Barrier Notes: Payoff Design, Volatility Exposure, and Issuer Funding

# Why do some principal-protected notes reset the gains to zero?


# Why do some principal-protected notes reset the gains to zero?












I was looking through the principal-protected notes issued by Lehman Brothers. One of them was the "100% Principal Protection Absolute Return Barrier Notes Linked to the S&P 500 Index". The description says:

> If the Index never closes 24.00% to 27.00% (to be determined on the Trade Date and to which percentage we refer to as the “Absolute Return Barrier”), above or below the Index Starting Level on any trading day during the Observation Period at maturity you will receive your principal plus a return equal to the absolute value of the Index return. Otherwise, at maturity you will receive only your principal.

Why in the world are the returns reset to zero? If the index performed well, say +25%, the investor in this scheme would receive a return of 0%. What is the reason for issuing such notes? Why was it necessary to impose such onerous conditions?

## Answer by Daneel Olivaw (score 3)

https://quant.stackexchange.com/a/53129

The payoff: you are missing one point. It's not only about the upside, it's also about the downside. Your performance $P_T$ at maturity $T$ (the Final Valuation Date) is basically: $$P_T=P_0\left(1+1_{\left\{\max_{0\leq t\leq T}\left|\frac{I_t}{I_0}-1\right|\leq B\right\}}\left|\frac{I_T}{I_0}-1\right|\right)$$ where $P_0$ is your Principal Amount (your initial investment), $I_t$ is the Index value and $B$ the Absolute Return Barrier. Hence, taking the values from the scenario analysis in your link, if the absolute return of the index over the period is never higher than 25.5%, then you will receive the absolute return at maturity as a payoff. Otherwise you only receive your principal.

Note that the actual barrier level $B$ will be definitely fixed closer to the effective trade date: this is to account for potential market changes between the date the product starts to be marketed and the date the deal is actually traded. The termsheet only states that $B$ will be between 24% and 27%.

What's the point for you? The point of this Equity Note is that if offers you capital protection (your investment is guaranteed) in exchange for exposure to some performance. This is actually a bet on market volatility remaining subdued over the period: if the market does not move much, then you will receive the absolute return, independently on whether the market went up or down. And if you are wrong, at least you get your money back.

What's the point for the bank? For the bank structuring these products, these notes usually work as a source of funding. Let us say the bank can fund itself in the market with \$92.5 now in exchange for paying \$100 at future time $T$ (so a 8.1% rate). Now, he can instead do the following:

- He observes he can trade options in the market, at a total cost of \$5, which pay the absolute return of the S&P500 over the period $[0,T]$ if the index's absolute performance has remained below $B$, and 0 otherwise;

- He has a client who is willing to pay \$100 today in exchange for recovering his investment at expiry $T$, plus the absolute performance of the S&P500 provided it has remained below $B$ over the period.

As you can see, the bank has secured cheaper financing: you paid him \$100 today, he only needs \$5 to get you the exposure, so he's actually getting \$95 of funding, namely a 5.3% rate over the period. He has therefore secured a cheaper rate of financing than the 8.1% he could get in the market. Plus he has probably managed to fit in some fees when selling the product (paid by the client of course).

## Answer by Martin Vesely (score 0)

https://quant.stackexchange.com/a/53128

It seems that issuer of these notes took a position in options and it has gains only when the value of index is between 24 % and 27 %.

Issued notes are probably used for funding some investment hedged by the options. The issuer is willing to pass part of its gains to a investor only in case the issuer has some gain.

It is hard to explain details if we do not have enough information about the issuer portfolio and its strategy.

## Answer by will (score 0)

https://quant.stackexchange.com/a/57854

This is a Structured Product. Desks issuing these products make money by selling the product for X when it is really worth X-Y (where X is usually 100%, and Y will be some number of the order 1%, depending on the complexity and maturity of the product).

In the case you reference, you can see from the link that Y was 1.75%. This means that, unhedged, the expected profit from selling this product is 1.75%. The reality is thata they will have (to some degree) hedged the product, which comes along with it hedging costs. For this product, a static hedge is unlikely, so it will be a dynamic hedging cost, where there can be an expected cost of dynamically hedging the product (where they would expect that cost to be lower than 1.75%).

The whole point of these desks is that they sell structured products, and then hedge the portfolio of sold products at a cost lower than the profit taken up front on the products.

So the next question, is where do the products they sell come from? There are a few ways it works:

- Structures create products which look attractive in various market scenarios.

- You have a shelf of preexisting products available to trade on exchange.

- End clients make specific requests for products.

if we just think about point 1, then the question is how are they created. Well it's not too complicated really, the first step is to notice that there is the notional - i.e. you pay 100% for the product, and you get back at the end that 100% + some payoff. The point here is that 100% in (in the case of the note above) 18 months is not worth 100% due to the time value of money. US rates in 2007 were around 5% p.a., so 100% at maturity would have been worth ~92.5% at inception - but you're paying 100% for it, which means that there is 7.5% - 1.75% = 5.75% which can be spent on the embedded derivative, in this case an option on SPX. Now, if the embedded derivative is too expensive, then the product can't be sold at 100% while still providing a profit to the issuer. At the time of issuance, SPX vol was around 20%, so the 18m option would have been worth about 15%. This is clearly more than the 5.75% they have available to spend. So they need to make it cheaper - there are a couple of ways of doing this. One is to add a cap in, i.e. payoff is `max(min(120%, Perf.), 100%)`, or you can do as they did and add in a barrier which if breached means you get nothing.

The reason you add in things like this is purely down to the structuring of the product. A couple of alternatives would have been to cap the payoff around 110%, or to have reduced the gearing of the option to about 30% (i.e. if SPX finishes up 130%, then you get 100% + (130%-100%)*0.3 = 109%). Of the three choices:

- 100% + max(0, Final/Initial-1) if no Kick Out

- 100% + 0.3 * max(0, Final/Initial-1)

- 100% + max(0, min(110%, Final/Initial) - 1)

Looks the most attractive to you?

The onerous condditions are there to make the note worth 100% at inception, and if you are not of the belief that SPX will increase by 25% in 18m, then maybe this doesn't seem too bad to you - at the very least you get your money back...