3 Exotic Options: The Log Contract (Matlab codes required) A log contract is an important building block in volatility derivatives and in the calculation of the VIX. As you will see later during this course, the log contract allows you to achieve pure exposure to fluctuations in volatility. The payoff from a log contract at maturity T is simply the natural logarithm of the underlying asset divided by the strike price, In(ST/K). The payoff is thus nonlinear and has many similarities with options. The value of this contract at time t=0 for a non-dividend paying asset is Lo=e*** (11 (*) +(3-7) 7] (1) In this exercise, you will find the price of a log contract using a binomial model. We will consider a log contract that pays LT = 1000 In(ST/K) (which is equivalent with 1000 individual log contracts). Report answers with 4 decimal digits. Use Matlab to write a computer code which takes as inputs: The initial stock price So The payoff function Lt = 1000 In(ST/K) The interest rates The length of the period h The up and down factors u and d The number of periods N and which calculates the log contract price as well as the composition of the replicating portfolio at every node of the tree. a. Apply your code to compute the initial value of a log contract with N = 4, p = 0.05, h = 1, u = erh+0.3Vh, d = erh-0.3Vh, So = 100, 0 = 0.3, and strike K = 90. Show in a table the values of the log contract on all nodes of the tree. What is the replicating portfolio at time 0? What is the difference between the closed-form price (use Eq. (1)) and the initial value of the log contract that you found above? b. Apply your code to compute the initial value of a log contract with N = 48, r= 0.05, h=1/12, u= erh+0.3Vh, d= erh-0.3Vh, So = 100, and strike K = 90. (I have underlined the parameters that are different from the previous point.) What is the replicating portfolio at time 0? What is the difference between the closed-form price (use Eq. (1)) and the initial value of the log contract that you found above? 3 Exotic Options: The Log Contract (Matlab codes required) A log contract is an important building block in volatility derivatives and in the calculation of the VIX. As you will see later during this course, the log contract allows you to achieve pure exposure to fluctuations in volatility. The payoff from a log contract at maturity T is simply the natural logarithm of the underlying asset divided by the strike price, In(ST/K). The payoff is thus nonlinear and has many similarities with options. The value of this contract at time t=0 for a non-dividend paying asset is Lo=e*** (11 (*) +(3-7) 7] (1) In this exercise, you will find the price of a log contract using a binomial model. We will consider a log contract that pays LT = 1000 In(ST/K) (which is equivalent with 1000 individual log contracts). Report answers with 4 decimal digits. Use Matlab to write a computer code which takes as inputs: The initial stock price So The payoff function Lt = 1000 In(ST/K) The interest rates The length of the period h The up and down factors u and d The number of periods N and which calculates the log contract price as well as the composition of the replicating portfolio at every node of the tree. a. Apply your code to compute the initial value of a log contract with N = 4, p = 0.05, h = 1, u = erh+0.3Vh, d = erh-0.3Vh, So = 100, 0 = 0.3, and strike K = 90. Show in a table the values of the log contract on all nodes of the tree. What is the replicating portfolio at time 0? What is the difference between the closed-form price (use Eq. (1)) and the initial value of the log contract that you found above? b. Apply your code to compute the initial value of a log contract with N = 48, r= 0.05, h=1/12, u= erh+0.3Vh, d= erh-0.3Vh, So = 100, and strike K = 90. (I have underlined the parameters that are different from the previous point.) What is the replicating portfolio at time 0? What is the difference between the closed-form price (use Eq. (1)) and the initial value of the log contract that you found above