Suppose that a stock index portfolio is currently worth $1,000. A bank can offer

Question

Suppose that a stock index portfolio is currently worth $1,000. A bank can offer clients an $1,000 investment opportunity that consists of (1) a 3-year zero coupon bond with a principal of $1,000 and (2) a 3-year European call options which are at-the-money. The option can be bought for the price c, using funds left over after buying the bond. Any remaining amount of funds will be the bank’s profit. Assume a 3-year risk-free rate of 3% with continuous compounding.

a) Find the upper limit for the option price c to make the bank profitable.

b) Suppose that the bank could not find any at-the-money call options to create the $1,000 principal-protected note. Suggest two different approaches to increase the feasibility of the principal-protected note. Briefly explain why each approach will work.

Explanation / Answer

The price of a 3 year zero-coupon bond when discounting at risk-free rate of 3% = Par Value / (1+r)^t

or, 1000/(1.03)^3 = $915.14

(Note: usually a bond issued by a corporate or bank is not discounted at risk-free rate because there is a risk of default, but in this question we do not have the market rate to discount the bond)

After purchasing the bond at $915.14, the remaining funds $84.86 ($1000-$915.14 = $84.86) can be used to invest in a three year European call option on the stock index portfolio, which is currently trading at $1000. The call option must have strike price of $1000 (i.e. at the money). If the stock index portfolio increases in value, the call option may be exercised and the client gains. If the stock index portfolio declines in value, the option expires worthless and at the end of three years, the bond matures to pay $1000.

a) The upper limit for the option price is $84.86. If the price of the option is more than that, the bank loses money. The lower the price of the option, the more the profit.

b) Two different approaches to creating a principal approach:

The first approach depends on the view of whether the stock index portfolio is expected to increase or decrease in value over the next three years. If the portfolio is expected to decrease in value, the same strategy as described above may be used; the only difference being buying an at the money put option, instead of a call option. The bond will mature to give $1000 and if the stock index portfolio declines (say by $100), the put option may be exercised to gain additional $100.

If however, the stock index portfolio is expected to increase, the strategy would be to buy the stock index portfolio at $1000, short the bond at market price (calculated earlier as $915.14) and buying an at the money put option on the stock index portfolio. Lets assume the put option is selling at $10. Then the total investment required will be ($1000 + $10 - $915.14 = $94.86). At the end of three years, if the stock index portfolio increased in value (goes above $1000) it will be sold to bear the $1000 liability of the zero coupon bond and the put option expires. If the stock index portfolio declines in value, the put option is exercised to sell the stock index portfolio at $1000 to meet the liability of the bond and the total loss on the strategy is the purchase price of the option.

The second approach requires buying the zero coupon bond and entering into a swap agreement on the stock index portfolio where the variable returns on the portfolio are swapped by a fixed stream of cash-flow (which should not be more than $84.86 to protect the principal). If the stock index portfolio returns are higher than the fixed payment of the swap, the investor gains, if not, the loss is limited to the fixed rate of cash outflow of the swap and the principal (of $1000) is protected since the zero coupon bond matures to pay $1000.

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