A$.$ Volatility is an important input in option valuation, but it is not an observed

variable and must be estimated. It can be estimated using historical data.

Alternatively, since option prices can be observed, we can use the observed option

price together with other market data to imply volatility. This volatility can then

be used to value other options. Suppose a European put option written on the

QQQ ETF with an exercise price of $450$ and a time$-$to$-$maturity of $12$ months is

currently trading at $$70\mathrm{.}4542.$ The interest rate and dividend yield are $4\mathrm{.}5\%$ and

$1\mathrm{.}5\%$ respectively $($both continuously compounded$).$ The current QQQ price is

Use a three$-$step tree to imply the volatility. $($Hint: set up the tree in a

spreadsheet and find the implied volatility by Solver.$)$

B$.$ In the same set$-$up as above, suppose everything remains the same except that

someone has already estimated the volatility to be $38\%,$ and the European put

is worth $$67\mathrm{.}3243.$ What is the implied dividend yield? $($Hint: to ensure better

convergence, set the Solver objective function as$,$ e$.$g$.,(p_O-p_T{)}^2{10}^6$ and try to

make it zero, where $p_O$ and $p_T$ are respectively the observed and tree values of the

options.$)$