YOUTUBE: https://wolfr.am/Kinlay-Stochastic-Processes
This lecture series on stochastic processes in Wolfram Language is now available on YouTube.
Below is an example lecture: Geometric Brownian Motion

Contents

◼
  • Basic properties of GBM processes
    • ◼
  • Examples of GBM processes
    • ◼
  • State properties for a GBM process
    • ◼
  • PDF and event probabilities
    • ◼
  • Autocorrelation, stationarity and independence
    • ◼
  • Applications
    • ◼
  • Modeling stocks and indices
    • ◼
  • Derivatives pricing and sensitivities

    • Geometric Brownian Motion

    https://youtu.be/hUFVkS_0yuc
    ◼
  • GeometricBrownianMotionProcess[μ,σ,
    x
    0
    ]
    represents a geometric Brownian motion process with drift μ, volatility σ, and initial value
    x
    0
    • ◼
  • Widely used for modeling stock price processes
    • ◼
  • A continuous time, continuous state model
    • ◼
  • The state satisfies an Ito differential equation: ​​x (t) = μ x(t)t + σ x(t) w (t)​​where w is a standard Wiener process
    • ◼
  • The state x(t) follows a
    LogNormalDistributionwithmeanμ-
    2
    σ
    2
    )t + Log[
    x
    0
    ]
    , and standard deviation
    σ
    t

    • Example

    In[]:=
    data=RandomFunction[GeometricBrownianMotionProcess[.1,0.5,100],{0,1,.01},4]
    In[]:=
    ListLinePlot[data,FillingAxis,ImageSizeLarge]

    Compare Paths for Different Drift Parameter Values
    

    Compare Paths with Different Volatilities
    

    State Properties

    Distribution of the Continuous State

    The state of a GBM process at time t follows a LogNormal Distribution
    In[]:=
    xGeometricBrownianMotionProcess[μ,σ,
    x
    0
    ][t]
    Out[]=
    xLogNormalDistributiontμ-
    2
    σ
    2
    +Log[
    x
    0
    ],
    t
    σ
    Let' s take a look at some examples of the LogNormally distributed price process x(t) at various times t, for a specific set of GBM parameters:
    In[]:=
    Plot[Evaluate@Table[PDF[GeometricBrownianMotionProcess[0,1,1][t],x],{t,{1/2,1,2}}],{x,0,3},FillingAxis,PlotLegends{"t = 1/2","t = 1","t = 2"},ImageSizeLarge]

    Mean

    The state mean grows exponentially over time, with growth rate governed by the process drift:
    In[]:=
    Mean[GeometricBrownianMotionProcess[μ,σ,
    x
    0
    ][t]]//Simplify
    Out[]=
    tμ
    
    x
    0
    In[]:=
    StateMean[t_]:=Mean[GeometricBrownianMotionProcess[μ,σ,
    x
    0
    ][t]]/.
    x
    0
    100//Simplify
    In[]:=
    Plot[{StateMean[t]/.μ0.1,StateMean[t]/.μ0.2,StateMean[t]/.μ0.3},{t,0,20},PlotLabel->Style["Mean",Darker[Red],Bold,14],PlotLabels->{"μ→0.1","μ→0.2","μ→0.3"},FillingAxis,ImageSizeLarge]
    Out[]=

    Standard Deviation

    The state standard deviation is a function of the initial value, time, drift and process volatility:
    In[]:=
    StandardDeviation[GeometricBrownianMotionProcess[μ,σ,
    x
    0
    ][t]]//Simplify
    Out[]=
    2tμ
    
    -1+
    t
    2
    σ
    
    
    2
    x
    0
    In[]:=
    StDev[t_]:=StandardDeviation[GeometricBrownianMotionProcess[μ,σ,
    x
    0
    ][t]]/.{μ0}/.
    x
    0
    100//Simplify
    In[]:=
    Plot[{StDev[t]/.σ0.15,StDev[t]/.σ0.20,StDev[t]/.σ0.25,StDev[t]/.σ0.30},{t,0,10},PlotLabels->{"σ→0.15","σ→0.2","σ→0.25","σ→0.3"},PlotLabel->Style["Standard Deviation",Darker[Red],Bold,14],FillingAxis,ImageSizeLarge]
    Out[]=

    Skewness

    State skewness grows exponentially over time with growth rate governed by the process variance:
    In[]:=
    Skewness[GeometricBrownianMotionProcess[μ,σ,Subscript[x,0]][t]]//Simplify
    Out[]=
    -1+
    t
    2
    σ
    
    2+
    t
    2
    σ
    
    
    In[]:=
    skew[t_]:=Skewness[GeometricBrownianMotionProcess[μ,σ,
    x
    0
    ][t]]//Simplify
    Plot[{skew[t]/.σ0.15,skew[t]/.σ0.20,skew[t]/.σ0.25,skew[t]/.σ0.30},{t,0,10},PlotLabel->Style["Skewness",Darker[Red],Bold,14],PlotLabels->{"σ→0.15","σ→0.2","σ→0.25","σ→0.3"},FillingAxis,ImageSizeLarge]
    Out[]=

    Kurtosis

    State kurtosis grows exponentially over time with growth rate governed by the process variance :
    In[]:=
    Kurtosis[GeometricBrownianMotionProcess[μ,σ,
    x
    0
    ][t]]
    Out[]=
    -3+3
    2t
    2
    σ
    
    +2
    3t
    2
    σ
    
    +
    4t
    2
    σ
    
    In[]:=
    kurt[t_]:=Kurtosis[GeometricBrownianMotionProcess[μ,σ,
    x
    0
    ][t]]
    In[]:=
    Plot[{kurt[t]/.σ0.15,kurt[t]/.σ0.20,kurt[t]/.σ0.25,kurt[t]/.σ0.30},{t,0,10},PlotLabel->Style["Kutosis",Darker[Red],Bold,14],PlotLabels->{"σ→0.15","σ→0.2","σ→0.25","σ→0.3"},FillingAxis,ImageSizeLarge]​​

    State PDF and Probabilities

    PDF

    We can calculate the PDF directly from the underlying process:

    Expectation of an Expression

    Probability of an Event

    Autocorrelation, Stationarity

    Modeling a Stock index

    Download daily data for the S&P 500 Index

    Estimate a GBM process representing the index:

    Simulate future paths for the next 100 days

    Calculate the mean of the simulated paths to find predicted values:

    Plot the sample paths and process mean

    Modeling Stock Returns

    Mean

    Variance

    Distribution of stock returns in GBM process

    Returns as a Wiener process

    Option Pricing and Sensitivity

    Call Option Payoff

    Next we look at how to derive the pricing formula and Greeks for simple option products.
    A call option with strike price k gives the holder of the option to purchase a stock at price k. He will do this providing the stock price at time t, x[t] > k.
    ​
    The payoff from a European call option at expiration is the expected value of Max[x(t)-K, 0], where K is the strike price.
    ​

    Option Expectation

    We can evaluate this expectation directly for a general GBM process :
    Next, we will price a specific call option and check that the pricing formula we derived matches the result given by the Wolfram Language FinancialDerivative function:

    Option Value vs Volatility

    Let’s take a look at how a call option value varies over time, with different levels of annual volatility in the range 10% to 50%.

    Option Sensitivities (“Greeks”)

    In order to derive the formulae for the option sensitivities (“Greeks”), we take the derivative of the option pricing formula with respect to each parameter.

    Delta

    Let’s check the delta for a specific option, using this derived formula:

    Theta

    We can do the same for theta, which measures the rate of decay in the option value :
    The values of delta and theta calculated from the above derivations are very close to the theoretical values:

    Summary

    ◼
  • We showed how to derive option pricing formula directly from the pdf of stock price, the state variable x(t)in a Geometric Brownian Motion Process
    • ◼
  • Likewise, we demonstrated how to calculate the option Greeks, by taking the derivative of the option value w.r.t. the option parameters x, t and σ
    • Geometric Brownian Motion 3D

    Option Greeks

    Call Option Greeks

    CITE THIS NOTEBOOK

    Lecture series on Stochastic Processes​
    by Jonathan Kinlay​
    Wolfram Community, STAFF PICKS, January 12, 2023
    ​https://community.wolfram.com/groups/-/m/t/2777429