Online Finance – Exotic Derivative products – Adjustable Strikes, Asian, Barrier options
Adjustable strike option
Key concepts
Adjustable strike options are options whose strike is reset either automatically or by the holder, depending on the path/level of the underlying. Depending on the terms of the reset mechanism they are also known as moving/floating strike options, indexed-strike options, periodic (reset) options, ratchet/ladder options and step-up/step-down options.
They are often combinations of vanilla and digital or barrier options and in the two chapters covering those instruments a number of products are explained which, because they actually consist of a number of options packaged together, appear to have similar resettable strikes. In this section we also include a small number of options whose unusual exercise conditions make them similar to adjustable strike options, namely fixed-strike lookbacks, lookforwards and shout options. All these options share one common characteristic: they enable the holder to create strike price conditions that more exactly suit their views especially their views on the dynamic path of spot not just its final resting place than conventional options.
Definitions
Deferred strike price option
Also known as a forward start option, this is an option that allows the holder to set the strike price at a predetermined time or during a predetermined period after its trade date. The strike price is usually expressed as a fixed ratio to or percentage of spot. The option’s premium is usually set on the trade date. These options allow the holder to lock in current levels of volatility in the expectation that volatilities will rise or fall without setting the delta of the option until the strike is set. These are more commonly embedded in structured assets than used as naked options.
Example
An investor might want a three-year bond whose annual coupon captured the appreciation of a currency, say sterling against the dollar, in each year. This would be constructed from a strip of two-one year forward start USD put/GBP call options plus a one year vanilla option. The first could be struck at-the money spot with the two forward starting options setting at 100% of spot on the first business day of the year and an expiry on the last business day of the year.
Hi-lo option
An option which pays out the difference between the high and the low price or rate reached by the underlying over the term of the option. Constructed from a combination of a lookback call and lookback put, the buyer is taking a view that the volatility of the underlying will be greater than the implied volatility of the component options. Because the expected payout is high, the premium is high, and the option buyer is taking a large, long position in gamma and vega.
Indexed-strike option
Also known as a periodic reset option, this is an option whose initial strike price moves up or down according to a preset schedule or depending on the path of a reference asset or index. The size, timing and direction of the reset mechanism can depend on almost any contingency required by the buyer/seller. It may rely simply on pre-set trigger points being hit by the underlying at any time during the life of the option; it may require that the underlying move a certain amount relative to the last fix within a given sub-period of the option’s life (sometimes called momentum or gap options); or it can be automatic with the option’s strike price resetting at a pre-agreed spread above or below the reference index or at a series of pre-agreed absolute levels for each successive period without the underlying having had to hit any predefined level (sometimes known as a moving strike option).
Many of these products are combinations of vanilla and exotic options. The holder of a momentum cap is long a conventional cap and short a series of digital caps. The benefits to him versus a vanilla cap will depend on the value of the sold options. In a positive yield curve environment, the steeper the curve the higher the chance that the trigger Libor rise will be breached and the higher the probability that the strike will be raised and the higher their value. Also known as step-up/-down options.
Example
Momentum options and gap options illustrate the subtleties available with resettable strike options. They enable the holder to hedge against or benefit from dramatic movements in the price of the underlying. An option on Libor struck at 7.50% would pay out if Libor rises by more than, say, 75 bp in the next three months. It therefore has two triggers, the gap trigger (75bp) and the speed trigger (one month). Regardless of whether Libor did rise by 75bp in the first three-month period, the strike would then be reset to current Libor at the beginning of the next payment period. This structure is usually altered so that the strike price ratchets up by a predetermined amount. So a borrower with a three-year US dollar loan based on three-month Libor could buy a three-year momentum cap with a 7.50% strike and a 75bp trigger amount. If In any three-month period three-month Libor rises by more than 75bp, then the cap strike is reset 25bp higher with a maximum cap rate of 8.50%.
Ladder option
An option whose strike resets automatically when the underlying hits predetermined levels (‘rungs’). When the strike is reset the intrinsic value of the option is automatically locked in regardless of whether the underlying subsequently moves disadvantageously. Ladder options are strips of capped/exploding options with the cap level of one option set equal to the strike level of the next and each cap level a rung in the ladder. Every time a rung/cap level is reached that option is exercised for its intrinsic value locking in that gain and a new option is triggered with a strike equal to the previous cap level and a new cap level higher (call)/lower (put) than the previous one.) As these options are sometimes known as cliquet options (because cliqueter is French for to knock and the automatic exercise became known as the cliquet clause) cliquet option can also be used as a name for ladder option. These options are more expensive than vanilla options, particularly if the put asset has a lower interest rate than the call asset.
Example
A ladder call on the EUR/USD rate with a strike of 1.0500 might have a rung every 1 cent up to a maximum of 1.0800 and have a payout of the greater of (i) the closing spot less the original strike and (ii) the highest rung reached less the strike. The more frequent the rungs, the more expensive it is. Other ladder options have only a minimum settlement level. Once the underlying has risen by, say, 10%, that gain is locked in regardless of the future path of the underlying price. If it subsequently rises above 10%, the investor still participates, but he also has a floor at 110. In exchange for this downside protection the maximum return is generally capped.
Lookback option
An option that allows the buyer, at maturity, to choose the most advantageous exercise conditions that have occurred over the life of the option, or in the case of a partial lookback option, during a pre-set sub-period (usually between one and three months) of the life of the option after which it becomes a standard European- or American-style option. A lookback period limited to the first part of the option’s life will help improve the timing of any market entry; one limited to the last part of the option’s life will help with market exit timing.
Lookbacks come in two varieties. The lookback strike option/floating-strike lookback, instead of having a specified strike price, allows the buyer at expiration to look back over the life of the option and set as the strike the most favourable price that has occurred during that time. A lookback call {put} allows the buyer to choose the lowest {highest} price that has occurred over the life of the option. These strikes are then compared with the spot price at expiration to determine the option’s payoff. The lookback spot option/fixed-strike lookback has a strike set at the outset but then at maturity allows the buyer to look back over the life of the option and choose the most favourable exercise point to maximize profit between strike and exercise. Lookbacks, like conventional options, are most profitable to the buyer (net of premium) if the realized volatility of the underlying price is higher than the implied volatility. If a buyer believes that there will be a sharp move in price but is not sure when and for how long the price will move, lookbacks are attractive. Because they allow the buyer to choose the best exercise conditions with perfect hindsight, lookbacks command much higher premiums than conventional options. Also known as hindsight options and lookforward options.
A fixed-strike lookback struck at-the-money spot is sometimes called a lookforward option. This gives the buyer the difference between the asset price at the beginning of the period covered by the option and its high (call) or low (put) over that period.
Ratchet option
A type of indexed strike option whose strike price resets favourably if the underlying moves out-of-the-money relative to the initial strike and hits certain trigger or ratchet levels but which does not reset in the other direction if the underlying subsequently moves in the other direction. A ratchet call option is a call struck at the ratchet option’s strike price, plus a series of bought knock-in put options each struck at a ratchet level and a series of sold knock-out puts with strikes staggered one rung behind the purchased options. Confusingly, like ladder options, ratchet options are also sometimes known as cliquet options because vilbréquin à cliquet is French for ratchet brace.
Roll-up option
An option whose strike price is favourably reset at the same time as the option itself is converted into knock-out option if the price of the underlying asset trades through a predetermined trigger point, usually struck at a point where the underlying has moved significantly against the original option. So, a roll-up put with an original at-the-money strike of 80 might be converted into an out-of-the-money knock-out (up-and-out) put with a strike of 100 and a knock-out level of 110 if the underlying traded to 100. The holder has a new, more favourable put strike, but if the underlying continues to rise (i.e. in his favour as long as his put is hedging an existing position) then the put is knocked-out (at a point where he does not need protection).
The roll-up put outperforms the standard put if the roll-up trigger is reached but the trigger is not. If the roll up trigger is not reached, then the roll-up put and vanilla put behave the same. Only if the roll-up trigger and the trigger are reached does the roll-up underperform the vanilla instrument. The trigger price for the up-and-out put is set in advance and is above the roll-up strike.
Shout option
Confusingly there are two completely different types of option that are called shout options. (i) A path dependent option that combines the features of lookback, ratchet and ladder options. A shout option allows the purchaser to lock in a minimum payout (the intrinsic value of the option at the time of the ‘shout’) while retaining the right to benefit from further upside. So-called because when the option holder thinks the market has reached a high (call) or low (put), he ‘shouts’ and locks in that level as the minimum and, with a one-shout option, still holds a European option with the original strike price for the remaining life of the option. If the market finishes higher (call) lower (put) than the shout level, the holder benefits further i.e. the payout of a shout option is the greater of the intrinsic value locked in by the shout and the intrinsic value at maturity.
This type of shout option is similar to a ladder option in which profits are locked in when the underlying rises/falls sufficiently to hit a pre-determined ‘rung’ level, but in the shout option the rungs are not set in advance. This makes the shout option more expensive than the ladder option, the more so when multiple shouts are allowed (multiple shout options are very expensive). As with a ladder option, the more shouts that are allowed, the more like a lookback the shout option becomes. The ability to lock in gains before expiry makes the shout more expensive than a standard European option, and the fact that even after a shout, the option holder effectively has another option struck at the shout level, makes it more expensive than an American-style option.
Example
A corporate treasurer might be bullish on EUR/USD rates but also expects the cross to be very volatile. He is worried that using a vanilla option will mean that he misses out on temporary highs. A shout call solves the problem. If the EUR/USD rate rises above the strike price, but ends up below the shout level (wherever the treasurer eventually chooses that to be), the treasurer receives a profit of the shout level less the strike level. If the exchange rate closes above the shout level, the investor will receive that additional profit as well. The payout is therefore the maximum of (shout strike) and (close strike).
(ii) The second type of shout option is a call or put option that gives the buyer or seller the right once and only once during a pre-specified period to ‘shout’ the option and reset the strike to the then prevailing spot rate (or some percentage thereof)). These shout options are therefore, in a sense, halfway between a vanilla and a lookback option. They are more expensive to buy and generate less premium when sold.
Surge options
An option whose strike price is reset on a daily basis to a fixed spread above or below a moving average. This hedges against the risk of rapid price changes rather than absolute price trends over longer periods. Commonest in the commodity markets, a put surge option on the price of crude oil could work like this whenever the spot oil price falls below the 45-day moving average less two cents, the option is in-the-money. The settlement amount is determined by the difference between the spot price and the strike price multiplied by the number of barrels to be priced each day. A call would move into the money if the spot price moved above the moving average plus a fixed spread.
Asian options
Key concepts
Asian or average rate options are options whose payout or strike price is based on an average of the price of the underlying over the life of the option.
The averaging process can begin at any point during the option period (for example a one-year option whose payout depends on the average underlying price in the final month). The sampling process frequency and interval of underlying price observations can also be tailored. The number and timing of price (strike) observations is determined in advance and may start at the beginning or near the end of the life of the option. Observations may also be weighted in favour of prices (strikes) observed on specific dates.
Unlike a straight American- or European-style option, an average option can be settled more than once over its life. So for example, the holder of a one-year average option can choose to settle the option monthly versus the average price of the underlying the previous month. Average options are nearly always cheaper than conventional options because the averaging process smoothes out the underlying price movements thereby reducing volatility and hence the premium of the option.
Definitions
average price/rate option (ARO)
Unlike a conventional option, which is settled by comparing the strike with the spot rate at expiration, an average rate option’s payout is the difference, if positive, between the predetermined, fixed strike price and the average of spot rates observed over the option’s life. This hedges against the average prevailing spot over the life of the trade. It also removes the reliance of the option’s expiration value to the underlying cash price on a particular day. Typically the volatility of an average rate option is about 58% of the volatility of a conventional option and so is cheaper.
AROs are cash-settled, not deliverable, so when hedging an underlying exposure, cash flows need to be converted in the underlying market on the relevant fixing dates. This ensures that the hedge instrument effectively offsets the aggregate FX rate of the cashflow conversions. There are three main varieties:
- Arithmetic Asian options are the most common. The arithmetic average is the sum of the price observations divided by the number of observations. These options cannot be priced using a closed-form model because the sum of lognormal components has no explicit representation the arithmetic average is not lognormally distributed even if its underlying is.
- Geometric Asian options’ payout is based on the geometric average price of a series of observed underlying spot rates. The geometric average is the nth root of the product of n quantities. These options can be priced using a closed-form option-pricing model because the product of lognormal prices is itself lognormal. They are rarer than arithmetic Asian options.
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Weighted Asian options are also available in which the weighting of each periodic price or rate used in the averaging process varies according to a predetermined schedule. These options are useful if the timing and magnitude of cash flows is known but the price or rate is unknown. A simple weighting scheme is normally used in which the weights add up to one.
Example
A hedger short EUR/long USD wishes to hedge on average at 1.0500 buys an ARO EUR call/USD put struck at 1.5000 and a fixing frequency of weekly every Friday for six months. With the forward at 1.0445 and 10% volatility the premium would be 1.12% EUR versus the 2.52% EUR of a vanilla European-style option. If the average were above 1.0500 on expiry, the underlying would be hedged at an effective rate of 1.0616 (strike + premium). If the average were below 1.0500, then the underlying benefits below an average rate of 1.0385 (strike – premium). If spot trades above the strike early on in the life of the option and then trades back down, the payoff from this ARO will exceed a vanilla option. However, if the spot is greater at expiry than its average until expiry, then the payoff of the ARO will be less than a vanilla. In general, the expected payoff of the ARO is lower, and this results in the lower premium.
Average strike option (ASO)
A moving/floating strike option whose payoff is determined by comparing the underlying price at expiration with a strike computed as the average of spot rates over the option’s life. The option is exercised against the spot rate prevailing at expiry and can be cash or physically settled. An ASO limits exposure and benefits to large movements of spot. It is equivalent to a strip of daily options struck at the average spot during each day and where the maximum loss on all these transactions is equal to the premium of the ASO. So an average strike call has a payoff equal to the difference between the asset price at expiry and its average over the option’s life if this difference is positive or zero otherwise.
Barrier or trigger options
Key concepts
Barrier or trigger options are conventional options except that they are cancelled or activated or, more generally, changed in a pre-determined way when the underlying trades at predetermined barrier/trigger levels. So a knock-in option pays nothing at expiry unless at some point in its life the underlying reaches a pre-set barrier and brings the option to life as a standard call or put. A knock-out option is a conventional option until the price of the underlying reaches a pre-set barrier price, in which case it is extinguished and ceases to exist. Barrier options have a strike price and a barrier price and the barrier can be above or below the strike price. In all variations the barrier can be made to be active for either part or all of the option’s life.
Because of the importance of the barrier event in determining the value of the option, users must ascertain at the outset of the transaction the definition of a barrier event. For example, is it to be based on quoted rates or transactions; how is the issue of crosses to be dealt with in illiquid currency pairings; when can barrier events occur (outside normal trading, only on hourly fixes, at the end of day fix, only on certain dates and so on. Where barrier events can only occur at certain times (and under certain circumstances) there is said to be barrier discontinuity. This makes the options more difficult to price and value).
The concept can be applied to every type of option and some option combinations- caps, floors, collars, digitals, swaptions and in any asset class. The two basic classes of barrier options are the standard (or out-of-the-money) barrier options and the reverse (or in-the-money) barrier options.
These options should not be confused with capped {floored} calls {puts} which are also sometimes known as trigger options. They are described in the chapter on ‘Vanilla Options.
Definitions
Balloon option
An option whose notional principal increases if a preset trigger level is breached. For example, an equity investor might believe that the FTSE-100 will rise from 4900 to 5000 and then, if it breaches this resistance level, rise strongly again. He could buy a 4900 call with a trigger of 5000 and a multiple of two, meaning that if the index stays below 5000 the option behaves like a vanilla call but if it rises above 5000 then the option’s notional principal doubles. The balloon option’s premium is more expensive on the original notional principal than a vanilla option because it is a combination of two options a vanilla call struck at 4900 and a knock-in call struck at 4900 with a knock in at 5000. However, if the trigger is reached, the premium on the ballooned premium is cheaper. The greater the ballooning the higher the premium; the further the trigger level is relative to the strike, the cheaper the premium.
Double barrier option
A general term for any barrier option incorporating two knock-out or knock-in levels, one either side of spot. These are commonest in the FX markets where users may have strong views on both a support and a resistance level. The illustrated of a knock-out rebate option in this chapter is an example of a double barrier option.
in-the-money/Reverse barrier option
A barrier option whose barrier is in-the-money relative to the strike. So, the barrier level would be above the strike for a call (up-and-in/out calls) and below it for a put (down-and-in/out puts). These are priced and behave very differently from standard barrier options since they have intrinsic value when they are knocked-in or out, making knock-ins relatively more expensive and knock-outs relatively cheaper for a given proximity to the strike. So, unlike standard options, which become more valuable as volatility increases, in-the-money knock-outs become cheaper. That is, they have negative vega: the probability of knock-out increases with increasing volatility, reducing the chance that the option will pay out and making them cheaper. In-the-money barrier options can be used both to hedge an underlying position (as in example 1 below) and to take outright speculative positions (as in example 2).
Example 1
In-the-money barrier option could be used by a USD-based exporter wishing to reduce the cost of protection against Euro weakness. The company could purchase a EUR put/USD call with an in-the-money knock-out struck at 1.0200 with the knock-out set at 1.0000. This option could be up to half as cheap again as the out-of-the-money knock-out, but carries the risk that if EUR/USD does trade down to the knock-out level, then the corporation has not only lost its hedge, but also has to re-hedge at much worse levels than with the standard knock-out. The hedger has to have a stronger view on exchange rate movements than with either vanilla options or standard knock-outs.
Example 2
An investor believes that over the next six months the dollar will strengthen against the yen by around 10% and by at the very most 12%. If dollar yen is trading at 125 he could buy a reverse knock-out USD call/JPY put struck at the money forward at, say, 121.50, with a trigger at 140.50, 0.5 yen above his predicted high of 140 and another knock-out at 119. Then if the dollar does strengthen, but trades either below 119 or above 140.5 over the life of the option, the call will disappear. If the dollar strengthens, but never reaches 140.50 or 119 over the life of the option, the call will behave like an ordinary call and the investor will exercise the call and make the same profit as the ordinary call. If the dollar does not close above the call strike, the option will expire worthless like an ordinary option.
Knock-in cylinder/collar
Barrier options are often used to modify simpler option spreads. The knock-in collar is an alternative to selling an out-of-the-money call {put} to finance the purchase of an out-of-the-money put/(call) to create a standard collar. Instead the holder sells a knock-in call {put} instead of a vanilla option. This allows the holder of the position to participate fully in the upside {downside} of any moves in the underlying, until the trigger level is breached.
Example
A USD-based exporter has EUR receivables due in six months. The current spot EUR/USD rate is 1.0310 and the six-month forward points are 130 so the six-month forward outright is 1.0440. The company wants protection against Euro weakening and favours options over selling Euros forward because of the high potential opportunity cost of locking in an outright forward rate. However it does not wish to incur any upfront premium at all or does not wish to be long volatility. In this case they could execute a zero premium standard risk-reversal/collar, selling a call at 1.0550 to finance the purchase of put protection at 1.0340. This risk-reversal can be as narrow or as wide as the corporation wishes. Alternatively, instead of selling a vanilla EUR call the hedger could sell a knock-in call struck at 1.0500 with a 1.0900 trigger level. Although the strikes of the call and put would be less advantageous than a regular collar, the corporation would have full protection and only give up its upside if the knock-in were triggered.
Knock-out cylinder/collar
Similar in concept to the knock-in collar, this is the substitution of the short call of a simple zero cost collar with a knock-out call. In the example above, the corporation would buy a 1.0240 EUR put and sell a 1.0500 EUR call with a knock-out trigger at 1.0000. If the knock-out were triggered, then upside is no longer limited.
Knock-out trigger option
A capped call {floored put} (see chapter 7) which incorporate an out-of-the-money knock-out level. If the underlying trades through this, the whole option is cancelled.
One factor barrier option
A barrier option whose barrier event and payout are based on the same underlying. Also known as inside barrier options. In a two-factor/outside barrier option the barrier event and the payout are based on two different underlying assets. So, the payout might be a function of a foreign exchange rate but the barrier event may be the breach of a level in the price of a commodity. See chapter 15.
Out-of-the-money/Standard barrier option
A barrier option whose barrier is set out-of-the-money relative to the strike. So the barrier level would be below the strike at the start of the option contract for a call (down-and-in/out calls) and above it for a put (up-and-in/out puts). These options cost less than standard options because the price of a vanilla option takes the entire probability distribution of possible prices for the underlying into account, while the knock-out feature removes many of those possible values. The exact premium reduction is determined by how likely the barrier event is to occur. The more likely the option is to be knocked out or the less likely it is to be knocked-in, the greater the premium reduction, and vice versa. The likelihood of the barrier event depends on how near the spot/forward level the extinguishing trigger level is, on the maturity of the option and on volatility.
Example
A USD-based exporter finds a vanilla EUR put/USD call too expensive. A knock-out EUR put/USD call struck at 1.0340 with a barrier level at 1.0600 costs less (and so has a lower break-even), gives the a guaranteed minimum of 1.0340 USD for every EUR of receivables and only disappears when the underlying is moving in the hedger’s favour, giving it the flexibility to examine other hedging strategies. The main risk is that if the option is knocked-out, the corporation is exposed to any subsequent dollar weakness and so may have to put on a second forward or option-based hedge that may cost more than the original vanilla put option.
Rebate option
An option that pays a fixed amount if it would otherwise have expired worthless due to some barrier event. So a knock-in rebate option pays a pre-set fixed amount if it has never been knocked-in (even if the option then expires out-of-the-money) and a knock-out rebate option pays the option holder a pre-set fixed amount if it is knocked out. The commonest structure is the knock-out rebate option a call or put with an in-the-money knock-out level and rebate almost always set equal to the initial premium paid. This is usually just called a rebate option. A second, out-of-the-money knock-out level can be incorporated which, if traded, cancels the entire option (see second diagram). This structure is sometimes, confusingly, called a knock-out rebate option.
Sloping/moving/jumping barrier option
A barrier option whose knock-in/knock-out level changes during the life of the option, for example to match moving technical levels in the underlying. This change may be either linear (i.e. sloping) or move in discrete steps (i.e. jumps). For example, our USD-based exporter could set the knock-out level for the first three months at 1.0500 and then 1.0600 for the last three months.
Step-up {down} barrier option
A barrier option whose barrier increases {decreases} over time.
Switchback option The simultaneous purchase of both a capped call {floored put} and a knock-in put {or call}. The trigger levels of the knock-in barrier options typically equal the cap/floor strike prices. If the underlying hits the trigger levels, the capped option is automatically locked-in and the knock-ins activated. The holder would typically set the strikes at a point he believed to be around a peak {trough} in the underlying. The position benefits from that level being reached and then switches back from call to put (or vice versa) as the underlying itself switches back, retreating {rising} from its peak {trough}.
Thomas A. Fetherston at the University of Albama put this together at some point in time – a mix of teaching notes, core concepts, a glossary and a 109 page handy desk reference that you would end up referring to if you work with derivatives in any shape and form.
I stumbled across this resource about 5 years ago and it had been stewing invisibly in one of the many resource folders I have on my hard drive. I believe it would be a crime to sit or hide on a resource like this. The Glossary is here and I will try and post the teaching notes over the next few days after turning them into bite sized pieces as and when I get time.
I looked for Tom’s home page but a Google search on Tom’s name only pulls up his authored books, no home page that I could possibly link to.
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