The binomial model assumes that stock prices can either go up with probability p or down with probability 1-p at times t=0,1,2...T. These probabilities are all independent of one another and make a probability tree like the one below:
We then wanted to price an option based on each of the outcomes of the probability tree. We say the that payout of an option is equal to: max{S(3) - 100, 0} where S(3) is the price of the stock in time t=3 and 100 is the price at the purchase of the option. We said at first that the price should be a measure of the expectation of the price at time t=3:
In this case, E is the expectation function and R^(-3) is simply the discount term discounting backwards 3 periods. However, this opens us up to a large amount of risk exposure. Therefore we use something called the utility function, which anyone with economics background should be familiar with. In order to explain the utility function fully, the lesson focused on the St. Petersburg Paradox.
St. Petersburg Paradox;
A fair coin is tossed until the first head appears and if the first head appears on the nth toss then you win $2^(n).
The expected payoff of the game is:
However, no one would pay an infinite amount of money to play this game, so we cannot use the expectation function. We must incorporate the utility function. We incorporate a rather simple log(.) utility function such that:
Similar to this explanation, we need to find an appropriate utility function and apply to to the compute the proper option price, which will be shown in later lessons.
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