 In this section, I'll explain a very famous Black Shoals model. So, we were talking about options. The prices of options, whether it's a call option or a put option, to determine its prices, in many areas, in many organizations, the model used, that is, like of all the different types of models that are used, one very prominent model is, or frequently used, or extensively used model is the Black Shoals model. Black Shoals model, we call Black Shoals Merton model, or it is called BSM model. So, it is important to know that Black Shoals Merton is given the Nobel Prize for presenting this particular model. So, this is the idea of this particular model. So, basically, what is this? With the help of a differential equation, we help to determine the price of the call or put option. So, it helps us in determining the price of the call or the put options. And the concept used in this is of differential equation. So, differential equation also means that a question in which there are derivatives, mathematical derivatives, they are included, we call this equation differential equation. Now, Black Shoals model, in its formula, we take the values of the five variables, and with the help of that, we can determine the price of the option. Now, what are the five things? First of all, we need to know the current stock price. What is the price of the stock at this time? Apart from that, we have to find out that how much time is left on the expiration. Next is, what is the risk-free rate? And the fourth important thing is, that is what we talked about volatility, that we have historical volatility and implied volatility. You should know both. Now, Black Shoals model is based on some assumptions. And these assumptions help us in explaining that why prices are different from the results of the real world. The prices it has defined, and when we look at the market, if you see the difference, then Black Shoals model tells, or Black Shoals-Merton model tells that if you follow these assumptions, then according to my formula, the prices are the same. But if those assumptions are violated, then the price you will get, that would be different from the price that has been calculated using the Black Shoals-Merton model. The standard BSA model is only used to price the European options. In the European option, there is a requirement that the maturity rate can be exercised. It cannot be done before that. So, in the BSA model, only the European style options can be calculated. Now, there are several assumptions based on which the BSA model is based. The first assumption is that no dividends are paid out due to the life of the option. So, the stocks you have defined and the contracts you have made, we have assumed that in the BSA model, no dividends are paid out. The companies whose stocks or shares are not paid out, this particular assumption has been applied to the analysis. So, if the dividend is paid out, then the BSA model cannot be paid due to the life of the option. If it is American style, then the model cannot be paid due to the life of the option. So, these are the assumptions which we need to see whether a certain option is fulfilling the assumptions of the BSA model only. Then you will be able to determine the price of that option using this model. Assumption number 2 says that the markets in which you are going to purchase and sell options, there is no transaction cost involved in that. What is the transaction cost? We have already discussed it. I will briefly tell you. The transaction cost is the amount of money that you are going to spend on the BSA model. So, the BSA model is the amount of money that you are going to spend on the BSA model. So, the transaction cost is the amount of money or expenditure that you incur. You have learnt information about investing in any different kind of financial instrument. You have made a payment for that. That will be our transaction cost or you have paid commissions or agents that you are going to buy. Other than that, the extra expenses that you pay are the amount to be and sum up as the transaction cost. So, we are taking another assumption of the BSA model that the options that we are talking about are not involved in buying the option. Next assumption is that the risk-free rate and the volatility of the underlying asset we already know are constant. Next assumption is the returns on the underlying asset are log-normally distributed. So, if you plot the returns then you will come up with a log-normal distribution of the returns. So, we know that if we plot the data of the returns then it can take up any distribution and it is not necessary that you will not get a log-normal distribution. The last assumption is the option is European and can only be exercised at expiration. If we look at the formula the call price given by capital C will be calculated by using this formula which is where we have we can see that C is equal to ST rest of the power minus RT and D2 Now, this D1 and D2 are written for this formula they have explained. Now, S is the spot price so, I have used all the rotations that I will explain to you. So, we have got D1 equal to these values and we have got the D2 for which the formula is written here. Now, I will explain that the black-shells call-option formula we multiply the stock price by the cumulative standard normal probability distribution function and what is that? So, what you have to do is to determine the call-option price you multiply the spot price standard cumulative standard normal probability distribution function and after that this section again, we are talking about we have assumed that the returns they will be following the log normal distribution so, next we next we calculate the net present value NPV of the strike price multiplied by the cumulative standard normal distribution and you subtract this particular value resulting value of the previous calculation so, what we have done we have taken the net present value of strike price and we have again we are assuming that it has standard normal distribution so, you have multiplied this by multiplying this particular value which is stock price multiplied by the cumulative standard normal probability distribution with this we have multiplied these two terms and then you have to subtract this term from the first term so, the value which you will get that will be the call price of that option so, the call price you can determine the formula in this way if you know then you can calculate the put price with the formula of black souls now, the symbols which I have used I have already explained the C I will explain it further capital C is the call option price capital S is the stock price the current stock price is capital K is the strike price and the R that is the risk free interest rate small t is the time to maturity it could be in days, it could be in weeks it could be in months or capital N that is specifying the normal distribution now, if we look at the formula you can see this R is your interest free rate and this T is the time which is remaining to maturity and here we said D2 is normally distributed then we said D1 is normally distributed both the formula you have to multiply this particular thing with the spot price with the current spot price and this is your strike price you have multiplied this risk free rate of interest multiplied by time is left to maturity and again we have multiplied by D2 this capital N is normally distributed and how will D2 be calculated that is obtained by using this particular thing I have told you about volatility this is sigma S that is an indicator or a mayor of volatility so together you are going to find out these first you are going to find out the values of these 5 numbers, 5 variables or 5 factors in this particular formula we will plug find out the call option price in this way you can calculate the put option price