The multi-period binomial model

The present time, denoted at n = 0 and the expiration time, denoted at n = N . The lower case letter n, k (and occasionally i, j,m) will be used to denote the time variable. The notation Sk (or Sn, Si, Sj · · · ) denotes the value of the stock (the underlying) at time k (or time n, i, j · · · ). An important convention we’ll use is that S0 will always be a constant, that is the present value of the stock is always known. For any k ≥ 1, Sk is a random variable. The specific distribution of Sk will be discussed below. Similarly, we’ll denote Vk to be the value of a specific financial product at time k. In particular, if V is the European call option with strike K and expiration N , then VN = (SN −K). We’ll also denote πk to be the value of a specific portfolio at time k. In particular, if the portfolio is replicating then VN = πN . Also note that V0, π0 are also constants, and for k ≥ 1, Vk, πk are random variables. We will suppose that the time intervals between any two discrete moments k, k+1 are the same, denoted as ∆T . Thus the expiration time can also be written as T = N∆T . For a replicating portfolio, we will denote the number of shares of S we hold at a particular time as ∆k (do not confuse this with the interval length ∆T . In general, ∆k will also be a random variable (which is easy to understand, as the number of shares we hold at time k will depend on the actual value of Sk at that time). The interest rate will be denoted as r.