Question 5 (Total 19 marks): Suppose the current price of a stock is $40, and over the next six months, it will either increase by 10% or decrease by 10% in each 3-month period, as shown in the figure of the two-step binomial tree below, in which the nodes are marked by letters A, B, ..., F. Assume that the risk-free interest rate is 9% p.a. with continuous compounding, and that the required rate of return on stock of this level of risk is 18% p.a. with continuous compounding. 48.4 D 44 B 40 39.6 E A 36 32.4 + F a) If you make the assumption that this is a risk-neutral world, regardless whether this is the right assumption or not, calculate the probability of the stock price going up at each step, with explanation why it is such a probability in the risk-neutral world. (5 marks) b) Explain the meaning of risk-neutral principle for derivative valuation. And then use this principle to calculate the value of a 6-month European call option on this stock with strike price of $42. Then, draw the figure of the above binomial tree with the value of this call option at each node written in the figure. (7 marks) c) Calculate the value of a 6-month American call option on this stock with strike price of $42. Specify the option value at each node of the tree. (4 marks) d) What is the true probability of the stock price going up in each step on the binomial tree. Show your calculations. (3 marks) Question 5 (Total 19 marks): Suppose the current price of a stock is $40, and over the next six months, it will either increase by 10% or decrease by 10% in each 3-month period, as shown in the figure of the two-step binomial tree below, in which the nodes are marked by letters A, B, ..., F. Assume that the risk-free interest rate is 9% p.a. with continuous compounding, and that the required rate of return on stock of this level of risk is 18% p.a. with continuous compounding. 48.4 D 44 B 40 39.6 E A 36 32.4 + F a) If you make the assumption that this is a risk-neutral world, regardless whether this is the right assumption or not, calculate the probability of the stock price going up at each step, with explanation why it is such a probability in the risk-neutral world. (5 marks) b) Explain the meaning of risk-neutral principle for derivative valuation. And then use this principle to calculate the value of a 6-month European call option on this stock with strike price of $42. Then, draw the figure of the above binomial tree with the value of this call option at each node written in the figure. (7 marks) c) Calculate the value of a 6-month American call option on this stock with strike price of $42. Specify the option value at each node of the tree. (4 marks) d) What is the true probability of the stock price going up in each step on the binomial tree. Show your calculations
