Forex Talking

The black box portfolio in section 2.5 is very contrived and was used just to illustrate a point. There are, however, real instruments that exhibit some of the same nonlinear type properties. One such example is a convertible bond (CB). CBs are issued by corporations as a way of raising capital. The simplest type of CB pays a fixed coupon per year and can either be redeemed like a straight bond or converted into shares. The holder of the CB has to choose whether to redeem into cash or convert into shares. No additional payments are required with either option. CBs have a fixed maturity date, so the holder eventually has to decide—cash or shares? Consider the following simple example.

A CB (Convertible Bonds), convertible into one share of stock, is about to mature. The redemption value of the CB is $100 and there are no further coupons due.

• Share Price at $110. What is the CB worth if the share price on maturity is $110? Obviously the holder would elect to take the shares and immediately sell these into the market and recoup $110. It would not make sense to redeem for the fixed cash sum of $100.

• Share Price at $120. What is the CB worth if the share price on maturity is $120? Obviously the holder would elect to take the shares and immediately sell these into the market and recoup $120. In fact at any price above $100 it would always make sense to convert into shares.

• Share Price at $90. What is the CB worth if the share price on maturity is $90? In this case converting into shares would not make sense. It would be better to take the cash sum of $100. In fact at any price below $100 it would always make sense to take the cash option.

Table gives the value of the CB on the maturity date for various stock price levels. The price profile is clearly nonlinear. At stock prices above $100, the slope of the profile is one—the instrument behaves exactly like one unit of stock. At stock prices below $100, the slope is zero—the instrument is completely independent and behaves like cash pool of $100.

This is the situation of all CBs on the maturity date. The nonlinearity, or kink at $100, is caused by the simple fact that the holder has the option to do one thing or another. The very fact that the holder can choose, on maturity, between having exposure to stock or cash has implications for the pricing of the instrument before maturity.