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EUROPEAN-TYPE OPTIONS IN THE DIFFUSION MODEL (B,S)-FINANCIAL MARKET ON THE BASIS OF EXTREME VALUE OF BASIC ASSET PRICE Andreeva U.V., Anikina A.V. 1, Dyomin N.S., Rozhkova S.V.
1

1

Tomsk State University, Department of Applied Mathematics and Cybernetics, 36 Lenin ave. 634050 Tomsk, Russia, Tel: (3822)-529-599, e-mail: svrhm@rambler.ru Tomsk Polytechnic University, Department of Natural Sciences and Mathematics 30 Lenin ave. 634050 Tomsk, Russia, Tel: (3822)-563-350, e-mail: rozhkova@tpu.ru

The investor builds up the investment portfolio with capital X t = t Bt + t St , t [ 0, T ] , where Bt is the price of riskfree asset, St is the basic (risk) asset price , t and t are shares Bt = B0 exp{ rt } , (quantities) of corresponding assets in the capital, 2 S t = S 0 exp{ ( µ - ( 2 ) ) t + Wt } , where B0 > 0 , S 0 > 0 , r > 0 , > 0 , i.e. a diffusion model of ( B, S ) - financial market [1] is under consideration. In the case of standard call and put options payment obligations (functions) are given by [1] fTc ( ST ) = max{ ( St - K ) , 0} , fTp ( ST ) = max{ ( K - St ) , 0} , (1) where K is the striking price of option. The subject of this work is research of call and put options, which belong to exotic options class [2] and are based on the extreme value of basic asset price, the payoff functions of which are given by f f
c max T p min T

= max max St - K , 0 ,
0 t T

=

{( max{ (

K - min S
0 t T

t

)} ) , 0} ,

f f

c min T

= max min St - K , 0 ,
0 t T

p max T

=

{( max{ (

K - max S
0 t T

t

)} ) , 0}

(2) (3)

.

In this work there has been found formulas which determine the value of options, corresponding to payment obligations (2), (3), as well as formulas which determine time * * * * evolution of portfolios t = ( t , t ) and capitals X t , which ensure the fulfillment of the payment obligation. References 1. Shiryaev A.N. Foundations of stochastic financial mathematics. M.: FAZIS, 1998. 1016 P. (in Russian) 2. Kozhin K. All about exotic options // Securities market. 2002. N. 15-17. (in Russian)