Option Pricing

IEOR4007 Homework #5

1. Option pricing in incomplete markets

Matlab codes:

close all

clear all

clc

%% Data

% number of assets

n = 10;

% number of states

m = 20;

% price today

p = [92  24  68  60  81  81  49  2   84  34]';

% strikes

strike = [100 98 101];

% S matrix

S = [117  151  176  141  34   78   178  175  78    170   11    68    155   2 27   39      38   115  52   38

1     35   21   43   22   20   39   25   10    31    39 1    32    18 39   24     33   20   15   9

24    83   37   66   19   85   46   105  104   72    61 109  100   79 99   80     42   36   42   65

71    31   82   56   36   69   54   44   68    61    78 93   51    86 17   96     27   25   85   72

25    3    160  36   54   118  51   84   12    176   104    76   92    60 78   41     104  136  95   115

32    58   120  104  71   87   121  9    92    8     64 47   133   3 117  148  151  120  67  76

20    59   30   88   67   38   68   25   41    85    63 20   77    58 13   19     56   58   34   53

2     1    1    2    1    2    3    1    1     3     3  1    1     1 3    1   2    3    3    3

14    75   73   58   26   111  115  120  79    91    20 74   118   147 45   42   142  39   132   149

21    22   5    7    58   17   76   16   16    89    40 31   28    33 35   53     11   4    41   78]';

%% compute a strictly positive state price deflator

f = zeros(1,m);

pi = linprog(f,[],[],S',p,zeros(1,m))

%% check if the call is attainable

K = 100;

payoff = max(S(:,1)-K,0);

equation = [S payoff];

rank_coefficient = rank(S)

rank_augmented = rank(equation)

if (rank_coefficient == rank_augmented)

    fprintf('security attainable\n\n')

else

    fprintf('security not attainable\n\n')

end

%% upper/lower bound and sensitivity analysis

% upper bound

[theta,~,~,~,u] = linprog(p,-S,-payoff);

upper_bnd = theta'*p

tol = 1e-6;

index = find(abs(u.ineqlin) > tol);

b99 = max(S(:,1)-99,0);

theta99 = S(index,:)\b99(index);

checksens_upper = (min(S*theta99 - b99)>=-tol)

new_upper_bnd = upper_bnd + (b99-payoff)'*u.ineqlin

% lower bound

[theta,~,~,~,u] = linprog(-p,S,payoff);

lower_bnd = theta'*p

index = find(abs(u.ineqlin) > tol);

theta99 = S(index,:)\b99(index);

checksens_lower = (max(S*theta99 - b99)<=tol)

new_lower_bnd = lower_bnd + (b99-payoff)'*u.ineqlin

Output:

Optimization terminated.

pi =

    0.1443

    0.0454

    0.0256

    0.0318

    0.1796

    0.0658

    0.0121

    0.1108

    0.0301

    0.0001

    0.0261

    0.0433

    0.0296

    0.0586

    0.0935

    0.0066

    0.0715

    0.0530

    0.0646

    0.0318

rank_coefficient =

    10

rank_augmented =

11

security not attainable

Optimization terminated.

upper_bnd =

   28.2009

checksens_upper =

     1

new_upper_bnd =

   28.6513

Optimization terminated.

lower_bnd =

   13.4915

checksens_lower =

     1

new_lower_bnd =

   13.8390

2. Pricing a gas supply contract