# managing options risk for exotic options - nbsp;· 3 here’s another table of exotic option types,...

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MANAGING OPTIONS RISK FOR EXOTIC OPTIONSAn exotic derivative is one for which no liquid market exists. As a general rule, the only liquid optionsare European-exercise calls and puts, including interest rate caps, floors, and European swaptions. Sinceexotic derivatives can rarely be marked-to-market based on publicly available prices at which they can bebought and sold, they usually have to be marked based on some hedging strategy which involvescombinations of dynamic hedging using forwards and European-exercise options.

The general approach we will be following is one of building up techniques for marking-to-market andcalculating risk on less liquid instruments by designing hedging strategies involving more liquidinstruments. A less liquid instrument is marked based on the cost of executing the hedging strategy, andrisk is then calculated by estimating the degree of uncertainty around this cost. We will do this insuccessive stages - at each stage, instruments for which we developed marks at previous stages will beconsidered available for use in creating hedging strategies to mark other instruments. However, we mustbe careful to realize that any risks due to uncertainty of how to mark an instrument will be inherited byinstruments which it is then used to mark. So we will often be faced (once again) with tradeoffs betweenbasis risk (employing a hedging strategy which uses more liquid instruments but which leaves a lot ofroom for uncertainty) and liquidity risk (employ a hedging strategy which does not introduce muchuncertainty by using less liquid instruments).

As one example of how the stages will work: USD/JPY FX We will use more liquid forwards to create a hedge for less liquid forwards on USD/JPY FX and

estimate the resulting basis risk.

We will use both less liquid and more liquid forwards on USD/JPY FX along with more liquidEuropean-exercise options on USD/JPY FX to create a hedge for less liquid European-exerciseoptions on USD/JPY FX.

We will use both less liquid and more liquid European-exercise options on USD/JPY FX to createa hedge for barrier options on USD/JPY FX.

We will use barrier options on USD/JPY FX to create a hedge for lookback options on USD/JPYFX.

The following table, taken from RISK Magazine, May, 2000, shows the principal forms of exoticproducts and how widely they are used in different markets. Those which are classified as correlation-dependent, as well as virtually all interest rate exotics, have payoffs based on more than one underlyingsecurity. We will address the risk management of these exotics in another section. In this section we willfocus on exotics whose payoff is based on just one underlying security, but many of the principles weestablish here are applicable to multi-security exotics as well.

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Intensity of use of option structures in various market

Interest rateoptions Forex options Equity options

Commodityoptions

OTC Exchanges OTC Exchanges OTC Exchanges OTC Exchanges

First-generation optionsEuropean-style A A A A A O A AAmerican-style A A R A A ABermuda-style A A R O

Second-generation optionsPath-dependent options

Average price (rate) A A A A ABarrier options A A A ACapped O O A OLookback R R O RLadder O O A ORatchet O O A OShout R R R O

Correlation-dependent optionsRainbow options R O O OQuanto options A A A ABasket options R A A A R

Time-dependent optionsChooser options R R R OForward start options R R A ACliquet options R R A O

Single payout optionsBinary options A A A AContingent premium options A A R A

A = actively used; O =occasionally used; R = rarelyused; blank = not used

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Heres another table of exotic option types, showing which ones can be hedged using staticcombinations of vanilla options and which require dynamic hedging with vanilla options. Inaddition to the options listed in the RISK Magazine chart, we have also added a few otherswhich have a reasonable amount of use. Dynamic hedging is divided up into the coverage of3 primary risks: shape of the volatility surface, liquidity risk, and correlation risk.Management of the first two risk types will be covered in this section; managementOf correlation risk will be covered in the section on multi-factor exotics.

Dynamic Hedge Needed

Static Hedgepossible

Shape ofvolatilitysurface Liquidity Risk

CorrelationRisk

First-generation optionsEuropean-styleAmerican-style Bermuda-style

Second-generation optionsPath-dependent options

Average price (rate) smallBarrier options Reverse barriersLookback by barriersLadder by barriersRatchet by barriersShout

Correlation-dependentoptions

Rainbow options Quanto options Basket options

Time-dependent optionsChooser options price risk only Forward start options Strikeless voCliquet options Strikeless volCompound options price risk only Volatility swaps Strikeless volVariance swaps

Single payout optionsBinary options Contingent premiumoptions

Power options

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It is reasonably straight-forward to see that the payoff of any derivative whose payments are afunction solely of the price of one underlying security at a single future time can be replicatedas closely as one likes with the payoff of a combination of vanilla forwards, calls, and puts,since any (smooth) function can be approximated as closely as you wish, by a piecewise-linear function, and vanilla options at different strikes can produce any desired piecewiselinear payoff. Hence, this portfolio of vanillas can be held as a nearly perfect hedge of theexotic derivative (see Carr & Madden , Towards a Theory of Volatility Trading, availableon Peter Carrs website). For a concrete example of how this can be used in practice, letsconsider gold-in-gold options.

A plain vanilla gold option would be, for example, an option to exchange 1,000 ounces ofgold for $300,000 in one year. Another way of viewing it is that the buyer has an option toreceive 1,000 x (G - 300), where G is the gold price in one year. If G = 330, the buyer canexchange $300,000 for 1,000 ounces of gold, under the option, and then sell the 1,000 ouncesof gold for $330,000, a net profit of $330,000 - $300,000 = $30,000.A gold-in-gold option calls for this price increase to be received in gold. So if the price ofgold goes up 10% from $300 to $330, the option buyer receives 10% x 1,000 ounces = 100ounces of gold. Since gold is worth $330 in ounces, this is a profit of 100 x $330 = $33,000,a payoff of 11% rather than 10%.

In general, the payoff is x% * (1 + x%) = x%+x2%.

This can be hedged with 101% of a vanilla option struck at par plus 2% of a call option struckat each 1% up in price, e.g.., to hedge a 1,000 ounces gold-in-gold contract with a strike of$300, a 1,001 ounce call with a strike of $300, a 2 ounce call at $303, a 2 ounce call at $306, a2 ounce call at $309, and so on. The payoff on these calls, if gold rises by x%, is

101% 2%1

1

+ =

x II

x

% = 101% 2%2

2

+

xx x

= x% + x2%

In practice, there is some degree of underhedging based on the spacing between strikes usedin the hedge and that there will be some highest strike above which you wont buy anyhedges. This leads to basis risk, which could be calculated fairly easily by an assumedprobability distribution of final gold prices multiplied by the amount of mis-hedge at eachprice level. This basis risk can be reduced as low as one likes, in principle, by going to asufficiently fine and broad range of strikes, but then we will be encountering liquidity risk infinding liquid prices for this large a set of vanilla calls. The basis risk can also be reduced bydynamic hedging in gold forwards the residual payments of the gold-in-gold option less thehedge package.

Even if this is not selected as a desirable hedge from a trading viewpoint, it still makes senseas a way to represent the trade from a risk management viewpoint, for the following reasons:

1) It allows realistic marking-to-market based on liquid, public prices. Alternative markingprocedures would utilize an analytic pricing model for the gold-in-gold option, which iseasily derivable using the PDE, but a level of volatility needs to be assumed and there isno straightforward procedure for deriving this volatility from observed market volatilities

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of vanilla options at different strikes. The hedge package method recommended willconverge to this analytic solution as you increase the number of vanilla option hedgesused, provided all vanilla options are priced at a flat volatility (it is recommended that thiscomparison be made as a check on the accuracy of the implementation of the hedgingpackage method). But the hedging package method has the flexibility to price the gold-in-gold option based on any observed volatility curve (in fact you dont use the volatilitycurve but the directly observed vanilla option prices, so the pricing is not dependent onany option model).

2) The hedge package method gives an easier calculation of remaining risk than the analyticmethod, which requires Monte Carlo simulation of dynamic hedging.

3) The hedge package method gives an easy means of integrating gold-in-gold options intostandard risk reports, such as vega exposure by strike and maturity.

4) It is sometimes raised as an objection to the hedge package method that it requirestransactions in vanilla options at strikes in which there is no available liquidity. But anytrading desk sophisticated enough to deal in exotic options should be sophisticated enoughto have a system for hedging and pricing desired positions in vanilla options at illiquidstrikes with vanilla options at liquid strikes (methods we discussed in our section onVanilla Options Risk).

Th

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