| We present option pricing models, for European call options, both in discrete-time (Chapters II-V) and in continuous-time (Chapters VI-XI) trading markets, under the assumption that transaction costs (commission fees), incorporated into the models through cost functions, have to be paid. In the following N denotes the number of shares of stock held and S or P the price of a share of stock.; In Chapter II, we present a binary discrete-time model, with one risky and one riskless asset, with cost function{dollar}{dollar}csb{lcub}ij{rcub} {lcub}:{rcub}{lcub}={rcub} alpha(leftvert Nsb{lcub}ij{rcub}-Nsb{lcub}i-1k(j){rcub}rightvert)+deltavert Nsb{lcub}ij{rcub}-Nsb{lcub}i-1k(j){rcub}vert Ssb{lcub}ij{rcub},eqno (1){dollar}{dollar}for time {dollar}i=0... n{dollar}, state {dollar}j=1...2sp{lcub}i{rcub}{dollar}, {dollar}k(j)=j/2{dollar} if {dollar}j mod 2=0{dollar} and {dollar}k(j)=(j+1)/2{dollar} if {dollar}j mod 2=1{dollar}. Moreover, {dollar}alpha(0)=0{dollar} and {dollar}alpha(x)=a>0{dollar} for {dollar}xnot=0{dollar} and {dollar}deltain(0{dollar}, 1). In Chapter III, we develop the corresponding ternary model, where the cost function is given by (1), {dollar}k(j)=j/3{dollar} if {dollar}j mod 3=0{dollar}, {dollar}k(j)=(j+1)/3{dollar} if {dollar}(j+1) mod 3=0{dollar} and {dollar}k(j)=(j+2)/3{dollar} if {dollar}(j+2) mod 3=0{dollar}. In Chapter IV, we study a binary model, with more than one risky assets and a riskless one and cost function given by (1). In Chapter V, the cost function is defined by{dollar}{dollar}csb{lcub}ij{rcub} {lcub}:{rcub}{lcub}={rcub} alpha(leftvert Nsb{lcub}ij{rcub}-Nsb{lcub}i-1k(j){rcub}rightvert)+delta(leftvert Nsb{lcub}ij{rcub}-Nsb{lcub}i-1k(j){rcub}rightvert Ssb{lcub}ij{rcub})spgamma,eqno (2){dollar}{dollar}with the notation of (1) and {dollar}gammainlbrack 0{dollar}, 1).; In Chapter VI, we develop a continuous-time model with one risky and one riskless asset. Let {dollar}tausb{lcub}i{rcub}, i=1... Nsp*{dollar} denote the time instants at which commission fees are paid. The individual has to pay a fixed amount of {dollar}thinspace{dollar} beta{dollar} at {dollar} tau sbi{dollar}. In Chapter VII, the amount paid at {dollar} tau sbi{dollar} is{dollar}{dollar}betasb{lcub}i{rcub} {lcub}:{rcub}{lcub}={rcub} lambdachi,eqno (3){dollar}{dollar}where {dollar} chi:= vert N sbsp tau sbi+P sb tau sbi-N sbsp tau sbi-P sb tau sbi vert+ left vert N sbsp tau sbi-P sb tau sbi-N sbsp tau sbi-1+P sb tau sbi-1 right vert{dollar} and {dollar} lambda in(0, 1).{dollar} In Chapter VIII, we study a continuous time model with more than one risky assets and a riskless one. The cost functions are as in Chapters VI and VII. In Chapter IX, the commission fees paid are{dollar}{dollar}betasb{lcub}i{rcub} {lcub}:{rcub}{lcub}={rcub} lambdachispGamma,eqno (5){dollar}{dollar}where {dollar} Gamma in(0, 1).{dollar} In Chapter X, the cost incurred is{dollar}{dollar}betasb{lcub}i{rcub} {lcub}:{rcub}{lcub}={rcub} alphasb0(chi)+lambdachi,eqno (6){dollar}{dollar}where {dollar} alpha sb0(x)= alpha sb0>0{dollar}, if {dollar}x not=0{dollar} and {dollar} alpha sb0(0)=0.{dollar} Finally in Chapter XI, the cost function is given by{dollar}{dollar}betasb{lcub}i{rcub} {lcub}:{rcub}{lcub}={rcub} alphasb0(chi)+lambdachispGamma,eqno (7){dollar}{dollar}where {dollar} Gamma in(0, 1).{dollar}... |
