路径事件数字期权（特例 r=σ2/2r=\sigma^2/2r=σ2/2） Derivative security that pays 2 专题 Finance / 金融 难度 L4 来源 QuantQuestion 题目详情 The price of an asset has lognormal distribution given by S(t)=S0eμt+σBtS(t) = S_{0}e^{\mu t + \sigma B_{t}}S(t)=S0eμt+σBt , where BtB_{t}Bt is a Brownian motion. The interest rate rrr satisfies r=σ22r = \frac{\sigma^{2}}{2}r=2σ2 . Determine the price of a derivative security that pays 1ifS(5)<S(2)1 if S(5) < S(2)1ifS(5)<S(2) and S(3)>S(2)S(3) > S(2)S(3)>S(2) , and 0 otherwise. 解析 当 r=σ2/2r=\sigma^2/2r=σ2/2 时，风险中性下 lnS(t)=lnS0+σBt\ln S(t)=\ln S_0+\sigma B_tlnS(t)=lnS0+σBt。 payoff 为 1{S(5)<S(2), S(3)>S(2)}=1{B5<B2, B3>B2}\mathbf{1}_{\{S(5)<S(2),\ S(3)>S(2)\}}=\mathbf{1}_{\{B_5<B_2,\ B_3>B_2\}}1{S(5)<S(2), S(3)>S(2)}=1{B5<B2, B3>B2}。 令 X,YX,YX,Y 独立标准正态，使得 B5−B3=2XB_5-B_3=\sqrt{2}XB5−B3=2X、B3−B2=YB_3-B_2=YB3−B2=Y，则 B5−B2=2X+Y.B_5-B_2=\sqrt{2}X+Y.B5−B2=2X+Y. 所求概率是平面中区域 {Y>0, 2X+Y<0}\{Y>0,\ \sqrt{2}X+Y<0\}{Y>0, 2X+Y<0} 的角扇形，角度为 arctan2\arctan\sqrt{2}arctan2，因此 Π0=e−5r arctan22π=e−5σ22 arctan22π.\boxed{\Pi_0=e^{-5r}\,\frac{\arctan\sqrt{2}}{2\pi}=e^{-\frac{5\sigma^2}{2}}\,\frac{\arctan\sqrt{2}}{2\pi}}.Π0=e−5r2πarctan2=e−25σ22πarctan2.