Tham khảo tài liệu 'financial calculus introduction to financial option valuation_7', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Deriving the parameters 153 Our task is to find V0 the option value at time zero. We may do this by working backwards through the tree. Suppose vn 1 n 10 are known that is we have the option values corresponding to time t ti 1 and all possible asset prices. Then consider the option value vn corresponding to asset price sln at time t ti. Because of our up down assumption about the asset price movement working from right to left the asset price sn comes either from Sn 1 with probability p or from .S n 1 with probability 1 p. Now recall the definition for the expected value of a discrete random variable. The big idea in the binomial method is to multiply the two possible values VÍ 1 and Vi 1 by their associated probabilities to get an expected value. In this way the option value Vi corresponding to asset price Sln is taken to be pV i 1 1 p vn 1 scaled by the appropriate factor that allows for the interest rate r. This gives the fundamental relation vi e rSt pVln 1 1 p vn 1 0 n i 0 i M 1. Once the parameters u d p and M have been chosen the formulas completely specify the binomial method. The recurrence shows how to insert the asset prices in the binomial tree. Having obtained the asset prices at time t tM T gives the corresponding option values at that time. The relation may then be used to step backwards through the tree until V00 the option value at time t t0 0 is computed. Deriving the parameters Since the discrete asset price model in the binomial method fits into the framework of by appealing to Exercise we could tune the parameters by asking for the corresponding Yi to have zero mean and unit variance. This would lead to two constraints. However to give more insight into the workings of the method we will derive those constraints from first principles. Exercise asks you to confirm that the two approaches lead to the same conclusion. As a means to write down an expression for the up down asset price model .
