路径事件数字期权：1S5<S3, S3>S2\mathbf{1}_{S_5<S_3,\ S_3>S_2}1S5<S3, S3>S2 Derivative security that pays 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 is rrr . Determine the price of a derivative security that pays 1ifS(5)<S(3)1 if S(5) < S(3)1ifS(5)<S(3) and S(3)>S(2)S(3) > S(2)S(3)>S(2) , and 0 otherwise. 解析 在风险中性下 lnS(t)=lnS0+(r−12σ2)t+σBt.\ln S(t)=\ln S_0+\left(r-\tfrac12\sigma^2\right)t+\sigma B_t.lnS(t)=lnS0+(r−21σ2)t+σBt. 事件 S(5)<S(3)S(5)<S(3)S(5)<S(3) 等价于 B5−B3<−2(r−12σ2)σ,B_5-B_3< -\frac{2\left(r-\tfrac12\sigma^2\right)}{\sigma},B5−B3<−σ2(r−21σ2), 事件 S(3)>S(2)S(3)>S(2)S(3)>S(2) 等价于 B3−B2>−(r−12σ2)σ.B_3-B_2> -\frac{\left(r-\tfrac12\sigma^2\right)}{\sigma}.B3−B2>−σ(r−21σ2). 两增量独立，且 B5−B3=2XB_5-B_3=\sqrt{2}XB5−B3=2X、B3−B2=YB_3-B_2=YB3−B2=Y，X,Y∼N(0,1)X,Y\sim N(0,1)X,Y∼N(0,1) 独立。 因此价格 Π0=e−5r Φ(−(r−12σ2)2σ) Φ((r−12σ2)σ).\boxed{\Pi_0=e^{-5r}\,\Phi\left(-\frac{\left(r-\tfrac12\sigma^2\right)\sqrt{2}}{\sigma}\right) \,\Phi\left(\frac{\left(r-\tfrac12\sigma^2\right)}{\sigma}\right)}.Π0=e−5rΦ(−σ(r−21σ2)2)Φ(σ(r−21σ2)).