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Exchange Options

专题
Finance / 金融
难度
L4

题目详情

若 S1,S2S_1,S_2 都服从相关 GBM（相关系数 ρ\rho），合约到期支付 max⁡(ST,1−ST,2,0)\max(S_{T,1}-S_{T,2},0)。其价值如何计算？

Value max⁡(ST,1−ST,2,0)\max(S_{T,1}-S_{T,2},0) if both S1,S2S_1,S_2 follow correlated GBM with correlation ρ\rho?

解析

使用 Margrabe 公式。

把支付写成

ST,1max⁡(1−ST,2ST,1,0),S_{T,1}\max\left(1-\frac{S_{T,2}}{S_{T,1}},0\right),

并利用比值的对数正态性。有效波动率为

σeff=σ12+σ22−2ρσ1σ2.\sigma_{\mathrm{eff}}=\sqrt{\sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2}.

随后可类比标准 BS 推导。

Original Explanation

Use Margrabe’s formula: rewrite payoff as ST,1max⁡(ST,2ST,1−1,0).S_{T,1}\max\bigl(\tfrac{S_{T,2}}{S_{T,1}}-1,0\bigr). The ratio is lognormal with some correlation structure. Solve similarly to standard BS with a modified volatility σ12+σ22−2ρσ1σ2\sqrt{\sigma_1^2 +\sigma_2^2 -2\rho\sigma_1\sigma_2}.