Q$4.$ Delta and Gamma with Uniform Distribution $(20$ points$)$

Current underlying price at $1000,$ and you expect price at expiration follows uniform distribution

with mean absolute deviation of $200.$

You LONG $100$ PUTs with strike at $900.$

Q$4$a$.$ What is the TOTAL delta of your $100$ LONG PUTs position? $(2$ points$)$

Q$4$b$.$ How many shares do you need in order to offset the option delta from Q$4$a$?$ Do you long or

short the underlying stock $(2$ points$)?$

Q$4$c$.$ What is the total gamma value of your hedged option positions from Q$4$a and Q$4$b$?(2$ points$)$

Q$4$d$.$ For the hedged option position $($LONG PUTs, hedged with shares$),$ what is the PnL from

$($starting$)$ delta, and from gamma respectively, when underlying moves from $1000$ to $950?(3$

points$)$

Q$4$e$.$ Calculate option price when stock moves $$950.$ How much of the total PnL of Q$4$d is from

option position, and how much is from the stock position? $(3$ points$)$

$($note: the TOTAL PnL from options and stocks should match the total PnL from delta and gamma$).$

Q$4$f$.$ If underlying moves down from $1000$ to $950,$ what is the new delta at stock price of $950$ for

your overall position? If you want to re$-$hedge to flatten the delta with underlying shares, what

trade do you need to do in the shares to achleve that? How many shares to trade? Long or short? $(2$

points$)$

Q$4$g$.$ What is the PnL$.$ from the delta and gamma when underlying moves from $1000$ to $1025,$

respectively? $(2$ points$)$

Q$4$h$.$ For Q$4$g$,$ how much of the total PnL is from option position, and how much is from stock

positions, when underlying moves from $1000$ to $1025?(2$ points$)$