Problem $16-07$

Consider the following price data for TanCo stock in two different subperiods:

Subperiod A: $168\mathrm{.}420$;$162\mathrm{.}580$;$162\mathrm{.}685$;$161\mathrm{.}590$;$160\mathrm{.}745$;$157\mathrm{.}815$;$157\mathrm{.}430$;$157\mathrm{.}795$;$161\mathrm{.}450$;$162\mathrm{.}550$;$157\mathrm{.}490$;$156\mathrm{.}645$;$157\mathrm{.}660$;$155\mathrm{.}575$; $150\mathrm{.}570$;$155\mathrm{.}900$;$154\mathrm{.}230$;$155\mathrm{.}515$;$156\mathrm{.}060$;$152\mathrm{.}840$;$150\mathrm{.}200$;$150\mathrm{.}335$

Subperiod B: $122\mathrm{.}510$;$124\mathrm{.}875$;$121\mathrm{.}810$;$120\mathrm{.}825$;$119\mathrm{.}690$;$118\mathrm{.}700$;$117\mathrm{.}810$;$119\mathrm{.}315$;$122\mathrm{.}380$;$121\mathrm{.}580$;$120\mathrm{.}205$;$117\mathrm{.}550$;$118\mathrm{.}800$;$115\mathrm{.}765$; $117\mathrm{.}605$;$117\mathrm{.}765$;$118\mathrm{.}855$;$117\mathrm{.}790$;$114\mathrm{.}915$;$110\mathrm{.}935$

a$.$ For each subperiod, calculate the annualized historical measure of stock volatility that could be used in pricing an option for Tanco. In your calculations, you may assume that there are $250$ trading days in a year. Do not round intermediate calculations. Round your answers to four decimal places.

Period A:

Period B:

b: Suppose now that you decide to gather additional data for each subperiod. Specifically, you obtain information for a call option with a current price of $$12\mathrm{.}40$ and the following characteristics: $x=114$;$S=121\mathrm{.}05$; time to expiration $=62$ days; $RFR=7\mathrm{.}78\%$; and dividend yield $=3\mathrm{.}45\%.$ Here the risk$-$free rate and dividend yields are stated on an annual basis. Use the volatility measure from Subperiod B and the Black$-$Scholes model to obtain the "fair value" for this call option. Based on your calculations, is the option currently priced as it should be$?$ Assume $365$ days in a year. You may use Appendix D to answer the question. Do not round intermediate calculations. Round your answer to the nearest cent.

$

The market price of $$12\mathrm{.}40$ is than the calculated BS price. This implies that if all of the other parameters of the model are correct, the implied BS volatility is $]$ than the historical volatility.