The aim of this Learning Curve is to explain the behaviour of a particularly interesting kind of barrier option, the Parisian, and to focus on its use in structuring exotic equity derivatives products.
Parisian barrier options were developed by a group of Paris-based academics as an extension to classical barrier options. The difference between a Parisian and a classical barrier lies in the way the barrier level is triggered. In the case of the Parisian, the closing price of the underlying assets must remain for a predefined number of trading days, the 'window,' above or below the assigned price level before the payoff disappears or comes into place. If the counting of the window's days resets every time the barrier is crossed, it is a standard Parisian option; if the counting is never reset from the first crossing, it is a cumulative Parisian option.
Using Parisian Options To Structure
Equity Derivatives
Parisian barriers have several advantages to plain vanilla barriers.
Parisian barriers lower the risk of the product. The use of a Parisian in structuring a reverse convertible with a down-and-in feature avoids the problem of 'exogenous hits,' which have caused many problems for retail investors and regulators in some European countries in the recent past. With a Parisian barrier the price decline should have the characteristic of a short negative trend, lasting for a certain spell of time, not an abnormal downward price movement.
Parisian barriers reduce the possibility of manipulating the underlying. They make any attempt to move the market so that the underlying price goes through the barrier hard to execute, even for illiquid stocks. This feature makes short volatility products safer from the investor's point of view.
Parisian barriers are optimal to build up a directional view. High volatility in the equity markets can mean it is complicated to consolidate returns, even when the client has a defined view on the price dynamic. But if the price trend is well supported the investor can lock-in returns, with structures like calls with rebates or ladders, at a lower cost by using Parisian rather than plain-vanilla barrier options.
Hedging Parisian Structured Products
Parisian barrier options are triggered with a reference price, usually the official closing price of the stock exchange where the cash underlying is traded. This is necessary for the 'window counting' mechanism, because if it was set on continuous prices the task of the over-the-counter calculation agent would become cumbersome and open to questioning.
However, using closing prices as triggers can cause some 'slippage' problem to equity derivatives hedgers, if the underlying moves into the 'triggering subset,' but closes on the other side.
On the other hand, 'window' triggering introduces some important smoothing effects on the barrier's Greeks, avoiding most of the delta and gamma jumps traders experience in standard barrier options when the underlying approaches the trigger price.
The Price Function Of A Cumulative
Parisian Down-And-In Put
For short volatility products, like reverse convertibles on single stocks, the expected return for the buyer depends on the value of the put option it sells to the issuer. The higher the value, the higher the final coupon and the lower the probability of having the capital eroded from an underlying price decreasing at maturity.
Thus, to retain the marketing appeal of short volatility structures like reverse convertibles, the value of the cumulative Parisian put option should be, ceteris paribus, comparable to the classical barrier option's value.
This statement is proved by analysing the three-dimensional chart on page 5.
The price axis contains the percentage price of a one-year at-the-money European put option with a down-and-in barrier, on a pan-European equity index. Given 100 as the at-the-money level, the barrier axis considers different levels of barrier triggers in percentage terms. The last axis shows several Parisian windows, from two to 30 trading days, one trading month is roughly 20 trading days. The first point, class, is the classical barrier price.
The graph reveals that the cumulative Parisian put is showing only a marginal price decrease in comparison to the classical one when consistent triggering windows are chosen. Considering two typical barrier levels for one-year products, 85% at-the-money and 75% at-the-money, we see that selecting 15 days and 5 days respectively as triggering windows, the difference in the coupon is not higher than 50 basis points.
References
[1] Chesney, Janblaque-Picqué, Yor "Brownian Excursion and Parisian Barrier Options," Advanced Applied Probability, 1997.
This week's Learning Curve was written by Maurizio Morini, country manager at KBC Financial Products Milan in Italy.
