Introduction Deregulation i creaseofvolatilityinfinancialprices technologicalprogressesin

Description

Introduction Deregulation, i creaseofvolatilityinfinancialprices, technologicalprogressesin informationsystemsandadvancesinfinancialtheoryarethemainreasonsofthe exponentialgrowthofderivativesmarkets: options, futures, swapsandmanyothers. The fiercecompetitionbetweenbanksandtheireagernesstodevelopnewproductscloseto theirclients'sneeds, makethemveryinnovative. Beinggenerallymoreinvolvedthan standardones, thenewfinancialproductsareoftendifficultoprice. Although sophisticatedmethodsareinvolvedintheirmodelling, asimplifiedeconomicenvironment isset, forexampleinthevaluationofcomplexderivativesonequityaflattermstructureof interestratesisassumed. Forshortermproductsthishypothesiscanbeconsideredasa reasonableapproximation. Formediumorlongtermcontractshisassumptionshouldnot bemadeandobviouslycan'tholdforinterestratederivatives. In thispaperwefocusour analysisonthevaluationoffinancialcontractswithbarriersi.econtractwhosepayoffs dependon, wetherornotthepriceoftheunderlyingassetbreachesapre-specifiedbarrier, fromaboveorfrombelow. Thesecontractscalledbarrieroptionshaveencountereda tremendoussuccessespeciallyinforeignexchangemarketsandonalessextentin fixed-incomemarketsandinequitymarkets. Undertheassumptionofauniqueand constantinterestrateclosedformsolutionsweregivenbyMerton[1973] fordownandout calls, byRubinsteinandReiner[1991] forvanillabarrieroptions. GemanandYor [1996] priceddoublebarrieroptions, Chesney, JeanblancandYor [1997] introducedparisian optionswhosepayoffsdependonthetimespentaboveorbelowthebarrier, morerecently Linetsky[1999] pioneeredstepoptions. Thispaperis, asfarasweknow, thefirst, in continuoustime, totakeintoaccountanon-flattermstructureofinterestrates. Todoso, weuseaonefactorgaussianHeathJarrowMorton[1992] model. Inourframeworkwe dealwiththreesourcesofuncertainty, thefirstonecomesfromthetermstructureitselfthe secondonefromthepriceoftheunderlyingassetandthethirdfromthebarriergivenbya secondaryunderlyingasset. Weillustrateourapproachinthepricingofatypeofstructured product: sharkoption. A structurenotelinksitsfinalpayoffonanindex, astockonefor instance, andguaranteesaminimumcapitalatmaturity. Capsandfloorsareverypopular contractswhichpermitostoptheeffectsofanincreaseordecreaseofaninterestrate, we showhowthesharkoptioncanbeusedtovalueexoticcaps. Thepaperisorganizedas follows. InSection2weexplainwhatisasharkindex. InSection3weexaminethecaseof sharksonequity, Section4isdevotedtosharkoptionsoninterestrates, Section5givesan applicationoftheseoptionstoexoticcaps. Weintroducefinancialcontractswithtwo underlyingassets, oneofthembeingthebarrierinSection6. Thesection7summarizesour results. TheappendixpresentsFortet'sequationandderivetherecursivealgorithmto approximatehefirstpassagetimetothebarrier. Shark Index In itsmostbasicform, asharkoptionisanoptionwhoseholderisentitledtoreceivea rebateassoonastheunderlyingindexhitsabarrierandaEuropeanpayoffotherwise. The latterdependsonthevalueoftheunderlyingindexatexpiryandmaytaketheformofa Europeancallorafunctionofit, astheexamplebelowwill show. Theunderlyingindex maybeafinancialasset, aninterestrate, anexchangerateoranequityindex. Here, we assumethatpaymentsarealwaysettledatexpiry, whichusuallyrangesfromsixmonths tofiveyears. Thepresenceofabarrierlowersthepremium, comparedtovanillaoptions. Thebarriercanbehitfrombeloworfromaboveandcanbeknock-inorknock-out. It can alsobeconstant, deterministicorstochastic. Letusgiveanexampleofasharkoption: a mediumtermstructurednotehasatwo-yearmaturityandtheinvestor(thepurchaserofthe

• thetermstructureisgiventhroughdefault-free zero-couponbonds


• t?


• inthet-forward-neutraluniverseweusethemartingale propertyoftherelativeprice


• proposition theshark'sarbitragequilibriumpriceis


• dst st


• letqt denotethet-forward


• theshark'spayoffatmaturityt

Informations

• Publié par: pefav
• Nombre de lectures: 7
• Langue: English

Extrait

Introduction Deregulation, increase of volatility in financial prices, technological progresses in information systems and advances in financial theory are the main reasons of the exponential growth of derivatives markets: options, futures, swaps and many others. The fierce competition between banks and their eagerness to develop new products close to their clients’s needs, make them very innovative. Being generally more involved than standard ones, the new financial products are often difficult to price. Although sophisticated methods are involved in their modelling, a simplified economic environment is set, for example in the valuation of complex derivatives on equity a flat term structure of interest rates is assumed. For short term products this hypothesis can be considered as a reasonable approximation. For medium or long term contracts this assumption should not be made and obviously can’t hold for interest rate derivatives. In this paper we focus our analysis on the valuation of financial contracts with barriers i.e contract whose payoffs depend on, wether or not the price of the underlying asset breaches a pre-specified barrier, from above or from below. These contracts called barrier options have encountered a tremendous success especially in foreign exchange markets and on a less extent in fixed-income markets and in equity markets. Under the assumption of a unique and constant interest rate closed form solutions were given by Merton [1973] for down and out calls, by Rubinstein and Reiner [1991] for vanilla barrier options. Geman and Yor [1996] priced double barrier options, Chesney, Jeanblanc and Yor [1997] introduced parisian options whose payoffs depend on the time spent above or below the barrier, more recently Linetsky [1999] pioneered step options. This paper is, as far as we know, the first, in continuous time, to take into account a non-flat term structure of interest rates. To do so, we use a one factor gaussian Heath Jarrow Morton [1992] model. In our framework we deal with three sources of uncertainty, the first one comes from the term structure itself the second one from the price of the underlying asset and the third from the barrier given by a secondary underlying asset. We illustrate our approach in the pricing of a type of structured product: shark option. A structure note links its final payoff on an index, a stock one for instance, and guarantees a minimum capital at maturity. Caps and floors are very popular contracts which permit to stop the effects of an increase or decrease of an interest rate, we show how the shark option can be used to value exotic caps. The paper is organized as follows. In Section 2 we explain what is a shark index. In Section 3 we examine the case of sharks on equity, Section 4 is devoted to shark options on interest rates, Section 5 gives an application of these options to exotic caps. We introduce financial contracts with two underlying assets, one of them being the barrier in Section 6. The section 7 summarizes our results. The appendix presents Fortet’s equation and derive the recursive algorithm to approximate the first passage time to the barrier. Shark Index In its most basic form, a shark option is an option whose holder is entitled to receive a rebate as soon as the underlying index hits a barrier and a European payoff otherwise. The latter depends on the value of the underlying index at expiry and may take the form of a European call or a function of it, as the example below will show. The underlying index may be a financial asset, an interest rate, an exchange rate or an equity index. Here, we assume that payments are always settled at expiry, which usually ranges from six months to five years. The presence of a barrier lowers the premium, compared to vanilla options. The barrier can be hit from below or from above and can be knock-in or knock-out. It can also be constant, deterministic or stochastic. Let us give an example of a shark option : a medium term structured note has a two-year maturity and the investor (the purchaser of the

shark) has 100% of his capital guaranteed with a linear link to an Equity Index; however this link is cut as soon as the index growth is equal or greater than 35 % during the shark’s life, in which case the investor receives 110% of his initial investment in the end. In formal terms the investor receives at expiry time T : S T ? S 0 Þ + if S M D 1 + Ý S 0 max < Ý 1.35 Þ D S 0 M D 1.1 otherwise Where : M is the capital invested at time 0, S t is the index at time t or the underlying asset’s price at time t , S max is the maximum of S during ß 0, T à . We call this structured product standard shark . Options on equity Let us consider a financial market with a primary asset on which a shark option is written. It is assumed that the underlying asset price follows a geometric brownian motion. The term structure of interest rates is described by a Heath, Jarrow and Morton one-factor model with a deterministic volatility structure, chosen to be either linear or exponential, in order to ensure the Markov property of the instantaneous interest rate.(Cf. El Karoui and Rochet [1989]). The market is complete and frictionless. The uncertainty is modelled by a filtered space Ý I , á F t â t ³ 0 , E Þ where I is the usual fundamental space, á F t â is the filtration collecting all the information available at time t , generated by brownian motions, and E is the historical probability measure. Trading takes place continuously and the prices of all assets follow correlated diffusions. The term structure is given through default-free zero-coupon bonds: P Ý t , T Þ is the price at time t of such a bond maturing at T . Its dynamics is given by the stochastic differential equation : Þ = d P P Ý Ý t , t , T T Þ J Ý t , T Þ dt ? a P Ý t , T Þ dZ 1 Ý t Þ J Ý t , T Þ is its expected return, a P Ý t , T Þ its volatility, and Z 1 Ý . Þ is a standard brownian motion. The asset price at time t , denoted by S t , is modelled by a geometric brownian motion : = W d S S t t dt + a dZ 2 Ý t Þ W and a are respectively the expected return and volatility of the asset, Z 2 Ý . Þ is a standard brownian motion correlated to Z 1 Ý . Þ and we write dZ 1 dZ 2 = _ dt .These dynamics are given in the historical universe. Using standard results from the risk-neutral analysis, we know from Harrison and Pliska that there exists a unique probability measure Q under which the discounted price of any security is a Q -martingale. After decorrelating the brownians, we can write : å d P P Ý Ý t t , , T T Þ Þ = r t dt ? a P Ý t , T Þ dZ 1 Ý t Þ and for the asset’s price : dS t = å S t r t dt + a _ dZ å 1 Ý t Þ + 1 ? _ 2 dZ 2 Ý t Þ å å Z 1 Ý . Þ and Z 2 Ý . Þ are now two uncorelated Q -brownian motions. Using the fundamental