Suppose you have written a derivative that pays the squared value of the stock price at maturity T=1; that is, it pays S2(1). The stock currently trades at S(0)=100. Your model is a single period binomial tree with up value for the stock equal to 102 and the down value equal to 98. One dollar deposited at time zero into the risk-free asset returns 1.01 dollars at maturity.
Enter the number of shares that the replicating portfolio holds (if selling short, use the minus sign):
correct
200
>>Enter the cost of the replicating portfolio:
correct
10102.97
I have an update, could you help me?
Suppose you have written a derivative that pays the squared value of the stock price at maturity T=1; that is, it pays S3(1). The stock currently trades at S(0)=100. Your model is a single period binomial tree with up value for the stock equal to 102 and the down value equal to 98. One dollar deposited at time zero into the risk-free asset returns 1.01 dollars at maturity.
Enter the number of shares that the replicating portfolio holds (if selling short, use the minus sign):
>>Enter the cost of the replicating portfolio:
To determine the number of shares in the replicating portfolio, we need to find the hedge ratio, which is the ratio of the change in option price to the change in stock price.
In a binomial tree model, the up and down factors are given as 102 and 98, respectively.
To calculate the hedge ratio, we divide the change in option price by the change in stock price. Since the derivative pays the squared value of the stock price, the change in option price is (102^2 - 98^2) = 416.
The change in stock price is 102 - 98 = 4.
Therefore, the hedge ratio is 416/4 = 104.
Since the stock currently trades at S(0) = 100, the number of shares in the replicating portfolio is 104 * 100 = 10400.
However, the question asks for the number of shares that the replicating portfolio holds, and if selling short, we need to use a negative sign.
Therefore, the correct answer is -10400.
To find the number of shares that the replicating portfolio holds, we need to use the principle of risk-neutral pricing in the binomial model.
The replicating portfolio consists of a combination of the stock and the risk-free asset such that it replicates the payoff of the derivative at maturity. In this case, the derivative pays S^2(1), where S(1) is the stock price at maturity.
To find the replicating portfolio, we first calculate the stock price at maturity in both the up and down states. The up state stock price is 100 * 102/100 = 102, and the down state stock price is 100 * 98/100 = 98.
Next, we calculate the probabilities of the stock moving up and down. The risk-neutral probabilities are calculated as (1 + risk-free rate - down value) / (up value - down value). In this case, the risk-free rate is 1.01, the up value is 102, and the down value is 98. So the risk-neutral probability of an up move is (1 + 1.01 - 98) / (102 - 98) = 0.505.
Now, we calculate the number of shares needed in the replicating portfolio by taking the difference in the derivative's payoff in the up and down states and dividing it by the difference in the stock prices in the up and down states. In this case, the difference in derivative's payoff is (102^2 - 98^2) = 400, and the difference in stock prices is 102 - 98 = 4. So the number of shares in the replicating portfolio is 400 / 4 = 100.
Since we are given that the derivative pays the squared value of the stock price, the number of shares in the replicating portfolio is 100. However, if the derivative had a different payoff, we would need to adjust the calculation accordingly.
Now that we have the number of shares in the replicating portfolio, we can calculate the cost of the portfolio. The cost of the portfolio is given by the number of shares multiplied by the stock price at time zero (P(0)). In this case, the number of shares is 100 and the stock price at time zero is 100. So the cost of the replicating portfolio is 100 * 100 = 10,000.
However, since we are told to use the minus sign for selling short, the replicating portfolio holds -100 shares. Therefore, the cost of the portfolio is -100 * 100 = -10,000.
In this case, the answer to the question is:
Number of shares in the replicating portfolio: -100 (selling short)
Cost of the replicating portfolio: -10,000