In the latest issue of Asia Risk there is an article which talks about the knock in equity linked structures that are being traded mostly in the Asian markets ("*Picking Up the Pieces" *, Asia Risk, September 2006, pp E6-E8). In this article, one Dickson Cheung of Calyon talks about "down and in" structures, "jump risks" and "expectation gap". We quote a line (ad verbatim): "These kinds of jump risks (in down and in puts) create an expectation gap among clients because of a sudden the clients will appear to lose their principal, which they do not expect." The context is of course, restructuring of equity linked products by the banks for their clients. This is an interesting issue and we would like to talk a bit about them as well and see if we can unravel what is meant by simple, yet esoteric words and phrases such as"jump", "expectation gap"

A knock-in option say, a knock-in call option, is an option that comes into existence when a particular price is hit in the market. A regular knock-in (as opposed to a reverse knock-in) comes into existence as it is out of the money which simplifies its trigger/barrier conditions. Knock-in options are instruments designed for a mean reverting market (or the expectation of a mean reversion in the market). These options come alive against the market direction and any operator, be it a trader, market maker or an investor, holding a knock-in structure will essentially have a view on the mean reversion of the market.

__Payoff profile and Delta of a Knock-In Options__

Knock-in options, whether naked or embedded in a structured product, often display scary and complex payoff profiles and therefore investors should be very careful while buying them. Let us look at a Down and In Call option with strike at 100 and knock-in trigger (barrier) at 98.

We use a sort of back of the envelope Monte Carlo pricing engine, with only 2,000 simulation runs on an Excel + VBA spreadsheet to price the above knock-in with strike price = 100, risk free rate = 4.5%, constant volatility = 15% and vary the spot around the barrier (also called the out-strike). You can also do this pretty easily on your Excel/VBA spreadsheet though of course, for an exact pricing you need to do at least 50,000 simulation runs with a either a Dupire volatility measure or the forward-forward vol estimated from the stopping time.

Anyway, the profile from our rather "stupid" spreadsheet model looks something like this:

If you were to run a 50,000 run simulation with Dupire volatility measure or the stopping time measure you will get a much sharper graph but the shape will essentially remain the same, rising exponentially with a pointed peak at the barrier and then a sharp exponential fall away from the barrier.

It can be seen from the above that the above payoff comprises two piecewise options of opposite deltas. In other words the delta of a knock-in will exhibit a jump - perhaps the "jump" that Dickson is talking above - around the barrier.

Let's now plot the delta of the knock-in against the underlying. We have actually taken a short cut and simply calculated the gradient of the option payoff with the underlying price and have used it as a proxy for the delta (it is not the delta and for calculation of delta we need to run at least around 100,000 simulation runs and re-price the option at each point on the spot axis).

Once again if you were to use a 100,000 simulation runs with the appropriate volatility measure (Dupire vol or the forward stopping time vol) you will get a very sharp graph with almost a straight line vertical drop in the delta at the barrier and smooth curves around it.

As can be seen from the plot of the gradient (which we are using as a proxy for delta of the option) with that of the underlying the jump in the deltas is of almost equal magnitude and its move is in the opposite sign, one positve and one negative. At around the barrier (trigger) the delta of the knock in moves up from a positve value of -0.2175 to 0.1908 which means that the operator has a gap delta of around 0.4 to make up for. (In the case of a knock out call there will be a jump delta down of around the same magnitude.) Perhaps this is the "(expectation) gap" that Dickson is referring to in his interview.

__A few points to Remember when trading or investing in Knock-in (and Knock-out) Structures__

- The first exit time (or Stopping time) plays a crucial role in the pricing of barrier options (first exit time is the time when a barrier is hit). It is important for traders to know where the expected first exit time is located on the volatiltiy term structure curve; the forward-forward volatility between the stopping time and the maturity of the option should be used to price a knock-in.Any vega hedge of the barrier is dependent on the first exit time;

- Parkinson's number can play an important role in the pricing process. If close-to-close volatility is lower than then Parkinson's number, the barrier component of the option can be underpriced by the normal method.

- An operator must understand fully comprehend whether he is long the barrier (short knock-out or long knock-in) or short the barrier (long knock-out or short knock-in). In a long barrier trade the operator benefits from the hitting of the barrier because either his liability gets knocked out or he experiences an increase in his wealth due to a knock-in. In a short barrier trade the opposite happens.

It is quite likely that the investors of the bank of which Dickson Cheung and others in that article are alluding to were short the barrier. Many a times the investors of barrier structures (whether in equity linked products or fixed income products) don't realize this fact or are too inexperienced, unsophisticated or simply lazy to comprehend this fact.

*Note: ** There is no greater options trader or for that matter option theorist than Nassim Taleb and there is not better book on options than Dynamic Hedging published by John Wiley & Sons. *

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