Geometric L evy Process Pricing Model

We consider models for stock prices which relates to random pro cesses with independent homogeneous increments Levy processes These models are arbitrage free but correspond to the incomplete nancial market There are many di erent approaches for pricing of nancial derivatives We consider here mainly the approach which is based on minimal relative entropy This method is related to an utility function of exponential type and the Esscher transformation of probabilistic measures Introduction We suppose that the nancial market consists of two assets bond Bt and stock St We use the notation r for the spot interest rate assuming that it is a constant Then under the assumption of continuous compounding the value of the bond at time t is Bt B expfrtg The classical di usion model Merton Black and Scholes for the process St is dSt Stdt StdWt where Wt is a standard Wiener process is the expected return and is the stock price volatility The solution of this equation is St S expf Wt gtg with g It is well know that contrary to aWiener process returns of stocks that is log St are neither Gaussian nor homogeneous and not having independent increments see e g Amaral et al In spite of these empirical observations the classical di usion model remains as a reference model due to its simplicity Trying to preserve this feature of simplicity one may keep the property of having independent homogeneous increments i e assuming that log St is a Levy process which could be a non Gaussian process So we will assume that St S expfZtg where Zt is a Levy process Z In particular it implies the distribution function of Zt belongs to the class of in nite divisible distributions see e g Bertoin or Sato Under this assumption the process Zt has the following representation Zt Wt Yt where Wt is a standard Wiener process Yt is a jump Levy process which is independent of Wt The process Yt has the following representation in terms of the counting Poisson measure N dt dx t x Rnf g generated by the jumps of Zt Yt bt Z t Z Rnf g xIfjxj gN ds dx Z t Z Rnf g xIfjxj g N ds dx dx ds Here dx is the so called Levy measure which satis es the following condi tion Z min x dx see e g or The characteristic function of Yt is given by E exp iuYt expft iu g where the cumulant function iu can be written in the form iu iub Z exp iux iuxIfjxj g dx it is a variant of the so called Levy Khintchine formula We will call the characteristic v dx b in the above representation the triplet of Zt Slightly di erent terminologies are used in monographs and Remark Chan considered a stock price model in a di erent form dSt Stdt St dXt where Xt is a Levy process satisfying some conditions This model is actu ally equivalent to the model under the assumption that jumps Xt which is necessary for positiveness of prices St To our mind it is more con venient to work with characteristics of the process Zt from Some particular cases Compound Poisson models