Rainbow options. A rainbow is an option on a basket that pays in its most common form, a nonequally

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1 Raibow optios INRODUCION A raibow is a optio o a basket that pays i its most commo form, a oequally weighted average of the assets of the basket accordig to their performace. he umber of assets is called the umber of colours of the raibow. At maturity, the assets are sorted by their performace. For istace, a raibow call with weights 50%, 30%, 0%, with a basket icludig a equity stock idex, a bod idex ad a foreig currecy, pays 50% of the best retur (at maturity) betwee the equity stock idex, the bod idex ad the foreig currecy, 30% of the secod best ad 0% of the third best. Oe of the commo features of raibow optios is to take a average for the uderlyig, as the Asia versio of the raibow is ofte cheaper tha the vailla oe. Also exotic raibow are packaged for retail busiess as structured otes with callable ad putable feature as well as capital guaratee features. A callable raibow ote allows the ote issuer to call back the ote while a putable oe allows the ote buyer to sell it back. Callable putable versios are becomig also commo as they allow flexibility o both sides. he ame of raibow comes from Rubistei (1991), who emphasises that this optio was based o a combiatio of various assets like a raibow is a combiatio of various colours. More geerally, raibow optios are multiasset optios, also referred to as correlatio optios. Raibow ca take various other forms but the combiig idea is to have a payoff that is depedig o the asset o the assets sorted by their performace at maturity.

2 Whe the raibow oly pays the best (respectively worst) performig asset of the basket, it is also called best-of, (respectively worst-of). Other popular optios that ca be reformulated as a raibow optio are spread ad exchage optios. Other variatios of the raibow (give for oly the simple case of payig the best performig asset 1 ) are described i table 1. MOIVAIONS Because of the accelerated cross-market itegratio ad greater globalisatio of fiacial markets, traders, ivestors or speculators have bee more ad more iterested i multi-assets also called correlatio products. Moreover, optios o may assets are appealig to ivestor for the followig reasos: Natural risk diversificatio. his risk diversificatio is all the more proouced that the assets take for the multi assets products are from very differet ad urelated markets. ypical example is to take a equity stock idex, a bod idex ad a currecy to have exposure to the equity the bod ad the foreig exchage markets. Cost efficiecy: Multi-asset products are cheaper tha the correspodig sum of the idividual optios. he effect of correlatio makes the product competitive. Better hedge for portfolio exposure. Correlatio products are better hedge for portfolios of ivestors, traders that icludes hudreds of ames or for corporate compaies that have risks to various market istrumets. Hedge agaist correlatio. his may seem stupid statemet, but correlatio risk is very hard to risk maage ad correlatio products eable 1 hese optios ca also be structured to be payig a weighted average of the first best, secod best etc. hese optios ca be call or put ad ca also be o the first worst, secod worst etc.

3 to lock up the implied correlatio. Proper delta-hedgig eables also to lock up the historical covariace. Raibow optio offers the additioal advatage to provide a weighed average o the best or worst performig assets. he best versio supplies attractive returs, while the worst versio is ofte very cheap optio. Raibow optio are either sold to ivestor as a speculatio product or packaged as a structured ote for retail market providig attractive multi-asset liked products. PRICING AND RISK MANAGEMEN Like ay complex correlatio products, the mai challege for pricig ad risk maagig raibow optios is the absece of orgaised markets. A price is therefore determied iterally after a close study of the model ad the associated risk. Mai risks of the raibows are: Correlatio risk betwee the various uderlyigs: various approaches to be take: stadard ad aïve approach of costat correlatio betwee Browia motio, time depedet correlatio ad stochastic correlatio, coitegratio ad other statistics tools to capture more robust measure of correlatio, copula to cotrol the measure depedece betwee the assets. Correlatio has a positive effect o raibow overall volatility but egative effect o the forward value of the raibow. Depedig o the maturity ad volatility of the uderlyig assets as well as the weights of the raibow, the optio ca be log, flat or short correlatio.

4 Smile risk: the hedge of the raibow uses various strikes. his eeds to be correctly represeted i the modellig of the assets. Quato risk for raibow with asset deomiated i other currecy tha the oe of the optios. his ca dramatically chaged the value of the raibow as the forwards are substatially modified. Although the academic literature has targeted successfully the case of raibow optio i Black Scholes - work of Stulz (198) for two assets, geeralisatio by Johso (1987) for optios with more assets usig multidimesioal ormal cumulative fuctios, optio pricig summary by Rubistei (1991) little has bee said for model with smile ad advaced correlatio models. Mote Carlo methods (with America Mote Carlo method to tackle backward lookig structures) are the most commo way to umerically evaluate raibow optios with a smile model. Raibow ca be see as a dyamic basket as the relative weight of each uderlyer ca chage with relative levels. A static hedge with a basket optio may perform poorly because of the various switch optios (optio to switch the weights betwee the iitial oes determied by the forwards ad other weights). Aother way of lookig at it is to see the Raibow as a basket optio with kow weights that are effective accordig to the relative performace of the uderlyig assets. his ca be see as the product of a basket optio time

5 digital optios. he digital optios idicate that we are sesitive to the spread volatility betwee the various uderlyig assets. Whe lookig at the raibow i terms of digital optios, it is appropriate to compute the rakig probability of each asset. hese rakig probabilities are the probability for each asset to be the most performig, secod most performig ad so o i order to assess the future compositio of the raibow basket. Part of the risk maagemet tool is the to provide the sesitivity of these probabilities to the various market ad model parameters like the forward value of the assets as well as their volatility ad correlatio. Name Payoff Multi-strike raibow optio Max 1 ( S K S K,..., S K ) 1, Pyramid raibow optio 1 Max( S K1 + S K S K K ) Madoa raibow optio 1 ( ) ( ) ( ) Max S K + S K S K K, 0 able 1: Variatio aroud the raibow optio 1 Etry category: optios Scope: correlatio products, Related articles: exotic optios, correlatio, copulas, multi-asset optios.

6 Eric Behamou Swaps Strategy, Lodo, FICC, Goldma Sachs Iteratioal he views ad opiios expressed herei are the oes of the author s ad do ot ecessarily reflect those of Goldma Sachs