# Kangaroo inc is a U.S. company

Kangaroo inc is a U.S. company whose shares are listed on and freely traded on the New York stock exchange. Let Stbe the price of Kangaroo inc shares in dollars, at time t (measured in years). Time zero is today.

You have just been appointed by Barclays Capital to be their new Global Head of Options trading. It is your first day on the job and you are keen to show senior management at Barclays Capital that they made the right decision to hire you and to show to Wall Street that there is a new kid on the block!

You are telephoned by Sally, a fund manager, at Tiger Capital (a hedge fund). Sally tells you that Tiger Capital today wants to buy a security (called a SQUARED DIFFERENCE contract), linked to the share price of Kangaroo inc. The SQUARED DIFFERENCE security has the following features:

It has a maturity of one year.

At maturity, the SQUARED DIFFERENCE security pays an amount in dollars equal to the amount

S12S02S02 . equation (*)

Here, S1 and S0 are, respectively, the Kangaroo inc share price one year from now and the share price today.

Assume that the risk-free interest rate is zero per cent and that Kangaroo inc shares pay no dividends. Assume that the share price, in dollars, today is S0=1.

Using Excel, build a four-step binomial tree (this means each time step corresponds to three months). (Hint: It is the same idea as we did in classes and assignments but whereas, before, we had one, two or three steps, now you will have four binomial steps).

Assume the absence of arbitrage throughout and assume that there are no transactions costs.

1. Assume (to begin with) that the volatility of Kangaroo inc shares is 1%. Using your binomial tree, what is the price today of this SQUARED DIFFERENCE security? (4 marks)
2. Still assuming the volatility of Kangaroo inc shares is 1%, and using the binomial tree, what is the delta hedge at each step. To answer this, do a screen-shot (Control-C then Control V on a pc) of the delta hedges. Do you see a pattern in the delta hedges? What is it? (3 marks)
3. Now assume instead that the volatility of Kangaroo inc shares is 2%. What is the price today of this SQUARED DIFFERENCE security? (1 mark)
4. Now assume instead that the volatility of Kangaroo inc shares is 3%, then 4%, then 5%, then finally 10% (skip 6, 7, 8 and 9% – you will (hopefully) already see a pattern emerging). For each case, what is the price today of this SQUARED DIFFERENCE security? (Hint: If you do this in excel in an efficient manner, this can be done very rapidly). (2 marks)

Give your answers to parts (a), (c) and (d) (in dollars) by filling out the table below (replacing x.yyyyyyy) giving every answer to 7 decimal places (you will see that the answers are quite small so that is why I am asking for 7 decimal places but this is no hassle – decimal places are “free” in excel since excel allows you to format up to 14 decimal places – Google this formatting feature if you have not seen it before):