Recently, in a discussion with the Head of Structuring for equity products of a bank here we came upon this product idea. The product is not new, but the context is. The product is quite appealing though it may be difficult to find a seller of such a product.

The structured note contains an embedded capped call option that is American in nature. The call lives on for an indefinite period of time which means that there is no maturity of the option. However, there is a barrier, a cap such that if the barrier is hit the call option is exercised (terminated). However, neither the buyer nor the seller of the option (structured note) can exercise the option voluntarily. The call will be exercised only when the barrier is it. If the barrier is not hit then the option goes on (lives on) indefinitely.

The structure of the note is something like this: the buyer of the note (structured product) gets a fixed coupon of say, 5% per annum, payable semi-annually as long as the spot (the underlying stock index) trades below the cap. As soon as the cap is hit, the structure terminates with a payoff from the capped call as the difference between the cap and the strike.

If
*
K
i*s the cap and X is the strike price, such that
*
K > X* and the current spot is below the cap level, i.e.
*
K < X* then the capped call pays
*
K - X* and terminates as soon as
*
S,** *hits *
K*. And the option has no maturity, i.e. it can be alive for as long as the spot remains below the cap. The note payoff will look like:

Of course, upon exercise, all accrued interest will also be paid to the buyer of the note. Note that in the above payoff there is no maturity.

**Investment Context**

Even though the idea is appealing, it may be difficult to find a seller (bank) who would be interested in selling such a structure to an investor.

Say, an investor holds a portfolio of stocks that mimics an equity index, say the S&P500. Given the recent sell off in the equity markets, he is under water and losing a lot of money on the portfolio. However, he is a long term investor and does not want to liquidate his portfolio. Therefore, he buys the above perpetual "capped" call note from a bank with a cap level that well above his breakeven level. For example, if he holds (long) the index (or the indexed portfolio) at 100 and the index is currently at 70 due to a massive sell off then he puts the cap at 120. The strike price could be 70. Therefore, as long as the market (index) is below the cap level he continues to get a fixed coupon from the bank, say 5% per annum payable semi-annually. This will compensate in some way the (paper) loss that he is sitting on because of a sell off. He may also have financing costs due to margin calls, etc. which can also be compensated by this fixed coupon payoff. It depends on the pricing and what is the appetite of the seller (the bank) but the fixed coupon can be structured to be as high as possible to benefit the buyer.

As soon as the market rallies back up and goes beyond his breakeven level he is back in the positive territory. And if the rally is good enough then the cap will be hit and the option will be terminated with a payoff to the buyer of 120 - 70 = 50. At this point the investor (buyer) no longer needs any protection because his portfolio or the long index position will be in the positive territory.

The product may be too good to be true, but indefinitely lived American style options are not without practical use. There could be many ways in which the above product can be made more realistic from both the buyer and the seller's point of view and the pricing can reflect that.

**Pricing of the Product**

The pricing of this product is actually quite simple. There exists a neat closed form solution for the price of the capped American call with no maturity. When there is no maturity, i.e. fixed termination date for an option, and the underlying - the index, in this case - follows a geometric Brownian motion with time-invariant volatility, and when the spot rate and the dividend yield are also constant through time the value of the derivative also becomes time-invariant*. And very interestingly the fundamental partial differential equation that describes the dynamics of the option becomes an ordinary differential equation.

Ordinary differential equations are very easy to solve. The ordinary differential equation for the above capped call option with no maturity is:

Where
*C*_{A} is the indefinitely lived American call option,
S_{t} is the instantaneous spot rate,
r ,q and
s are interest rate, dividend yield and volatility respectively. Since the call option has a cap on the upside the upper boundary condition of the differential equation (1) becomes:

Where, is the strike price. Lower boundary condition of course remains . The solution of equation (1) is trivial and it reduces to a quadratic equation with two roots.

The price of the American style capped call option with no maturity is then given by**:

Where, u is one of the roots of the solution of (1) and is a function of the volatility, interest rate and the dividend yield of the underlying.

Of course, upon exercise, all accrued interest will also be paid to the buyer of the note. Note that in the above payoff there is no maturity.

** *See Pricing Derivatives Securities, 2nd Edition, by T W Epps (World Scientific Publishing)
*

** We have kept the same notation as T W Epps.

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