Compute the price (mathrm{e}^{-(T-t) r} mathbb{E}^{*}left[mathbb{1}_{left{R_{T} geqslant kappa ight}} mid mathcal{F}_{t} ight]) at time (t in[0, T])

Compute the price \(\mathrm{e}^{-(T-t) r} \mathbb{E}^{*}\left[\mathbb{1}_{\left\{R_{T} \geqslant \kappa\right\}} \mid \mathcal{F}_{t}\right]\) at time \(t \in[0, T]\) of a cashor-nothing "binary" foreign exchange call option with maturity \(T\) and strike price \(\kappa\) on the foreign exchange rate process \(\left(R_{t}\right)_{t \in \mathbb{R}_{+}}\)given by

\[d R_{t}=\left(r^{l}-r^{\mathrm{f}}\right) R_{t} d t+\sigma R_{t} d W_{t}\]

where \(\left(W_{t}\right)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion under \(\mathbb{P}^{*}\).

We have the relation

\[\mathbb{P}^{*}\left(x \mathrm{e}^{X+\mu} \geqslant \kappa\right)=\Phi\left(\frac{\mu-\log (\kappa / x)}{\sqrt{\operatorname{Var}[X]}}\right)\]

for \(X \simeq \mathcal{N}(0, \operatorname{Var}[X])\) a centered Gaussian random variable.