$(16$ points$)$ Consider a stock index option that expires in $75$ days. The stock index is currently at $1240\mathrm{.}89$ and makes no cash payments during the life of the option. Assume that the stock index has a multiplier of $1.$ The risk$-$free rate is $3\%.$ a$.$ Calculate the lowest and highest possible prices for European call options on the above stock index with exercise prices of: i$.12251.$ Lower bound $2.$ Upper bound ii$.12551.$ Lower bound $2.$ Upper bound b$.$ Calculate the lowest and highest possible prices for European put options on the above stock index with exercise prices of: i$.12251.$ Lower bound $2.$ Upper bound ii$.12551.$ Lower bound $2.$ Upper bound