Financial Options & Option Valuation REVISITED Week 7 IMBA 2017 ACF FALL 1 CLASS ASSIGNMENT SOLUTION

Class assignment (hand in) Consider AAPL after the iPhone X announcement:

And related Option prices for Oct 27 2017

Required: 1) Calculate the theoretical value of the Call Option X=$155 listed at market price $8.70 Assume: So=$ 161, X=$155, Rf=3% per year, Su=$200 and Sd=$120 2) Use the put call parity to calculate the P(ut) value with X=$155 for expiration 27 Oct. 2017 3) Use the BS model to calculate C and P and compare the above values: what is your conclusion? Note the dividend yield on AAPL stock is 1.56% (DPS/stock price) Market values : C= $8.70 and P= $2.68

Solution… 1) Calculate the theoretical value of the Call Option X=$155 listed at market price $8.70 Assume: So=$ 161, X=$155, Rf=3% per year, Su=$200 and Sd=$120 Hedge Ratio: (Cu-Cd)/(Su-Sd)= ($45-$0)/($200 - $120)=$45/$80=9/16 (buy 9 stocks AAPL and sell 16 calls) Set up the table and find: -$1449+16C+$1080/(1.0034)=0 so C= $23.29 Note 3% per year is about 0.34% per 41 days (the option expires 27 october 2017) today is 16 september 2017

2) Use the put call parity to calculate the P(ut) value with X=$155 for expiration 27 Oct. 2017 Put Call Parity we found in class: P=C-So+PV(X)+PV(dividends) Note that option prices for stocks are effected by dividends on stocks…(see the BS model for instance) You found C=$23.29 in 1) P= $23.29-$161+$155/1.0034+(1.56%*$161* 41/365)/1.0034= To be entirely correct we will need to consider what part of the dividend is actually paid between 16 september and 27 october… Above equation leads to: P=$13.06

3) Use the BS model to calculate C and P and compare the above values: what is your conclusion? Note the dividend yield on AAPL stock is 1.56% (DPS/stock price) All the data were given your only task is to fill them in in the YELLOW area. Note the dividend yield is put in as a ratio, the time to expiration is 41/365, the standard deviation is in the option price table last column “volatility”…. Use the EXCEL file that was sent to you… REMINDER: Bring it for the final exam…

You have 3 prices for C and P Market Price (given in question) Theoretical Value Binomial Model Theoretical Value Black Scholes EXCEL CALL Option on AAPL expiring 27 Oct X=$155 $ 8.70 $ 23.29 $ 9.15 Put Option on AAPL expiring 27 Oct X=$155 $ 2.68 $ 13.06 (based on C = $ 23.29 above) $ 2.91 The market may be wrong… These values are far off the market values These values are close to market values… Of course you trust the market BUT the BS model came quite close. The Binomial model offers very high values due to the assumed high STDEV of ($200 - $ 120) some students used the BS EXCEL to see at what STDEV the model predicts a C=$23.29 and found: 95.4%... (see next page)

AAPL’s STDEV is high but not 95% so we don’t trust this result….