Question $1$: Basic Concepts on Options $(2/10)$

$(1)$ Keeping all other characteristics the same, which one do you think cost more, European

options or American options? Why? $($Two sentences at most$)$

$(2)$ Is a European put option is just a short position of European call options? Explain

your claim. $($Two sentences at most$)$

$(3)$ In binomial tree model, what is the fictitious risk$-$neutral probability $q^{**}?$ How is it

related to risk$-$adjusted discount rate? $($Two sentences at most$)$

$(4)$ Define a self$-$financing dynamic trading strategy. $($One sentence$)$