A building society issues a one-year bond that entitles the holder to the return on a weighted-average share index (ABC500) up to a maximum level of 30% growth over the year. The bond has a guaranteed minimum level of return so that investors will receive at least x% of their initial investment back. Investors cannot redeem their bonds prior to the end of the year.

i.      Explain how the building society can use a combination of call and put options to prevent making a loss on these bonds. [4 marks]

ii.    The volatility of the ABC500 index is 30% pa and the continuously compounded risk-free rate of return is 4% pa . Assuming no dividends, use the Black-Scholes pricing formulae to determine the value of x (to the nearest 1%) that the building society should choose to make neither a profit nor a loss. [6 marks]

 

i. The bond provides a guaranteed minimum level of return (i.e., the investor's loss is capped, irrespective of how badly the index performed over the year). The investor would need to be extended a put as this would give them the right to sell the bond at the strike (x% of initial investment). If they were short a put option (as you suggest), the investor would be obligated to buy (and not sell) the index at the put option's strike price

ii. The Payoff will range between -5.68% and 24.32%. If you want the maximum gain to be 30%, you will have to find a higher strike to sell, pay more net premium, and invests a smaller amount at 4%, which will increase the maximum loss. 

Step-by-step explanation

Let The Index be Normalized to 100 So the ATM call strike is 100 and the Call 30% out of the Money I struck at 130. The price of the call 130 call 4.44. the price of the 100 calls is 13.75. The difference is 13.75-4.44 =9.31, So if they buy a call spread on an initial investment of $100. the invest the difference (90.89) at 4% in, and you will have 94.31 if the return on the 100 will be the proceeds from the bond investment plus the gain on the option = 94.31/100-1 = -5.68% plus of to 30%. 

 The investor earns a return up to a maximum of 30% growth over the year. Therefore, they would need to purchase/be long the index between the put option's strike price and the strike those correspondents to the maximum 30% index growth rate. This would be achieved by investing the initial investment into the index directly.

 The investor does not participate in the growth above 30% over the year. A short call option obligates the call writer to sell stock at the strike price. If they were long a call (as you suggest), the investor would have the option to buy (and not sell) the index. Combining a long put (at stike K1) with the underlying (index) and a short call (at strike K2) is called a collar. The building society eliminates its risk of loss by replicating its exposure/payoff profile of the investor. It makes money via the spread