Problem $3(25$p$)$

$($i$)(8$p$)$ For a binary European call option which pays $$1$ if $S_T>K$ and nothing else, draw on three separate figures

$(2$p$)$ the pay$-$off function of the option as a function of $S_T$

$(3$p$)$ the graph of the "delta" of the option as a function of current stock price $S_t$ for some arbitrary $t>0$ less than $T.$

$(3$p$)$ the graph of the "delta" of the option as a function of current stock price $S_{T-t}$ for a small $t>0.$ Explain what happens with the "delta" of the option when $t0.$

$($ii$)(7$p$)$ At time $t$ the stock price is trading at $S_t=$$$40.$ An option on that stock has a delta ${}_t=0\mathrm{.}4127$ and gamma ${}_t=0\mathrm{.}1134.$ Using this information, find $($an approximate$)$ new value of delta, ${}_{t+t},$ if $S_{t+t}=$$$42\mathrm{.}75$ at time $t+t,$ for a small $t>0.$

$($iii$)(5$p$)$ What is riskier: a call option or the underlying? Provide all the necessary derivations. $($Assume Black$-$Scholes model and consider a one$-$day time horizon and compute which has bigger Delta as a fraction of value.$)$

$($iv$)(5$p$)$ State the variational inequality for the value function of the American put option, and all the boundary conditions. How your results change when you consider a perpetula American put option $($American put which never expires$)-$ there is no need to derive the price of the option.

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