 The two-state binomial model has some deficiency in the sense that it assumes only two values taken by any stock or an asset on a given day, whereas in reality any asset or stock may take any value over the two values on any given day. Now to correct this deficiency, we can add more intervals to the two-state model or to the two-date binomial model. Now let's take an example of three-date binomial model. We have an example where on September 1, we assume that sale price per gallon of an oil is $2 and we assume that oil prices vary from September 1 till December 1. We have a large buyer that offers the seller buying on December 1 6 million gallons of oil at a price of 2.10 per gallon. Now if we see the price movement of the heating oil per gallon over the two months of one and half month interval each, we see that there is an upstate price of $2.5 per gallon on October 15 which gives a return of 25% to the seller and on the other side there is a downstate price of $1.60 on October 15 which is giving a negative return of 20% to the seller. We also assume that same variation prevails from October 1 to December 1. Now let's see what will be the price per gallon on December 1 with reference to the price of $2.50 per gallon. On upstate side there is the price of $3.12 per gallon with the increment of 25% and on the downstate side there is the price of $2 per gallon with the decrease return of 20%. On December 1 with reference to the price per gallon of $1.60 the new price will be as an upstate price of $2 with the increment of 25% and a downstate price of $1.28 with the negative return of 20%. Now there are two types of possibilities of prices. The one type is that there are two possible prices on October 15 and the second class is that there are three possibilities of prices on December 1st. There are two paths to reach a price of $2 on December 1, two paths are there. The first path is that the price could rise to $2.5 on October 15 before it falls down to $2 on December 1st and the second path is that the price could fall to $1.60 on October 15 before it falls to before it goes up to $2 on December 15. Now using this data we can value this contract for the seller in three steps. These three steps are at first we need to determine the risk neutral probabilities and then we need to determine the value of the option on October 15 and then we need to value the option on October 1st. Let's first determine the risk neutral probabilities. We determine the probability of a price rise while taking the required rate of return equal to the risk less interest rate and here we assume 8% interest rate which implies the 1% required rate of return over the next one and half month's interval. Using the unknown values of price of probability of rise in the equation we determine the probability of rise as to 47%. This means that the probability of fall is now equal to 53%. So at the probability of rise of 47% the required rate of return on the oil is 1%. It is notable that these probabilities are consistent with the world of the risk neutrality that assume that a required rate of return is equal to the risk less rate of interest. Now at the probabilities of 47% and 53% holding for both the period we have an upstate return of 25% from September 1 to October 15 and using these two probabilities we have downstate return a negative return of 20% from October 15 to December 1. In the second step we need to determine the value of the option on October 15. The options worth for the seller on December 1 will be equal to 1.02 dollars and given the value of sale value of 3.12 and purchase value of 2.10 for the seller and given the value of 1.28 the option will worthless for the seller because the buyer will not buy at the promised price of 2.10 rather he will go in the open market and will buy the oil at the price of either 2 or 1.28 dollars per gallon. So the option is here out of money for the seller. Now the value of call option on October 1st 15 we can compute let see the price per gallon of 2.50 the value will be equal to 0.474 dollars per gallon that value we have computed using the respective probabilities of rise and fall with the corresponding returns and at the price of 1.60 per gallon the value of the option is equal to 0. So what we can conclude here we can conclude that the call must be ending up out of money on December 1 if the oil price is 1.60 per dollar on October 15 and it must have a zero value on October 15 if the oil price is again 1.60 on that given date. Finally we need to determine the value of the options as of September 1st let see if we use the call values of 0.474 and 0 on October 15 then the call value of September will be equal to 0.220 per gallon and the value of the contract now for the seller with respect to because that the value of the contract for the seller will be that he will be receiving only 1 million dollar as the upfront amount against the sale of 6 million gallons of oil to the buyer so the worth will be equal to negative 320,000 and that is the negative value for the seller in this contract so far as the buyer is concerned let's see in fact the value of the contract for the seller we need to calculate first the seller is giving away the option at which is worth 0.220 per gallon and that is a 0.220 dollar per gallon of oil and in this given up he is receiving the 1 million dollar of upfront and overall his loss is equal to 320,000 dollars as we have seen earlier so far as the buyer is concerned the sale the loss of the seller is the gain of the buyer this means that the loss of 320,000 that has been earned by the or the suffered by the seller is basically the gain of the buyer so for the buyer the worth of this call is equal to 320,000 dollars