Problem $3(25$p$)$

$($i$)(8$p$)$ For a binary European put option which pays $$1$ if $S_T{S}_{t}t>0T{S}_{T-t}t>0t0t{S}_t=$$$50{}_t=0\mathrm{.}4127{}_t=0\mathrm{.}1134{}_{t+t}{S}_{t+t}=$$$52\mathrm{.}75t+tt>0S_T$ and nothing else, draw $on$ three separate figures

$(2p)$ the pay$-$off function $of$ the option $as$ a function $of{S}_T$

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

$(3p)$ 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 $t0.$

$(ii)(7p)At$ time $t$ the stock price $is$ trading $at{S}_t=$$$50.An$ option $on$ that stock has a delta ${}_t=0\mathrm{.}4127$ and gamma ${}_t=0\mathrm{.}1134.$ Using this information, find $(an$ approximate$)$ new value $of$ delta, ${}_{t+t},if{S}_{t+t}=$$$52\mathrm{.}75at$ time $t+t,$ for a small $t>0.$

$(iii)(5p)$ 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.$)$

$(iv)(5p)$ 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 $isno$ need $to$ derive the price $of$ the option.

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