Accurate and Efficient Computations of the Greeks for Options Near Expiry Using the Black-Scholes Equations

Algorithm 1

Algorithm of adaptive time-stepping strategy.

Require: Set the initial conditions and , the expiry time , the maximum and minimum

time steps and , tolerance tol, truncation error , safety

factor , time , iteration number .

 While do

  if then

   Set .

  else

   Set .

  end if

  if then

   Set .

   Solve (4) with time step and set its approximation as .

  end if

  Solve (4) twice with time step and set its approximation as .

  Solve (4) with time step and set its approximation as .

  Calculate where and is the total number of .

  while do

   Set .

   if then

    Set .

   end if

   Solve (4) twice with and set its approximation as .

   Solve (4) with and set its approximation as .

   Calculate

  end while

  Set .

  Set and .

end while