Letting α = β, our positive process reduces to Geometric Brownian motion. I am working on a problem and had a quick question. f��=e и�L��?��D�|�:����ϳ�驤B �b�f��[�54�eMU��b vF�@� ?V:޻�J*�y�kV�Oz]�J�����jQZ.��� �q����eY�VS�1�2��(׽�.Z)������:}��Hw`O�>:?����N�� o�L��,KE�W������{����͛�g����{�����~�>��Uޘ]`�$? %�����ha���l��c��8~kZX6̻.Ɛ��p�p 9"'��$��ַ�I�,��ʔ�m����U)�t%j�nʄV�}H%��F�=J�S�@�����6�VRR�~�>M����EV��?0!d�E�']��N�E��K��)�$����Ct}��R�����Ne��R�� [�"��X�0�B;b5��T4y�Kn�P(���#��NJ�zI�N~xF�[9��_t �2IZg�j+W��-H�1�drW�υht3�&p�|� (Ov&�_�r*��j��-B J�����x��"����! :���-����9/��Z,�] Geometric Brownian motion is a mathematical model for predicting the future price of stock. 5 0 obj M���`|c�S����6 ͨ!�/�� �Zh�ƕR�c��|ר�ǖg�z����r�F��J�1�r�$j�m^�T�}��(S�Uiè�K�����O�Я�l6�����=���u�UpU)�_��D}A��2�ZG�@#��_8�l�!tɪ��t�9�$_�܀��Dyn�-=Z�Ғ���x4N��JƓ!^4�W��ڠ�4���ɦ $�D;�h�1��p��� ����Va�rE��Jfk���� ���Jn�hD�-���&����Y����`�n&M�y{� In the absence of dividends, the surplus of a company is modeled by a Wiener process (or Brownian motion) with positive drift. endobj stream 25 0 obj �ȹ��b��IDqP:�ăƘd�a���� YQ����o��,6h#\�@�;��eKVM;�̇7g:]1����� �ls`D��e6��צsU�G���\o@��d�R��"g3�C��_������!w�6p�Gğ����k��FBi/�"�ش�rN3�F�DMZj�|}�(51J��6,��G����S�7���y�ri�c��0r��2����a����`��Fj72��a��\�F���ʒ�?k�!�ے��: Wش�F�c�TB�:`pR��L��� 1. I understand that for Geometric Brownian Motion we use the formula: X t n = X t n − 1 + μ X t n − 1 Δ t + σ X t n − 1 ϵ n Δ t. To my understanding this is for modeling non dividend paying stocks. x��ZIs]�M;-���$��aAUb�� T�"d!$O�����s���ջbHYP�Q�o�7�o�n�ܩE���?/���������噖/��������ٽ/�Ψ%v���.��:-��B4K6���g��tPKN:Ÿy0KT�����Q-F�h��)�*��\ܿ>�_����hr^L��durч��x���h��q�'��Uf�y��hk��_��A|S�&������K�&﯅�3i�V�3�B(�α���������W"�VV����a��H��~��������8�R�$WJ��V{:S��W�*d�1��k^qu�)����p ���b]0�`��������98���>>�����I/*�Z�H��C!G�`��i���Ъ؆�������� ���~1>Μn. *w�濌P>���ʤ��wE(�_��"B9iٱ�T���a���P�n#>ռt�Z�c��@�6����c;Z���m� @�*��颫*s�9�6u�D�����#܎`�2q��1;E{���⎆咎`G?���t���F.ZDqKR���'J���f��ُ?8W؈YV�n9f����lh)�����Z��]�y�1;v?��ְn��r�tF�>I�)�=ޤ�Y�M�^ӣ��kz��c�֡�����|���]o�>?�6,��endstream 3344 Assuming a security’s dynamics are driven by these processes in risk neutral measure, we price several derivatives including vanilla, barrier and lookback options. �jgvyɀ�|��İ�X�c,��w8�%��7��]���ؐ��/!�Qz�K�j]��:?OB/T���v.g}��KR�����3��i�b�N���P�VuK��Q���gd�\�N�n��#DU��:�2Jſ�U�����U� j,�ݵ�\� geometric Brownian motion. ?ˣM��]�HJ�]�=yQ�*�~>=59��1��y�v�.țX���D�F|0��Nz�"�7�yb5 endobj A geometric Brownian motion is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion with drift. <> �m�g��/�T-NT��+7�3C��2$E�4o��62(��+5������F��i��׍T5�D�1�� ����1�3�Z�n+����f�SN|UNI^|1�n�r���]~Q]#e=\��3�wMoT���:� ˇmIq�R e��:�Mv) ��9E��oy�v��g�˙)g�C멮]�� �#�&���ӜT�? What we shall do instead is to take for the dividend process a Markov-modulated geometric Brownian motion, and derive the resulting dynam-ics of the stock price from an equilibirum analysis for a CRRA representative agent. They then evaluate the stock as the net present value of future dividends, but in contrast to what we do here, they compute the expectation in the original measure, and not in the risk-neutral measure aris- It is an important example of stochastic processes satisfying a stochastic differential equation; in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model. stream By adding a jump to default to the new process, we introduce a non-negative martingale with the same tractabilities. � �M�th�����l�$�#�t^����d��I���m�50F�f��Wc��n�"h�a���M0��h�d�Oi2�y�F>� �1�BD0�YD'�lc¼�q0�^�a�(%��! we model the dividend process as a geometric Browniam motion in the pricing measure; so if we take this model for the dividend process, we will come up with nothing new. n�1�+��&�Փ7��=�t}��&eT�;��)���k�E/�q���Zm��px}��� -~�3^[xu /2^�ó|�r�]+������7%��~��\����u��z�N[��R~s�������ޤ�to� qS�e� i���k��`��[��/z��f���Op��� ߏ��]���l2_���ր�+\@�N }�h]j�����=y56�0)uq`�]�c5f@�FU’����s�6̞�&:,�.~C�+P�|��Ԟ�}ׅ#���'���p�+r��S[{�CA$��fW"�aG��� ��G�V7��ܥ ����d����}DuIr�3;����M���WU;���OE.�䄠Aj��W$�`���S��LM�+Q���Z(�vF�iQ��$���I!ϣ�'���>�"m|�L +�caA>n��B:��f���v�a�h7�eм�D�*]h��*5�I�n� ��_����y��O��*�a�š?��%������ƥi���s��:����¬>�߿�6a3�(r{~׸�FnB�P�u����. %PDF-1.2 %�쏢 x��]I�\�q����9�=�S�K�DٲB�N"��� `H �zg֚U/���Z�o�%�/�̬��Jl�J���o��?��^>���L�\�^���ݓG��J�+)�h��z���|�� {���)V1���J�˕�m��_/7���7Y�3�qC.��?�i��,����Z,⭐~r��� 7���8�v��ļ��������D���X_V�-� ����4:��w Lȉ�&�:����7@�:�Ӯ��W����^� �� �r涛��;��`x�N�`j�[�:�mU��SA�ZIj=����cx�$h}s��[��3���!�Uk�m��e`''o6�+�Ȕ3�����=da~q_y��~k���=��:�d�Q���L�U�p�o�.Y�"�7�5>���-n��֊�a�����nf��*GW��`W���=�]��>JY�ʯp��H�nuZX��k�_�t���o�<2 ��y�R6����i�Ѭ�{yW;��f^f�]�^���&�N�=I|9�z��뇃Е� �-X�z�3us�;R�mw{%�U�����+�vvO�mg�S`_�l���SL˱++-�.���_uJq4�2W��F�xȢmx��#x#rD¶�N�5�ţGSKġ[g��Z�R�x�Q��@�"[�lF� ��8,HJm�Ki������h�l��2�=Ό�;t8����;�Vܝzv�&*�;�,r1Ƚ�����}ކ(�۴6�Ѫ(G��Ȟע�>mnIA�+F���_ Geometric Brownian Motion • Consider the geometric Brownian motion process Y(t) ≡ eX(t) – X(t) is a (µ,σ) Brownian motion. Geometric Brownian Motion with Dividends. • As ∂Y/∂X = Y and ∂2Y/∂X2 = Y, Ito’s formula (51) on p. 453 implies dY Y = µ+σ2/2 dt+σdW. 1 Geometric Brownian motion Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. <> �� �}� There are other reasons too why BM is not appropriate for modeling stock prices. Now dividends are paid according to a barrier strategy: Whenever the (modified) surplus attains the levelb, the “overflow” is paid as dividends to shareholders. 6 0 obj