Question $1.$ Answer the following from the paper $$Real Options in Finance$.$

b$.$ What are the weaknesses and challenges of real option valuation according to the paper? Describe

each. $(9)$

Question $2.$ The price of a stock is currently $$33\mathrm{.}46$ and the historic annualized volatility of that stock is

$32\mathrm{.}68\%.$ Assume risk$-$free interest rate with continuous compounding to be $3\mathrm{.}00\%$ per year. The value of a $2-$

month European call option with a strike price at $$35\mathrm{.}00$ is to be found using a two$-$step binomial option

pricing model. Answer the following to help find the value of the option.

a$.$ What are the values of u$,$ d$,$ and q$?($To find the value of $,$ use the Excel function exp with values

of $$ and $(3)$

b$.$ Draw stock tree using the information provided. Indicate expected values of stock at expiration as well

as at intermediate nodes. $(2)$

c$.$ What are the values of payoffs at expiration $($at $2$ months$)$ for the option? $(2)$

d$.$ What are the values at the intermediate nodes $($at $1$ month$)$ for the option? $(2)$

e$.$ What is the value of the option at present? $(2)$

f$.$ Draw a tree$-$diagram for the option. Indicate the value of the option at each node. $(2)$

Question $3.$ Answer questions c to f as above for a $2-$month European put option using a two$-$step binomial

model with strike price at $$33\mathrm{.}50.$ The values of u$,$ d and q will be same as in part a above. The stock tree will

also be the same for this as part b above. $(8)$

Question $4.$ The price of a stock is currently $$76\mathrm{.}64,$ and the historic annualized volatility of that stock is

$34\mathrm{.}28\%$ per year. The risk$-$free interest rate with continuous compounding is $4\mathrm{.}35\%$ per year. Suppose you

are planning to value a $4-$month European put option using a four$-$step binomial option pricing model. Strike

price for the option is $$77\mathrm{.}50.$

a$.$ What are the values of u$,$ d$,$ and q$?(3)$

b$.$ Draw stock tree using the information provided. Indicate value of stock at expiration as well as at the

intermediate nodes. There should be $14$ values as shown in the following. $(6)$

c$.$ What are the values of payoff for the option at expiration? $(2)$

d$.$ What are the values of the option at each of the nine intermediate nodes? $(9)$

e$.$ What is the current price of the option? $(1)$