 Apart from using the black school model for valuing options, there is another approach that can also be used in place of the black school model and that approach is termed as the two-state or the binomial model. This approach is an alternative to the black school model and is also considered as a superior approach to the black school model. To see the working of two-state or the binomial model, let's see an example. We have an example of two-date period where we have an organization that is heating oil. This is the firm that trades in heating oil. It buys at a wholesale rate and sells at the retailing rate to the customers. Its large revenue comes from the winter sales. On September 1, we assume that the sale price per gallon is $2 and it is assumed that the oil prices vary from September 1 to December 1. The owner of this firm believes that the oil prices will either be $2.74 or $1.6 per gallon on December 1. We see that there is a wide price range that creates an uncertainty at the greater level for the firm. But this greater uncertainty is not much risky for the owner because as the price changes, it will be duly passed on the customers who buy oil from heating oil. In the sense that the honor will be charging more if he ends up in buying at $2.74 gallon then ends up buying at $1.846 per gallon. He can do so easily because individual customers have no negotiation power. But let's consider the case of a larger buyer who has the large negotiation power offering buying on December 1 at $6 million gallons of oil at a price of $2.10 per gallon. An implication of this deal is that at $2.74 per gallon there will be a loss for the seller as he needs to buy at $2.74 whereas he will be selling at $2.10 per gallon. But at $1.46 per gallon the buyer will not buy at the promised rate of $2.10 per gallon because he will be willing to buy at the open market rate of $1.46 in the open market. So the buyer now wants a call option on oil. He in fact agrees to compensate the seller for the loss by paying rupees 1 million as an upfront money for the right of the call. So by $6 million gallons at a rate of $2.10 per gallon now the question is that is it a fair deal for any of the side? The answer is that it can be evaluated quantitatively by using the binomial model assuming the risk neutral pricing approach. We see that change in oil price from number one to the number one give two scenarios at $2.74 it will give a gain of 37% upstate rise and at a price of $1.46 per gallon it will give a downstate loss of 27% negative return. This means we can now consider the 27% and negative 37% as the possible returns on these two trades and these can be noted as mu which is equal to 1.37 and d which is equal to 0.73. So we can value this contract in two steps. At first we need to determine the risk neutral probabilities and then with these risk neutral probabilities we can determine the value of the whole contract. Let's see first determine the risk neutral probabilities while taking the required return on the oil being equal to the risk less rate of interest we assume that at 8% is the rate of interest which implies the 2% return over the next three months period. This means that if we develop an equation to determine the probability of rise with the help of this equation we can determine the probability of rise equal to 47% which other means that the probability of fall is equal to 55%. So at the probability of rise of 45% there is the expected return on this oil is equal to 2%. So these probabilities are consistent with the risk neutrality assumption because the risk required rate of return on any asset is assumed to be equal with the risk less rate of interest and it is so because no investor can demand for the required rate of return in excess of the risk less rate of return and that is the reason that the risk neutral individual will not need to be compensated for the loss he bears. Now let's compute the value of the contract to see that on December 1st what will be the gain or loss on the trade of the oil and if the sale price is 2.10 gallon then the seller will be bearing a loss of 0.64 dollars per gallon and if the sale price is 1.46 dollar the buyer will go out of the game he will not buy the oil at this price at the price of 2.10 which he has promised because in the open market he will be willing to buy oil at the market price of 1.46 dollars. So the seller will only be receiving 1 million dollar being the upfront money. Now using these probabilities we can determine the value of the contract which is equal to minus 694 thousand dollars and that is the negative value of this contract which is termed as a negative NPV of the contract and due to this the seller will reject to sell the oil. The seller in fact has sold the call option to the buyer and the overall value of this contract is equal to negative 1.69 million. This is negative for the seller and in contrary for the buyer the figure is positive. If you see that the negative number is because this equation is for the sellers and for the buyer the equation is positive which is plus 1.694 million dollars and if we determine the per gallon value of the call this is equal to 0.282 so this is the value that is in fact we have computed using the risk neutral pricing and that is the value of the call options which is 0.282 dollars per gallon.