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Valuation of Barrier Options in a Black Scholes Setup with Jump Risk

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Valuation of Barrier Options in a Black Scholes Setup with Jump Risk View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Research Papers in Economics Discussion Pap er No. B{446 Valuation of Barrier Options in a Black{Scholes Setup with Jump Risk Dietmar P.J. Leisen This version: January 12, 1999. Abstract This pap er discusses the pitfalls in the pricing of barrier options using approx- imations of the underlying continuous pro cesses via discrete lattice mo dels. These problems are studied rst in a Black{Scholes mo del. Improvements re- sult from a trinomial mo del and a further mo di ed mo del where price changes o ccur at the jump times of a Poisson pro cess. After the numerical diculties have b een resolved in the Black{Scholes mo del, unpredictable discontinuous price movements are incorp orated. Keywords binomial mo del, option valuation, lattice{approach, barrier option JEL Classi cation C63, G12, G13 Stanford University, Ho over Institution, Stanford, CA 94305, U.S.A., and University of Bonn, Department of Statistics, Adenauerallee 24{42, 53113 Bonn, Germany, email: [email protected] We are grateful to Dieter Sondermann and Holger Wiesenb erg for comments and assistance. Financial supp ort from the Deutsche Forschungsgemeinschaft, Sonderforschungsb ereich 303 at Rheinische Friedrich-Wilhelms-Universitat Bonn is gratefully acknowledged. 1 1 Intro duction Over time, barrier options have b ecome increasingly p opular to reduce the cost of plain vanilla options while incorp orating individual views of market participants concerning the asset evolution in an easy way: the payo of barrier options dep ends on whether the asset price path will not cross a presp eci ed b oundary. Of course, similar to standard options, they are also used to insure against price drops b elow the barrier. Standard barrier options do then either pay o a call or a put at the maturity date. For all these eight standard barrier options with single barrier, closed form solutions are available in the Black{ Scholes setup see Rubinstein and Reiner 1991 and Carr 1995. For double barrier options, however, analytical approximations are known only in some sp eci c cases see Kunitomo and Ikeda 1992. Numerical pro cedures haveto be applied to come up with prices, esp ecially in those generalizations of the Black{Scholes setup, in which jumps are p ossible. In this pap er we are interested in the \ecient" pricing of barrier options us- ing approximations of the underlying pro cess. It is well known that binomial mo dels su er from numerical de ciencies with barrier options: increasing the re nement, prices converge erratically in a saw{to oth manner to the contin- uous time price and even high re nements do not ensure adequate accuracy. Many authors addressed this problem and suggested adjustments. Ritchken 1995, and Cheuk and Vorst 1996 constructed trees where the no des lie on the barrier. Figlewski and Gao 1998 re ne the tree further at the barrier. Our improvements are related to the nite di erence approach see Boyle and Tian 1998: we start from a discretization of the asset space, instead of dis- cretizing time rst, as previous approaches did. With barrier options this is a straightforward way to have no des on critical levels by construction. We re- cover binomial mo dels with the sp eci c re nementofBoyle and Lau 1994. To incorp orate our approach in full generalitywe make use of trinomial mo dels. Since discretizing the asset space breaks up the time discretization, we al- low for random trading, similar to Leisen 1998b for American put options, and Rogers and Stapleton 1998 for barrier options. However, the latter did not recognize the numerical de ciencies resulting from the strike at maturity, which will be prevented in our trinomial mo del. We also di er from their approach by assuming that trading dates are the jump times of a Poisson pro cess, which results in simple valuation formulas. Our parameters are set in accordance with a convergence theorem of the pro cesses which also en- sures convergence of prices. The mo del smo othes the convergence structure even b etter. The order of convergence increases from 1=2 to 1 and removes the wavy patterns. Using extrap olation we are able to obtain even quadratic order. 2 After the numerical de ciencies have b een removed in the Black{Scholes setup, we turn to the framework of Merton 1976, adding a comp ound Poisson pro- cess to the Black{Scholes mo del. This mo dels the empirical observation of sudden strong price changes. An extension of binomial mo dels to this setup was prop osed by Amin 1993, which unfortunately inherits from there its poor convergence prop erties, esp ecially with barrier options. Our mo del with random jump times driven by a Poisson pro cess allows us in a simple way and intuitive way to incorp orate jumps. It inherits the extraordinary conver- gence prop erties from the trinomial mo del in its randomized version. Again, a theorem ensures convergence to the continuous time solution. The remainder of the pap er is organized as follows. In section 2 we present the jump{di usion setup as well as the barrier option contract. Section 3 explains the pitfalls in discretizing according to CRR. In section 4 these diculties are circumvented using a suitable trinomial mo del. This mo del is then random- ized. Section 5 incorp orates \strong" jumps. Section 6 discusses barrier option valuation and the price accuracy in our framework. Section 7 concludes the pap er. 2 The Setup On a probability space ; F ;P we study an economy B; S , consisting of the Bond B and the sto ck S . The interest rate r is assumed to b e constantover time B = exp frtg and the sto ck price evolves under the ob jective measure t P according to S = S G J ; 1 t 0 t t where G = exp ft + W g ; 2 t t N t Y V : 3 and J = i t i=1 N isaPoisson pro cess with constant parameter ,V a sequence of non{ t t i i + negative iid random variables, 2 IR, 2 IR . The pro cess G is the continuous pro cess known as geometric Brownian t t motion and has stationary, Gaussian returns. It was suggested by Samuelson 1965 to mo del the evolution of a sto ck. Since Black and Scholes 1973 it is used as one of the standard nancial mo dels. S evolves according to G until the next jump time of the Poisson pro cess at which N changes from, say, i to i +1. We then observe a p er{cent change V 1, i.e., the sto ck changes i value from S b efore the jump to S V . i 3 So, the two parts can be interpreted as follows: G mo dels the \typical" t t evolution of the sto ck under the \normal" arrival of information, whereas J mo dels jumps in the sto ck prices, due to some rare strong information t t sho ck. Since the Poisson pro cess is \memoryless," the exp ected time until the next sho ck o ccurs is equal to 1=, indep endent of current time. Merton 1976 studied this mo del under the assumptions that the V are lognormally i i distributed random variables and and N are sp eci c for the rm under t t consideration. The Black{Scholes setup is a so{called complete market, i.e., any contingent claim on the sto ck can be hedged see Due 1992. The no{arbitrage prin- ciple is sucient to price any claim. In the language of Harrison and Kreps 1979 this means that there is a probability measure Q, equivalent to P , and under which G =B is a martingale. Such a measure gives rise to a t t t linear pricing op erator; it is completely sp eci ed for the B; G market by G 2 = r =2. In our setup, however, we are in an incomplete market; the unforeseeable jump can not hedged. Under the assumption of no{arbitrage there isamultiplicity of equivalent martingale measure EMM under which agents do evaluate this risk. The mathematical nance literature suggests the use sp eci c EMM, de- rived from hedging criteria see Follmer and Sondermann 1986 and Follmer and Schweizer 1991. When our setup is used to mo del \market crashes," i.e., J represents jumps in the aggregate or market p ortfolio, the pro ce- t t dure taken in the mathematical nance literature seems appropriate. If the asset J would be traded in the market, the three assets B ; S; J would t t constitute a complete market. As it is, however, not traded, under the EMM Q chosen by the market for valuation, only S has to be a martingale, but t t J not; the description of J under P and the choice of Q together with t t t t the Girsanov{Theorem are equivalent. Our view on this is that of sp ecifying under an appropriate measure the pro cess J by t t N t Y E [V 1]t i J = J e V ; 4 t 0 i i=1 for some 2 IR, which will be explained b elow. Using the description of the t sto chastic exp onential we derive E [J =J ] = e see Protter 1990 or Jaco d t 0 and Shiryaev 1987. Using the Girsanov{Theorem as in the Black{Scholes setup gives then the EMM for the B ; S; J market as a change in the Wiener S 2 measure for which = r =2 .
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