On a generalized Cox-Ross-Rubinstein option market model (Q5946858)

This paper deals with an option market defined by the two following processes: a risk free asset process given by \(B_{n}=B_{n-1}(1+r_{n})\), where \(B_0\) is known and \(r_{n}=r_{n}(N)\), \(n=1,2,\ldots,N\); a stock price process \(S_{n}=S_{n-1}(1+\rho_{n})\), where \(S_0\) is known and \(\rho_{n}\) satisfies \(p_{k}=P\{\rho_{k}=u_{k}\}\) and \(q_{k}=P\{\rho_{k}=d_{k}\}=1-p_{k}\), \(-1<d_{k}<u_{k}\), \(u_{k}= u_{k}(N),\;\rho_{k}=\rho_{k}(N),\;p_{k}=p_{k}(N)=1-q_{k}(N)\). The authors prove that under some conditions on \(u_{k}, d_{k}, p_{k}\) the random variable \(\ln(S_{n}/S_0)\) will be asymptotically normal as \(N\to+\infty\). The approximation of rational price of standard European call and put options is obtained and the functional convergence in the space \(D\) of cadlag functions with Skorokhod's metric is proved. This convergence results are used for hedging some contingent claim.