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交换期权

Exchange Option

专题
Finance / 金融
难度
L4

题目详情

金融数学题：交换期权。

英文原题

The payoff of an exchange option at expiry is

Ex⁡(T)=max⁡(S1(T)−S2(T))+\operatorname {Ex}(\mathrm{T}) = \max \left(S_{1}(T) - S_{2}(T)\right)^{+}

Calculate the price of an exchange option at t=0t = 0 when ρ\rho is the correlation between S1S_{1} and S2S_{2} and σ\sigma and rr are constant.

解析

交换期权 payoff：(S1(T)−S2(T))+(S_1(T)-S_2(T))^+。

Margrabe 公式（无分红；若有分红可把 SiS_i 换成贴现/远期形式）：

令

σM=σ12+σ22−2ρσ1σ2,τ=T.\sigma_M=\sqrt{\sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2},\quad \tau=T.

则

C0=S1(0)N(d1)−S2(0)N(d2),\boxed{C_0=S_1(0)N(d_1)-S_2(0)N(d_2)},

其中

d1=ln⁡(S1(0)S2(0))+12σM2τσMτ,qquadd2=d1−σMτ.d_1=\frac{\ln\left(\frac{S_1(0)}{S_2(0)}\right)+\tfrac12\sigma_M^2\tau}{\sigma_M\sqrt{\tau}}, \\qquad d_2=d_1-\sigma_M\sqrt{\tau}.

英文解析

σM=σ12+σ22−2ρσ1σ2,τ=T.\sigma_M=\sqrt{\sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2},\quad \tau=T. C0=S1(0)N(d1)−S2(0)N(d2),\boxed{C_0=S_1(0)N(d_1)-S_2(0)N(d_2)}, d1=ln⁡(S1(0)S2(0))+12σM2τσMτ,qquadd2=d1−σMτ.d_1=\frac{\ln\left(\frac{S_1(0)}{S_2(0)}\right)+\tfrac12\sigma_M^2\tau}{\sigma_M\sqrt{\tau}}, \\qquad d_2=d_1-\sigma_M\sqrt{\tau}.