Финансовая математика - Зуев В.П., Пыжев И.С
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Hkgh\gu_ aZ\bkbfhklb b iZjZf_lju ijhkluo b keh`guo nbgZgkh\uo hi_jZpbc ijb gZqbke_gby ijhp_glh\ |
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<b^ hi_jZpbb ih gZqbke_gbx ijhp_glh\ |
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Ijhklu_ ijhp_glu |
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S = P(1+ ni ) |
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P = S (1 − nd |
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d |
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G = − 3 |
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3 |
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I = |
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6 |
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P − 23 |
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LZ[ebpZ |
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Wd\b\Ze_glgu_ aZ\bkbfhklb f_`^m jZaebqgufb \b^Zfb ijhp_glguo klZ\hd |
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<b^u |
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Ijhklu_ ijhp_glu |
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Keh`gu_ ijhp_glu |
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klZ\hd |
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glu |
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i = n 1+ ni |
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−1 |
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d = 1− n |
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1 |
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1− mn |
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1 + ni |
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Ijhklu |
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i = n |
1 |
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1 |
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d =1− n 1 |
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= m(1 − mn 1 − nd |
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s |
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j = m mn |
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n |
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Keh`gu |
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1− (1− d ) |
n |
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1 |
− (1− d ) |
n |
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1− d |
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1− f |
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LZ[ebpZ |
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Hkgh\gu_ aZ\bkbfhklb ^ey ki_pbZevguo nbgZgkh\uo hi_jZpbc k\yaZgguo k gZqbke_gb_f ijhp_glh\ |
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<b^ hi_jZpbc |
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<b^ ki_pbZevghc nbgZgkh\hc hi_jZpbb |
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ih gZqbke_gbx |
GZeh] gZ ijhp_glu |
Bgneypby |
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Dhg\_jkby \Zexl |
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ijhp_glh\ |
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KD<→ |
Jm[→ Jm[→ |
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KD<→ |
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1 + ni |
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glu |
SC |
= P[1+ n(1− g )is ] |
C = P (1+ h100s |
)n |
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(1+ nis )(1+ h100)n −1 |
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ijhp |
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1 |
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s KD< |
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Ijhklu |
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G = Pnis g |
(1 |
1+ nr |
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1 |
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is = |
100 |
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(1+ i)n |
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SC =P[(1−g)(1+i)n +g] |
C = P (1+ h |
100 |
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Keh`gu ijhpgluih wnn dlb\ghcklZ\d |
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r = i + |
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100 |
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G = P[(1 + i)n |
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i = |
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Ihlhdb ieZl_`_c |
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DeZkkbnbdZpbhggu_ ijbagZdb |
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DeZkkbnbdZpbhggu_ ijbagZdb |
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- \k_ qe_gu jZaf_ju hl^_evguo ieZl_`_c |
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ke_^mxl mklZgh\e_gghfm ijZ\bem |
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g_ ke_^mxl mklZgh\e_gghfm ijZ\bem |
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- bgl_j\Zeu f_`^m ieZl_`Zfb h^bgZdh\u |
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Qe_g j_glu |
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iheh`bl_evgu_ \_ebqbgu |
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ieZl_` |
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NbgZgkh\uNbgZgkh\u__ jj__gluglu |
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Ih h[s_fm |
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Dhebq kl\h ieZl ` c Klhbfhklgub b gZqbke gbc \ ]h^m GZjZs ggZy kmffZ kjhdbgluj |
OQ = OQ |
65 |
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S = P = L |
S= P > |
S = P → ∞ 5=6 H |
S > P = S> |
S> P = S OQPM |
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S > P > P ≠ S |
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,9K:FHKLHYL?EVG:Y IH>=HLH<D: KLM>?GLH<
JZkq_lgh_ aZ^Zgb_
Ij_^ijbylb_ \ l_q_gb_ ^\mo e_l hkms_kl\ey_l ijhba\h^kl\_ggmx b dhf- f_jq_kdmx ^_yl_evghklv kh\_jrZ_l hi_jZpbb k p_ggufb [mfZ]Zfb bf__l dj_- ^blu \ [Zgd_ _`_d\ZjlZevgh gZqbkey_l b mieZqb\Z_l gZeh]b GZqZeh \k_o \b- ^h\ ^_yl_evghklb ij_^ijbylby±] hdhgqZgb_ ]
DZ`^uc \b^ ^_yl_evghklb oZjZdl_jbam_lky ke_^mxsbfb ihdZaZl_eyfb
1.< ijhba\h^kl\_ \j_fy hl ijbh[j_l_gby j_kmjkh\ ^h ihemq_gby ^_- g_] hl j_ZebaZpbb ijh^mdpbb khklZ\ey_l t1 ^g_c \ dZ`^uc ijhba\h^kl\_gguc pbde gZijZ\ey_lky J luk jm[ j_glZ[_evghklv ijhba\h^kl\Z khklZ\ey_l j ijh- p_glh\
2.< dhff_jq_kdbo hi_jZpbyo lhj]h\uc h[hjhl kh\_jrZ_lky aZ t2 ^g_c lhj]h\Zy gZ^[Z\dZ khklZ\ey_l Z ijhp_glh\ i_j\hgZqZevgh \ lhj]h\u_ hi_jZpbb gZijZ\e_gh L luk jm[ b kj_^kl\Z ba lhj]h\eb g_ baufZxlky
3.Hi_jZpbb k p_ggufb [mfZ]Zfb ih\lhjyxlky dZ`^u_ t3 ^g_c b \ wlb hi_jZpbb gZijZ\ey_lky < luk jm[ ^hoh^ghklv h^ghc hi_jZpbb b ijhp_glh\
4.Kh]eZkgh dj_^blgh]h ^h]h\hjZ gZ kmffm D luk jm[ kjhdhf k
] ih ] ij_^ijbylb_ \ dhgp_ dZ`^h]h f_kypZ \uieZqb\Z_l ijhp_glu ih ijhklhc ijhp_glghc klZ\d_ 28 ]h^h\uo
KZfhklhyl_evgh ^hihegblv ij_^hklZ\e_ggmx bgnhjfZpbx g_h[oh^bfu- fb qbkeh\ufb ^Zggufb b hp_gblv
Z \Zeh\hc ^hoh^ \ujmqdm ij_^ijbylby aZ ]
[ ^hoh^ghklv ij_^ijbylby ih dZ`^hfm \b^m ^_yl_evghklb b \ p_ehf ih ij_^ijbylbx ^hoh^ghklv hp_gb\Zlv \ \b^_ ijhklhc ]h^h\hc ijhp_glghc klZ\- db aZ ]
\ gZjZs_ggmx kmffm hlqbke_gbc ij_^ijbylby \ [x^`_lu \k_o mjh\g_c b \g_[x^`_lgu_ nhg^u aZ ^\Z ]h^Z ^_yl_evghklb ij_^ijbylby Ij_^iheZ]Z_lky qlh hlqbke_gby ij_^ijbylb_f ^_eZxlky ijbf_jgh jZ\gufb kmffZfb _`_d\Zj- lZevgh \k_ hlqbke_gby khklZ\eyxl n ijhp_glh\ hl ^hoh^Z ijb[ueb ij_^ijb- ylby ^hoh^ghklv [x^`_lh\ b \g_[x^`_lguo nhg^h\ jZ\gZ klZ\d_ j_nbgZgkb- jh\Zgby P; JN ]h^h\uo
AZ^Zqb ^ey kZfhklhyl_evgh]h j_r_gby
1. Ij_^ijbylb_ aZljZqb\Z_l gZ ijhba\h^kl\h ijh^mdpbb feg jm[ I_jbh^ \j_f_gb hl fhf_glZ \eh`_gby ^_g_`guo kj_^kl\ \ ijhba\h^kl\h ^h ih- emq_gby \ujmqdb hl j_ZebaZpbb ijh^mdpbb jZ\_g ^_gv >hoh^ghklv ijhba- \h^kl\Z \ \b^_ ijhklhc ijhp_glghc klZ\db wd\b\Ze_glgZ ]h^h\uo Hij_^_-
14
eblv kmffm ^_g_`guo kj_^kl\ \ h[hjhl_ ij_^ijbylby d dhgpm i_j\h]h ]h^Z jZ- |
Z Hij_^_eblv ^hoh^ghklv nhg^Z \ \b^_ ijhklhc ijhp_glghc klZ\db |
|
[hlu _keb |
[ Hij_^_eblv ^hoh^ nhg^Z |
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Z ijb[uev hl j_amevlZlh\ ijhba\h^kl\Z jZkoh^m_lky aZ ij_^_eZfb ijh- |
\ Hij_^_eblv m[uldb ij_^ijbylby |
|
p_kkZ ijhba\h^kl\Z |
6. |
GZeh]h\Zy bgki_dpby gZqbkey_l i_gb aZ g_k\h_\j_f_ggh_ i_j_qbk- |
[ ijb[uev hl j_amevlZlh\ ijhba\h^kl\Z iheghklvx gZijZ\ey_lky gZ _]h |
e_gb_ \aghkh\ \ k\hc Z^j_k \ jZaf_j_ ]h^h\uo hl g_mieZq_gghc kmffu aZ |
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jZa\blb_ |
dZ`^uc ^_gv Ij_^ijbylb_ g_ ieZlbl gZeh]b \ kmff_ luk jm[ \ l_q_gb_ |
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\ ijb[ueb hl j_amevlZlh\ ijhba\h^kl\Z gZijZ\ey_lky gZ _]h jZa\b- |
^g_c |
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lb_ |
Z Hij_^_eblv ^hoh^ghklv [x^`_lZ \ \b^_ ijhklhc ijhp_glghc klZ\db |
2.Ij_^ijbylb_ aZljZlbeh gZ ijhba\h^kl\h feg jm[ I_jbh^ \j_- [ Hij_^_eblv ^hihegbl_evguc ^hoh^ [x^`_lZ
f_gb hl fhf_glZ \eh`_gby ^_g_`guo kj_^kl\ \ ijhba\h^kl\h ^h ihemq_gby \u- |
\ Hij_^_eblv m[uldb ij_^ijbylby |
||
jmqdb hl j_ZebaZpbb ijh^mdpbb jZ\_g ^gy J_glZ[_evghklv ijhba\h^kl\Z |
7. |
<_dk_ev ghfbgZevghc klhbfhklvx luk jm[ dmie_g ij_^ijb- |
|
jZ\gZ Hij_^_eblv kmffm ^_g_`guo kj_^kl\ \ h[hjhl_ ij_^ijbylby d |
ylb_f aZ hl _]h ghfbgZeZ Q_j_a ^gy \_dk_ev [ue hieZq_g \_dk_e_^Zl_- |
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dhgpm i_j\h]h ]h^Z jZ[hlu _keb |
e_f |
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Z ijb[uev hl j_amevlZlh\ ijhba\h^kl\Z jZkoh^m_lky aZ ij_^_eZfb ijh- |
Z Ih dZdhc mq_lghc klZ\d_ [ue \_dk_ev mql_g" |
||
p_kkZ ijhba\h^kl\Z |
[ Q_fm jZ\gZ ^hoh^ghklv wlhc hi_jZpbb \ \b^_ ijhklhc ijhp_glghc ]h- |
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[ ijb[uev hl j_amevlZlh\ ijhba\h^kl\Z iheghklvx gZijZ\ey_lky gZ _]h |
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jZa\blb_ |
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\ Q_fm jZ\_g ^hoh^ wlhc hi_jZpbb" |
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\ ijb[ueb hl j_amevlZlh\ ijhba\h^kl\Z gZijZ\ey_lky gZ _]h jZa\b- |
8. |
Ij_^ijbylb_ h`b^Z_l ihemqblv ih dhgljZdlm aZ ihklZ\dm k\h_c |
|
lb_ |
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ijh^mdpbb luk jm[ GZ ba]hlh\e_gb_ ijh^mdpbb ihljZq_gh luk jm[ |
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3. |
Ij_^ijbylb_ ijh^Z_l lh\Zj klhbfhklvx luk jm[ aZ ^g_c |
kdhevdh ^g_c fh`_l ihljZlblv ij_^ijbylb_ gZ \_kv pbde hl gZqZeZ ^h fhf_glZ |
|
ihemqZ_l ^hoh^ \ kmff_ luk jm[ |
hieZlu ijh^mdpbb _keb mjh\_gv ^hoh^ghklb gZ ij_^ijbylbb jZ\_g" |
||
Z Hij_^_eblv ^hoh^ghklv nbgZgkh\hc hi_jZpbb \ \b^_ ijhklhc ]h^h\hc |
9. |
Ij_^ijbylb_ h`b^Z_l ihemqblv ih dhgljZdlm aZ ihklZ\dm k\h_c |
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klZ\db ijhp_glh\ |
ijh^mdpbb luk jm[ Hl fhf_glZ gZqZeZ ijhba\h^kl\Z ^h hieZlu ijh^md- |
||
[ Hij_^_eblv ^hoh^ghklv wlh]h \b^Z ^_yl_evghklb \ \b^_ ijhklhc ]h^h- |
pbb ijhoh^bl ^g_c Kdhevdh kj_^kl\ fh`_l ihljZlblv ij_^ijbylb_ gZ ba]h- |
||
\hc klZ\db ijhp_glh\ _keb nbgZgkh\Zy hi_jZpby ih\lhjy_lky \ l_q_gb_ ihem- |
lh\e_gb_ ijh^mdpbb _keb mjh\_gv ^hoh^ghklb gZ ij_^ijbylbb jZ\_g" |
||
]h^Z b \_kv ^hoh^ gZijZ\ey_lky gZ m\_ebq_gb_ lhj]h\uo hi_jZpbc |
10. <_dk_ev ghfbgZevghc klhbfhklvx luk jm[ mql_g [Zgdhf aZ |
||
\ Hij_^_eblv ^hoh^ghklv wlh]h \b^Z ^_yl_evghklb \ \b^_ ijhklhc ]h^h- |
^g_c ^h kjhdZ _]h hieZlu \_dk_e_^Zl_e_f ih ijhklhc mq_lghc klZ\d_ ]h^h- |
||
\hc klZ\db ijhp_glh\ _keb nbgZgkh\Zy hi_jZpby ih\lhjy_lky \ l_q_gb_ ihem- |
\uo Q_fm jZ\gZ ^hoh^ghklv wlhc hi_jZpbb \ \b^_ ijhklhc ]h^h\hc ijhp_glghc |
||
]h^Z b ^hoh^Z gZijZ\ey_lky gZ jZa\blb_ lhj]h\uo hi_jZpbc |
klZ\db" |
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4. |
Ij_^ijbylb_ _`_^_dZ^gh \deZ^u\Z_l \ p_ggu_ [mfZ]b luk jm[ |
11. <_dk_ev ghfbgZevghc klhbfhklvx luk jm[ mql_g [Zgdhf aZ |
|
b ijh^Z_l bo gZ ^hjh`_ |
^g_c ^h kjhdZ _]h hieZlu \_dk_e_^Zl_e_f ih ijhklhc ijhp_glghc klZ\d_ |
||
Z Hij_^_eblv ^hoh^ghklv \ \b^_ ijhklhc ijhp_glghc klZ\db ^Zgghc nb- |
]h^h\uo Q_fm jZ\_g ^hoh^ hl wlhc hi_jZpbb" |
||
gZgkh\hc hi_jZpbb |
12. <_dk_ev ghfbgZevghc klhbfhklvx luk jm[ mql_g [Zgdhf aZ |
||
[ Hij_^_eblv ^hoh^ghklv \ \b^_ ijhklhc ijhp_glghc klZ\db _keb hi_- |
^g_c ^h kjhdZ _]h hieZlu \_dk_e_^Zl_e_f ih ijhklhc ijhp_glghc klZ\d_ |
||
jZpbb ih\lhjyxlky jZa b \_kv ^hoh^ dZiblZebabjm_lky |
]h^h\uo Kdhevdh kj_^kl\ \eh`be [Zgd \ wlm hi_jZpbx" |
||
\ Hij_^_eblv ^hoh^ghklv \ \b^_ ijhklhc ijhp_glghc klZ\db _keb hi_- |
13. <_dk_ev ghfbgZevghc klhbfhklvx luk jm[ mql_g [Zgdhf aZ |
||
jZpbb ih\lhjyxlky jZa b ^hoh^Z dZiblZebabjm_lky |
^g_c ^h kjhdZ _]h hieZlu \_dk_e_^Zl_e_f ih ijhklhc mq_lghc klZ\d_ ]h^h- |
||
5. |
I_gkbhgguc nhg^ gZqbkey_l i_gb aZ g_k\h_\j_f_ggh_ i_j_qbke_- |
\uo Q_fm jZ\_g ^bkdhgl wlhc hi_jZpbb" |
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gb_ \aghkh\ \ k\hc Z^j_k \ jZaf_j_ hl g_mieZq_gghc kmffu aZ dZ`^uc ^_gv |
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Ij_^ijbylb_ g_ i_j_qbkey_l \aghku \ kmff_ luk jm[ \ l_q_gb_ ^g_c |
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15 |
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16 |
14.GZ kdhevdh ^g_c ^he`_g baf_gblvky kjhd hdhgqZgby k^_edb _keb iZjlg_j ij_^eh`be aZieZlblv gZ f_gvr_ i_j\hgZqZevgZy kmffZ dhgljZdlZ
luk jm[ mjh\_gv ^hoh^ghklb dhgljZdlZ ]h^h\uo"
15.K^_edZ aZdZgqb\Z_lky gZ ^g_c iha`_ q_f ij_^mkfZljb\Zehkv dhgljZdlhf i_j\hgZqZevgh iZjlg_j ij_^eZ]Z_l aZieZlblv gZ f_gvr_ i_j- \hgZqZevgZy kmffZ dhgljZdlZ luk jm[ Q_fm jZ\gZ ^hoh^ghklv dhgljZdlZ \ nhjf_ ijhklhc ]h^h\hc ijhp_glghc klZ\db _keb k^_edZ m^h\e_l\hjy_l h[_ klh- jhgu"
16.GZ kdhevdh ^he`_g baf_gblvky kjhd hdhgqZgby k^_edb _keb iZjl- g_j ij_^eh`be aZieZlblv gZ [hevr_ i_j\hgZqZevgZy kmffZ dhgljZdlZ
luk jm[ mjh\_gv ^hoh^ghklb dhgljZdlZ ]h^h\uo"
17.IZjlg_j ij_^eh`be h[t_^bgblv ieZl_`b ih dhgljZdlm aZdexq_g- ghfm ] \ h^bg Q_fm jZ\_g wlhl ieZl_` _keb h[t_^bgyxlky ieZl_`b
luk jm[ hl ] b luk jm[ hl ] \ ieZl_`
] ijb ^hoh^ghklb dhgljZdlZ \ ]h^h\uo"
18.IZjlg_j ij_^eh`be h[t_^bgblv ieZl_`b ih dhgljZdlm aZdexq_g- ghfm ] \ h^bg Dh]^Z gm`gh k^_eZlv wlhl ieZl_` _keb h[t_^bgyxl- ky ieZl_`b luk jm[ ] b luk jm[ ] \ ieZl_` jZ\guc luk jm[ ijb ^hoh^ghklb dhgljZdlZ \ ]h^h\uo"
19.Hp_gblv \_ebqbgm ieZl_`_c luk jm[ hl ] b luk jm[ hl ] gZ ^Zlm \lhjh]h ieZl_`Z _keb ieZl_`b h[_ki_qb\Zxl ^h- oh^ghklv ]h^h\uo
20.GZclb gZjZs_ggmx kmffm ieZl_`_c luk jm[ hl ] b
luk jm[ hl ] gZ ] _keb ieZl_`b h[_ki_qb\Zxl ^h- oh^ghklv ]h^h\uo
21.GZclb kh\j_f_ggmx klhbfhklv ieZl_`_c luk jm[ hl
] b luk jm[ hl ] gZ ^Zlm i_j\h]h ieZl_`Z _keb ieZ- l_`b h[_ki_qb\Zxl ^hoh^ghklv ]h^h\uo
22.Hp_gblv \_ebqbgm ieZl_`_c luk jm[ hl ] b luk jm[ hl ] gZ ] _keb ieZl_`b h[_ki_qb\Zxl ^hoh^ghklv
]h^h\uo
23.Hp_gblv k ihfhsvx fh^_eb ihklhygghc ]h^h\hc j_glu ihklgmf_- jZg^h kh\j_f_ggmx klhbfhklv ieZl_`_c luk jm[ hl ] b luk jm[ hl ] _keb ieZl_`b h[_ki_qb\Zxl ^hoh^ghklv ]h^h\uo
24.Hp_gblv klhbfhklv ih khklhygbx gZ ] ieZl_`_c
luk jm[ hl ] b luk jm[ hl ] _keb ieZl_`b h[_ki_- qb\Zxl ^hoh^ghklv ]h^h\uo
25.Hp_gblv k ihfhsvx fh^_eb ihklhygghc ]h^h\hc j_glu ihklgmf_- jZg^h gZjZs_ggmx klhbfhklv ieZl_`_c luk jm[ hl ] b luk jm[ hl ] _keb ieZl_`b h[_ki_qb\Zxl ^hoh^ghklv ]h^h\uo
17
26.Ij_^ijbylb_ ih dj_^blm \ kmff_ luk jm[ \aylhfm \ [Zgd_
] k^_eZeh ke_^mxsb_ ieZl_`b ih ih]Zr_gbx hkgh\gh]h ^he]Z
] ±luk j ] luk jm[ ] luk jm[ Hp_gblv kmffm ^he]Z gZ ] _keb \ dj_^blghf ^h]h\hj_ ijhklZy ]h^h- \Zy ijhp_glgZy klZ\dZ khklZ\ey_l b
Z j_Zebah\Zgh©ijZ\beh lhj]h\pZª [ j_Zebah\Zg ZdlmZjguc f_lh^
27.Ij_^ijbylb_ ih dj_^blm \ kmff_ luk jm[ \aylhfm \ [Zgd_
] k^_eZeh ke_^mxsb_ ieZl_`b ih ih]Zr_gbx h[s_c aZ^he`_gghklb
] ± luk jm[ ] luk jm[ ] luk jm[ Hp_gblv kmffm ^he]Z gZ ] _keb \ dj_^blghf ^h]h\hj_ ijhklZy ]h^h\Zy ijhp_glgZy klZ\dZ khklZ\ey_l b
Z j_Zebah\Zgh©ijZ\beh lhj]h\pZª [ j_Zebah\Zg ZdlmZjguc f_lh^
28.Ij_^ijbylb_ ieZgbjm_l ]Zkblv h[smx aZ^he`_gghklv ih dj_^blm \ kmff_ luk jm[ \aylhfm \ [Zgd_ ] ih ijhklhc ]h^h\hc ijhp_gl- ghc klZ\d_ jZ\gufb kmffZfb ih luk jm[ _`_f_kyqgh DZdh_ \j_fy [m^_l kms_kl\h\Zlv aZ^he`_gghklv [Zgdm"
29.Ij_^ijbylb_ ieZgbjm_l ]Zkblv hkgh\ghc ^he] ih dj_^blm \ kmff_
luk jm[ \aylhfm \ [Zgd_ ] ih ijhklhc ]h^h\hc ijhp_glghc klZ\d_ jZ\gufb kmffZfb ih luk jm[ _`_f_kyqgh b dZ`^uc f_kyp \u- ieZqb\Zlv ijhp_glu DZdh_ \j_fy [m^_l kms_kl\h\Zlv aZ^he`_gghklv [Zgdm"
9DHGLJHEV AG:GBC KLM>?GLH<
AZsblZ hlq_lZ ih jZkq_lghfm aZ^Zgbx \uiheg_ggh]h \ ibkvf_gghc nhjf_
AZq_l
<hijhku d aZq_lm \ mklghc nhjf_
1.Ij_^f_l f_lh^ b h[t_dl nbgZgkh\hc fZl_fZlbdb __ f_klh \ dheb- q_kl\_gghf ZgZeba_ nbgZgkh\uo hi_jZpbc
2.Ijhklu_ b pbdebq_kdb_ nbgZgkh\u_ hi_jZpbb iZjZf_lju nbgZg- kh\hc hi_jZpbb
3.DZiblZebaZpby ^hoh^Z dj_^blu b ^jm]b_ nhjfu ^he]Z
4.Ijhp_glu ijhp_glgZy klZ\dZ b ^hoh^ghklv nbgZgkh\hc hi_jZpbb
5.Kj_^gyy ^hoh^ghklv nbgZgkh\uo hi_jZpbc
6.GZjZs_gb_ ih ijhklhc klZ\d_ ijhp_glh\ JZkq_l ijhp_glh\ ^ey djZldhkjhqguo hi_jZpbc
7.Keh`gu_ ijhp_glu
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18
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19.Wd\b\Ze_glghklv ieZl_`_c
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23.IZjZf_lju j_glu
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26.Hij_^_e_gb_ iZjZf_ljh\ j_glu
27.K[ZeZgkbjh\Zgghklv ihlhdh\ ieZl_`_c
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32.:dlmZjguc f_lh^
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