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foreign-domestic symmetry

One can directly verify the relationship

\begin{displaymath} \frac{1}{x}v(x,K,T,t,\sigma,r_d,r_f,\phi)=Kv(\frac{1}{x},\frac{1}{K},T,t,\sigma,r_f,r_d,-\phi). \end{displaymath} (37)

This equality can be viewed as one of the faces of put-call symmetry. The reason is that the value of an option can be computed both in a domestic as well as in a foreign scenario. We consider the example of $S_t$ modelling the exchange rate of EUR/USD. In New York, the call option $(S_T-K)^+$ costs $v(x,K,T,t,\sigma,r_{usd},r_{eur},1)$ USD and hence $v(x,K,T,t,\sigma,r_{usd},r_{eur},1)/x$ EUR. This EUR-call option can also be viewed as a USD-put option with payoff $K\left(\frac{1}{K}-\frac{1}{S_T}\right)^+$. This option costs $Kv(\frac{1}{x},\frac{1}{K},T,t,\sigma,r_{eur},r_{usd},-1)$ EUR in Frankfurt, because $S_t$ and $\frac{1}{S_t}$ have the same volatility. Of course, the New York value and the Frankfurt value must agree, which leads to (37).



2000-06-11
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