Quantile hedging call option ina diffusion (B, S)-market in case of dividends payment on a risk active.

Risk and risk free assets, circulating in a financial market, have current prices{( ( 2 )) }St=S0 exp ƒ− ƒ 2 t+ƒWt and Bt =B0 exp{rt} , t[0,T], where ( ) t t 0 W W ≥ = is a standardWiener process, ƒR =(−,+), ƒ > 0, r > 0, S0 > 0, B0 > 0 . Capital value of investoris Xt=ƒtBt+ƒtSt, where ƒt=(ƒt, ƒt) is an investment portfolio consisted of two Ft -measurable process. For holding of assets dividends are paid at the rate dDt= ƒƒtStdt , ƒ > 0 .The payoff function is fT (ST K) max(0,ST K) = − + = − , its payment liability is fulfillied withprobability P(A) =1− ƒ , 0 < ƒ <1 .Theorem. Let( ) ( )xx ydy−ƒ =  ϕ ,( ) 1 exp 22 2y y⎧ ⎫ϕ = ⎨− ⎬ƒ ⎩ ⎭,( ) ( ( 2 ))0ln ƒ ƒ 2 ( )( , )ƒttKS r T ty T t ST t− − − −− =−,1 ( ) 2 ( )( ) ƒ ƒ ƒ ƒƒ ƒA = ƒ ⎛⎜ ⎛⎜bT−t − −r+ T− t ⎞⎟ T − t ⎞⎟ + ƒ⎛⎜ ⎛⎜ −bT−t + −r+ T − t ⎞⎟ T − t ⎞⎟⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎠P ,where Ф(x) is Laplace's function, y0(T, S0), 1bT and 2bT are y0 (T−t,St ) , 1bT−t and 2bT−t , whent = 0. If the condition ⎡⎣(ƒ − r + ƒ) ƒ2⎤⎦>1 holds, an option price, the hedging strategy and capitalare defined by formulasƒ (( ) ) ( ( ) )0 T 1T ƒ 2T ƒ ( 0( , 0 ) ƒ )CT=Se−⎡⎣ƒ b T − T +ƒ−b T + T −ƒy TS − T⎤⎦−( ) ( ) −Ke−rT⎡⎣ƒ b1T T +ƒ −b2T T −ƒ(y0(T,S0 ))⎤⎦,* ƒ( ) (( ) ) ( ( ) )ƒ T t 1T t ƒ 2T t ƒ ( 0( , ) ƒ )t =e− − ⎡⎣ƒ b− T −t − T −t +ƒ − b − T −t + T −t −ƒ y T−t St − T−t ⎤⎦ ,* ( ) ( ) ( )ƒ r T t 1T t 2T t ( 0( , ))t tte K b T t b T t y T t SB= − − − ⎡⎣ƒ − − + ƒ − − − − ƒ − ⎤⎦,* ƒ( ) (( ) ) ( ( ) )T t 1T t ƒ 2T t ƒ ( 0( , ) ƒ )Xt=Ste− − ⎡⎣ƒ b − T−t − T−t +ƒ − b − T−t + T−t −ƒ y T−t St − T−t ⎤⎦−( ) ( ) ( )r T t 1T t 2T t ( 0( , ))−Ke− −⎡⎣ƒ b − T −t +ƒ −b − T −t −ƒ y T −t St⎤⎦.Proposition. Let 00SСT = СTS and KСT= СTK be sensitivity coefficients, determiningdependences of call-option price with dividends payment on asset's initial price S0 and on strikeprice If ⎡⎣(ƒ − r + ƒ) ƒ2⎤⎦>10 ƒ ( ) ( ) ( ( ) )S T 1T ƒ 2T ƒ 0 , 0 ƒСT =e−⎡⎣ƒb T− T+ƒ −b T+ T−ƒ y T S − T⎤⎦,( ) ( ) ( ( )) K rT 1T 2T 0 , 0СT = −e−⎡⎣ƒb T+ ƒ −b T− ƒyTS ⎤⎦,and S0 0CT > , K 0CT < , i.e. the call-option price is an increasing function of asset's initial priceand decreasing function of strike price. The solution of perfect hedging problem as extreme caseof quantile hedging, when ƒ  0 , is obtained.

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