${p}_{0,\mathrm{call}}={X}_{2}{\mathrm{e}}^{-r{T}_{2}}M\left(-{d}_{2},{y}_{2};-\sqrt{\frac{{T}_{1}}{{T}_{2}}}\right)-{S}_{0}M\left(-{d}_{1},{y}_{1};-\sqrt{\frac{{T}_{1}}{{T}_{2}}}\right)+{X}_{1}{\mathrm{e}}^{-r{T}_{1}}N(-{d}_{2})$

${c}_{0,\mathrm{put}}={X}_{2}{\mathrm{e}}^{-r{T}_{2}}M\left(-{d}_{2}-{y}_{2};\sqrt{\frac{{T}_{1}}{{T}_{2}}}\right)-{S}_{0}M\left(-{d}_{2}-{y}_{2};\sqrt{\frac{{T}_{1}}{{T}_{2}}}\right)-{X}_{1}{\mathrm{e}}^{-r{T}_{1}}N(-{d}_{2})$

${p}_{0,\mathrm{put}}={S}_{0}M\left({d}_{1}-{y}_{1};-\sqrt{\frac{{T}_{1}}{{T}_{2}}}\right)-{X}_{2}{\mathrm{e}}^{-r{T}_{2}}M\left({d}_{2}-{y}_{2};-\sqrt{\frac{{T}_{1}}{{T}_{2}}}\right)+{X}_{1}{\mathrm{e}}^{-r{T}_{1}}N({d}_{2})$

Let *c*_{t,call}, *p*_{t,call}, *c*_{t,put}, and *p*_{t,put} denote the values of a call on a call, a put on a call, a call on a put, and a put on a put at time *t*, respectively. To obtain the ...

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