交换期权定价

The asset prices

专题: Finance / 金融
难度: L4
来源: QuantQuestion

题目详情

The asset prices $X(t)$ and $Y(t)$ satisfy the stochastic differential equations

\begin{array}{r}dX(t) = \mu_{x}X(t)dt + \sigma_{x}X(t)dW_{x}(t) \\ dY(t) = \mu_{y}Y(t)dt + \sigma_{y}Y(t)dW_{y}(t) \end{array}

with $dW_{x}(t)dW_{y}(t) = \rho_{xy}dt$ and $\rho_{xy} \in (- 1,1)$ . Let $r$ denote the risk- free interest rate. Find the price at time 0 of the exchange option with payoff

V(T) = \max \{X(T) - Y(T),0\}

解析

交换期权 payoff： $(X_T-Y_T)^+$ 。

其无套利价格仍为 Margrabe 形式，只依赖相关与波动：

令

\sigma_M=\sqrt{\sigma_x^2+\sigma_y^2-2\rho_{xy}\sigma_x\sigma_y}.

则

\boxed{V_0=X_0N(d_1)-Y_0N(d_2)},

其中

d_1=\frac{\ln(X_0/Y_0)+\tfrac12\sigma_M^2T}{\sigma_M\sqrt{T}}, \qquad d_2=d_1-\sigma_M\sqrt{T}.

（在有分红情形，把 $X_0,Y_0$ 换成相应贴现/远期形式即可。）