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Modeluvannya ekonomiki - Vіtlіnskiy V.V.
16.4. Розв'язання рівняння арбітражу
I will glance at the visibility of the arbitrage (16.2) in a formal glance, togo yak rivnyannya, rozv'yazki kotrogo potrebno vidshukati. Нехай rate дохідності й купонний дохід є відомими функціями від часу, і рівняння (16.2) written in wikis:
(16.3)
The rogue rozvjazyok zvichaynogo odnogorіdnogo differentsionnogo rivnyannya first order (16.3) mozhna vidshukati, zokrema, method varіаcії довільної константи у вигляді
(16.4)
De A ( t ) - nevіdoma postійна, що є functцією to hour, and
Differentials (16.4) are the result of (16.3), knowable for the tasks of current minds V (0) = V 0 zagalniy rozvjazok rivnyannya arbitrazhu u viglyadі:
. (16.5)
Відшуканий розв'язок є ніби «поглядом у майбутнє»: ватість активу в бы-який момент майбутнього визначається як зміна почакових умов (початкової вартості active) під впливом rate відсотка и купонних сплат. Economical sence rozv'yuku є dos shame: in Maybutnomu vartizst activist zrostaє vidpovidno up to the rate vidsotka, and the coupon payments coupons Maybutnyu vartit active zmenshut.
Significantly, for the sake of economical processes, you should gain access to the process not at the time of the hour, but at the t = T end .
The investment can be assigned to the contract by the active contractual moment at one hour in Maybutnom, for example, yak vartit burgu (suma, scho is taken into the borg at once in time) V ( T ) = VT . In the case of vipadk rozv'azok (16.3) vidshkuyut nemovbi zinci, yak "look at the end", for the visible value of the value process. With the t ? = T - t рівняння арбітражу to be brought up to the vigil:
(16.6)
І meaning, in the course of an inexpensively small period, dt is the wartness of the borgo, in which the value of Vt is reduced to the value r ( t ?) V ( t ?) Dt ? For the visibility of the arbiterary affluence. Rozv'yazok (16.6) vidushuyutsya analogіchnim way varіаcії довільної зміnoї, yak to lead up to the function of functions
(16.7)
Zokrema, for the post-value of the coupon to the income C ( t ?) = C і the rates of the overlap r ( t ?) = R is off
Rozv'yazok (16.7) show, shocho rinkova vartіst borgu is discounted for the rate of the value of the wartost borg, scoring by the amount of discounted coupon coupons.
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