Black-Scholes pricing

Question

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.

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.

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

Answer 1

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 od the call 130 call 4.44. the price of the 100 call is 13.75. The difference is 13.75-4.44 =9.31 So if the buy a call spread on an initial invest of $100. the invest the difference (90.89) at 4% in and yiou will have 94.31. If the The return on the 100 wil be the proceeds from the bond investment plus the gain on the option = 94.31/100-1 = -5.68% plus of to 30%. The Payoof 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 iinvets a smaller amount at 4% which will increase the maximum loss. If you have any question reach out

Lenny