{1}

 BLACK F, SCHOLES M. The pricing of options and corporate liabilities[J].

 Political Economy, 1973, 81: 637-659.
{2} BALLESTER C, COMPANY R, J\'{O}DAR L, et al. Numerical analysis and simulation of

 option pricing problems modeling illiquid markets[J]. Computers and Mathematics with Applications, 2010, 59(8): 2964-2975.
{3} LIU H, YONG J. Option pricing with an illiquid underlying

asset market[J]. Journal of Economic Dynamics and Control, 2005, 29:

2125-2156.
{4} COMPANY R, J\'{O}DAR L, PINTOS J R. Numerical analysis and

computing for option pricing models in illiquid markets[J].

Mathematical and Computer Modelling, 2010, 52: 1066-1073.
{5}BAKSTEIN D, HOWISON S. An arbitrage-free liquidity model with

observable parameters for derivatives[R]. Working paper,

Mathematical Institute, Oxford University, 2004.
{6}HOWISON S. Matched asymptotic expansions in financial

engineering[J]. Journal of Engineering Mathematics Computers, 2005,

53: 385-406.
{7}CASAB\'{A}N M C, COMPANY R, J\'{O}DAR L, et al. Numerical

analysis and computing of a non-arbitrage liquidity model with

observable parameters for derivatives[J]. Computers and Mathematics

with Applications. 2010, doi:10.1016/j. camwa. 2010.08.009.
{8}BARLES G, SONER H M. Option pricing with transaction costs and

a nonlinear Black-Scholes equation[J]. Finance Stoch, 1998, 2:

369-397.
{9} COMPANY R, NAVARRO R, PINTOS J R, et al. Numerical solution of

linear and nonlinear Black-Scholes option pricing equations[J].

Computers and Mathematics with Applications, 2008, 56: 813-821.

{10} COMPANY R, J\'{A}DAR L, PONSODAR E. Numerical solution of

Black-Scholes option pricing with variable yield discrete dividend

payment[J]. Banach Center Publ, 2008, 83: 37-47.
{11} COMPANY R, J\'{O}DAR L, PINTOS J R. A numerical method for

european option pricing with transaction costs nonlinear

equation[J]. Mathematical and Computer Modelling, 2009, 50: 910-920.
{12} COMPANY R, J\'{O}DAR L, PINTOS J R, et al. Computing option

pricing models under transaction costs[J]. Computers and Mathematics

with Applications, 2010, 59: 651-662.
{13} JANDA\v{C}KA M, \v{S}EV\v{C}OVI\v{C} D. On the risk-adjusted

pricing-methodology-based valuation of vanilla options and

explanation of the volatility smile[J]. J Appl Math, 2005(3):

235-258.