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“Now talk about the heston model. The heston model consists of two correlated brownian motion. Motion. So there are two stochastic differential equations in the heston model.
So the one the geometric brownian motion. Which talks about evolution of stock price. And the second is a cir model. Which talks about evolution of volatility.
So you have your data set your data. Set consists of two series. One scene is a stock price. And the second series is volatility so this mu is long term rate of return for the asset.
So this is long term rate of return so this kappa as in a cir model. This is a rate of mean reversion. So theta is our long term variance. So when you say rate of mean reversion.
The mean you re talking about is this theta. So that is the speed at which you will go to theta. Then sigma t. Obviously you have to compute and this eita is nothing but wolof this series on volatility so this kappa is not arbitrary.
So what you essentially try to do is you try to put this fellow condition that is kappa theta has to be greater than 8 a square. So this condition will ensure that your volatility is always positive. And yeah. That is pretty much it so the model here has correlated brownian motion that is this d 1 t.
And dv 2 t. They are both correlated so both correlated means that this t v. I comma t. That is 1 and 2 are both drawn from a multivariate normal distribution.
.
So multum area normal distribution has mean 0. And variance covariance matrix is sigma so sigma is the variance covariance matrix so shown sq decomposition of this variance covariance matrix will give you a times a transpose a sigma and as you know this is important because now you can take a random variable from normal 0 1. So you can take and random variable from normal 0 1 multiply them with this a and you will get a random variable with mean 0. And variance covariance matrix sigma so say you start with variance covariance matrix.
Which is fed in. Right here we ll use it this in the code that is 27 075. So obviously this variance covariance matrix is about two variables v. 1 and v 2 and that is why you have a 2 by 2 matrix.
So. The first column is clear it comes from delta s by s. So that is geometric brownian motion. The se column is a matter of qualitative judgment.
You can take it as delta sigma by sigma or just delta sigma or something else so once you have this matrix you have to do a cheol su decomposition of it so you will construct a new matrix. Which is a cheol su decomposition of it so now let us whatever. We have said let us see this in action so first step is you rewrite this these two set of correlated stochastic differential equations in this model. So left hand side is very easy just comes down like this then you have mu s t.
This comes right here and this term comes right here. Then you have ddt and here. Again. You have sigma t.
Sd square root of sigma t st. And then 8r square root of sigma t. This is just we have copy pasted right here in matrix form the second step is you know that these two are correlated and we want to convert them to normal 0 1. So what you do is you write the covariance matrix takes as tallis q.
Decomposition. So this is the chola skew decomposition. So this matrix. I am writing right here.
.
So cho leske decomposition is always upper triangular and then you can use normal 0 1 so these two are now normal 0 1. So this is normal 0 1. And this is normal 0 1 random variable now you just multiply these two matrices you get this matrix everything. Else is copies down as it is and this now you can feed it into your our and simulate the heston model.
So what we are going to do is we are going to call this variable s 1. For stock price and s. 2. For volatility.
So we are going to solve for s 1. And s. 2. Your drift.
Is mu s. 1. And k. Times.
Theta minus s. 2. So theta minus s. 2.
Because this. You re calling. S. 1.
This we are calling s. 2. Similarly. Here.
.
This will be square root of s 2 square. Root of s 2 square root of s 2. And that s what we are going to write you know the c 1. S.
1 square. Root of s. 2. Then c.
2 s. 1. So. This is s 1 c.
2. S 1. Square root of s 2. S 2 to the power of 05.
Then this 0 is coming here and then c 3 theta. The square root of s 2. So that is your model and then it s just a matter of setting. The model up your drift is drift.
So this is our drift. Which we set here this is our drift. This is a word so this is our drift. This is your diffusion matrix so that is what you have you re gonna solve for s 1.
And s 2. And initial values. So off say s 1. Your stock price is 50 and initial value of your wall is 5.
.
And then you are going to just simulate it i m just giving some parameters here obviously parameters you have to have theta you have to give a tire you have to give mu. I m giving his one kappa is 2 and then there s c1. I said you know this comes from the cholan scheme matrix so chill s key matrix. This is the chola scheme matrix of c1 is the first row first column.
So this is 1 1. So that is what i have read c1 is this joe liske. Decomposed 1 1 c. 2.
Is cho leske decomposed first row second column so 1 2. Then you have just have 0. But c. 3.
Is second row second column. So this is 2 2. And that s why you have 2 2. Here.
So you just put this into our and run. It and let us see the results. So you just select here. Everything you have and hit the run button.
And you get the stock price on the top starting at 50 and volatility starting at 5. So you get the evolution. So you can do as many simulations as you like for the heston. ” .
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