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Stochastic Differential Equations
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The purpose:
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We begin our discussion with a simple observations about Stock Market daily returns:
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The returns appear to be normally distributed.
But is this actually true? Of course not! However, we can pretend that it is Normal as a first approximation. If you'd like to explore some stock market data, feel free to look at the python code: "Stock_simulations.py" at my github page: Ricard0000. Below, I provide you with a plot of the returns of Apple "AAPL", the claim is that this is approximately normally distributed.
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The astute reader will likely notice the following about the above distribution:
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The distribution of returns appears to have higher peaks when compared to the Normal distribution.
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The tails are "fatter" than that of the exponentially decaying Normal distribution.
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Assuming the asset price has normally distributed randomness we can formulate the simple model:
This model can be implemented using a few lines of code on Python:
S=np.zeros([N],dtype=float)
S[0]=start
for I in range(0,N-1):
S[I+1]=S[I]*(1+mu*dt+np.sqrt(dt)*std*np.random.normal())
This is how we generate the plots below:
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We can even use this to for forecasting. Below is an example of the Wiener process using AAPL closing values.
