Paskalis Glabadanidis and Yufeng Han
1. Portfolio insurance -- fee paid separately.
Set up a portfolio insurance strategy for a 2-period model with u = 1.2, d = 0.7 and r = 1.05. Assume the floor is 100 (which is the same as the initial investment) and the fee is paid separately. Compute the cost of the put and the optimal initial proportion to be invested in the stock.
The artificial (risk-neutral) probabilities are pi_u = (1.05-0.7)/(1.2-0.7) = 0.7 and pi_d = 1 - pi_u = 0.3. The state prices are p_u = 2/3 and pi_d = 2/7 respectively. Stepping back through the tree we arrive at a put price of 10.258503.
Using the hedge ratio (or the replicating portfolio) for the put initially we obtain that its replicating portfolio contains an amount of (4.571428-25.238095)/(1.2-0.7) = -41.3333 in the stock and an amount of (10.258503-(-41.3333)) = 51.5918 in the riskless asset. Given that the initial investment in the underlying is 100 we find that the insured portfolio has (100-41.3333) = 58.6667 in stock and 51.5918 in the riskless asset which gives us an initial proportion of 53.2083% in the stock.
2. Portfolio insurance with a built-in fee.
Set up a portfolio insurance strategy for a 2-period model with u = 1.2, d = 0.7 and r = 1.05. Let the initial investment be 100 and assume the floor is 80. Compute the proportion k of initial investment to allocate to the underlying portfolio, the cost of the put, and the initial optimal proportion to be invested in stocks. (Hint: The parameters are chosen so that the floor kicks in only in the worst state when the stock has gone down twice.)
The risk-neutral probabilities and the state prices are the same as in the previous exercise. However, since the fee is built-in in this case we have to find a value of k such that the cost of the put P(k) and the initial position in the stock (100k) equals the total investment which is 100. Setting up the terminal payoff of the put and discounting back using the state prices we find that P(k) = 4(80-49k)/49. To solve for k we need to solve the following:
4(80-49k)/49 + 100k = 100
which produces a value of k = 0.973639. Hence, 97.3639% of the initial investment should be devoted to the underlying portfolio. To find out the price of the put we just need to plug in the value of k we obtained in P(k) and get
P = 4(80-49*0.973639)/59 = 2.636054
To get the initial optimal proportion to be invested in the stock we again use the hedge ratio of the put. That is how we find that initially the put's replicating portfolio contains an amount of (0-9.226197)/(1.2-0.7) = -18.452394 in the stock and (2.636054+18.452394) = 21.088448 in the riskless asset. Finally, the insured portfolio contains (100 x 0.973639 - 18.452394) = 78.911552 of stock for an optimal proportion of 78.9115%.