Q$2.$ Underlying priced at $500$ with MAD of $100.(15$ points$)$

Q$2$a$.$ What is the probability of option expiring ITM for a $550$ CALL? $(3$ points$)$

Q$2$b$.$ What is the average underlying price when CALL expires ITM? $(3$ points$)$

Q$2$c$.$ What is the average CALL option payment conditional on that the call expires in the money? $(3$

points$)($not this is asking for OPTION payment NOT average stock price when option expires ITM$)$

Q$2$d$.$ How much should the CALL be priced at today based on Q$2$a and Q$2$c$?(3$ points$)$

Q$2$e$.$ Out of the price in Q$2$d$,$ how much of that is intrinsic value and how much is time value? $(3$

points$)$ Q$3.$ Implied MAD with Uniform Distribution

Underlying price currently at $500,$ and follows a uniform distribution. You observed $660$ strike

CALL priced at $$36.$ What is the implied MAD? $(10$ points$)$ Q$4.$ Delta and Gamma with Uniform Distribution $(20$ points$)$

Current underlying price at $200,$ and you expect price at expiration follows uniform distribution

with mean absolute deviation of $25.$

You SHORT $10$ CALLs with strike at $210.$

Q$4$a$.$ What is the TOTAL delta of your $10$ short CALL position? $(2$ points$)$

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

the underlying shares $(2$ points$)?$

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

points$)$

Q$4$d$.$ For the hedged CALL position $($short $10$ CALLs, hedged with shares$),$ what is the PnL from

$($starting$)$ delta, and from gamma when underlying moves from $200$ to $190,$ respectively? $(3$ points$)$Q$4$e$.$ Calculate option price when stock moves $$190.$ How much of the total PnL of Q$4$d is from

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

Q$4$f$.$ If underlying moves down from $200$ to $190,$ what is the new delta at stock price of $190$ for your

overall position? If you need to re$-$hedge to flatten delta with underlying shares, what trade do you

need to do in the shares to flatten the delta? 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 $200$ to $205,$

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 $200$ to $205?(2$ points$)$

Q$4$i$.$ If underlying moves up from $200$ to $205,$ If you need to re$-$hedge to flatten delta with

underlying shares, what trade do you need to do in the shares? $(2$ points$)$Assume $rf=0$ for the entire exam

Q$1.$ You believe stock price by year end will have the following multinomial distribution $(15$ points$)$:

Q$1$a$.$ What should be the stock price TODAY? $(2$ points$)$

Q$1$b$.$ what is the prob that a $110$ strike PUT will expire ITM? $(2$ points$)$

Q$1$c$.$ what is the conditional average price of underlying stock when $110$ strike PUT expires ITM? $(3$

points$)$

Q$1$d$.$ what is the conditional average payment from the $110$ strike PUT option when the PUT expires ITM?

$(2$ points$)$

Q$1$e$.$ based on Q$1$b$-$Q$1$d$,$ how much should the $110$ PUT be priced at today? $(3$ points$)$

Q$1$f$.$ Out of the total value of the put from Q$1$e$,$ how much is intrinsic value and how much is time value?

$(3$ points$)$