rate of mean reversionΒ = volatility of the volatilityμ = drift of the stock
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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
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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 termα = rate of mean reversionΒ = volatility of the volatilityμ = drift of the stock
Where m = f ( ρσma,σ , , X )
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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.
• Because of their reliance on repeated computation and random or pseudo-random numbers, Monte Carlo methods are most suited to calculation by a computer.
• Monte Carlo methods tend to be used when it is infeasible or impossible to compute an exact result with a deterministic algorithm
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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
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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
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The EndSpecial 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