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

RECAP KLP’s FINC 5880 Week 1: Intrinsic Valuation Week 2: Capital Budgeting (Disney Brasilia Case) Week 3: Capital Structure (Disney Case) Week 4: Business Analysis (integrating your knowledge) Week 5: Mid Test & Leasing… Week 6: Financial Options and Option Valuation Week 7: BS model and more on options Week 8: REVIEW & GUIDANCE FINAL EXAM

Japanese cars in China…

USD decline against CNY… CNY rises against USD… In just 6 months time…

Today’s Agenda (16 September) Financial Options DEMO for homework session 6 (Apple INC.) Break Financial Options & Option Valuation: Black Scholes (EXCEL) With some class assignments…to hand in Session 8: Subject Overview & Final Exam guidance

Apple & butterfly option 11 sept. Expectations about the new product announcements September 2017 have been strengthening the stock price…

We found (screenshot)

Spot price Apple Inc. $ 158.63 Reasoning: prices have been building up to the current level over the past 6 months… If the announcement include revolutionary news this will give the stock price a further boost to above $ 160 If there is no unexpected news about iPhone 8, the new Apple watch etc. then the stock price will decline to under $ 155 Expecting this to be the most likely range in the short term (October 2017) we choose; Call X=155, Call X=160 and 2 Calls in between 157.50

We GET : $ 7.90+$4.98 - $12.80 (bought)=$ 0.08… Short or Long? We short the butterfly: Thus buy 1 call X=$155 Sell 2 calls X= $157.50 Buy 1 call X=$ 160 According to attached table: We GET : $ 7.90+$4.98 - $12.80 (bought)=$ 0.08… We are now ready to make the table like we did in class last Saturday…

Butterfly…. You conclude this does not look good…

How about a Long Butterfly? This looks ok?

Let’s test the premiums The Call 27 Oct X=$155 was $ 7.90 What would the Binomial model calculate? What would the BS Model calculate?

Binomial Model Su= $175 Sd=$135 So=$158 Rf=3% per year (for this period 0.5%) Time 48 days= 0.13 year X=$155 Hedge ratio: $20/$40=1/2 C= ($158-$134.33)/2= $ 11.84 (we neglected dividends)

Black Scholes

So what happened? (12 sept.)

And thus the option prices changed…

Comparing premiums…11-12 sept Call Option 11 Sept So=$158.63 12 Sept So=$161.50 X=$155 bought (1) $ 7.90 $ 9.23 X=$157.50 sold (2) $ 6.40 $ 7.80 X=$160 bought (1) $ 4.98 It is absolutely logical that if the stock price increases from $158.63 to $161.50 that Call options with same X and t are more expensive… You could close your position and thus…you lost money because your butterfly was hoping for a smaller change…so you’d better stick to it as you still have time until 27 Oct to make money on it…

And on 15 Sept…

And thus option prices…

Compare with 11 and 12 sept. Call Option 11 Sept So=$158.63 12 Sept So=$161.50 15 Sept So=$158.20 X=$155 bought (1) $ 7.90 $ 9.23 $ 6.80 (15 sept) X=$157.50 sold (2) $ 6.40 $ 7.80 $ 4.83 (15 sept) X=$160 bought (1) $ 4.98 $ 6.40 $ 3.70 (15 sept) Your long butterfly: (did you make money?) X=$155 you make $3.20 but paid $7.90 (loss $4.70) X=$157.50 you loose $0.70 (twice) but received $12.80 (profit $11.40) X=$160 you loose the premium $4.98 TOTAL: -$4.70+$11.40-$4.98= $ 1.72 profit (your investment was almost zero)…

Of course you already knew that at So=$158. 20 your profit would be $1

What determines option value? Stock Price (S) Exercise Price (Strike Price) (X) Volatility (σ) Time to expiration (T) Interest rates (Rf) Dividend Payouts (D)

Try to guestimate…for a call option price… (5 min) Stock Price ↑ Then call premium will? Exercise Price ↑ Then…..? Volatility ↑ Time to expiration↑ Interest rate ↑ Dividend payout ↑

Answer Try to guestimate…for a call option price… (5 min) Stock Price ↑ Then call premium will? Go up Exercise Price ↑ Then…..? Go down. Volatility ↑ Then…..? Go up. Time to expiration↑ Interest rate ↑ Dividend payout ↑

Your answer should be: Call Putt So up up down X up Rf up D up Time up STDEV up

Binomial model Key to this analysis is the creation of a perfect hedge… The hedge ratio for a two state option like this is: H= (Cu-Cd)/(Su-Sd)=($75-$0)/($200-$50)=0.5 Portfolio with 0.5 shares and 1 written option (strike $125) will have a pay off of $25 with certainty…. So now solve: Hedged portfolio value=present value certain pay off 0.5shares-1call (written)=$ 23.15 With the value of 1 share = $100 $50-1call=$23.15 so 1 call=$26.85

What if the option is overpriced? Say $30 instead of $ 26.85 Then you can make arbitrage profits: Risk free $6.80…no matter what happens to share price! Cash flow At S=$50 At S=$200 Write 2 options $60 $ 0 -$150 Buy 1 share -$100 $50 $200 Borrow $40 at 8% $40 -$43.20 Pay off $ 6.80

Class assignment: What if the option is under-priced Class assignment: What if the option is under-priced? Say $25 instead of $ 26.85 (5 min) Then you can make arbitrage profits: Risk free …no matter what happens to share price! Cash flow At S=$50 At S=$200 …….2 options ? ….. 1 share Borrow/Lend $ ? at 8% Pay off

Answer… Then you can make arbitrage profits: Risk free $4 no matter what happens to share price! The PV of $4=$3.70 Or $ 1.85 per option (exactly the amount by which the option was under priced!: $26.85-$25=$1.85) Cash flow At $50 At $200 Buy 2 options -$50 $ 0 +$150 sell 1 share $100 -$200 Lending $50 at 8% +$54 Pay off $4 $ 4

Now we can play around… Assume more price volatility in the underlying asset Change X Change So Change Rf And if we change them one by one we will see the impact on C value Let’s use the Binomial model to show you the effects on C…

Increased Volatility in the Price of the Stock (Let’s assume Su= $225 and Sd=$25) Recall Volatility was: Su=$200 and Sd=$50 All other data are the same…(c.p.) Hedge ratio: (Cu-Cd)/(Su-Sd)=$100/$200 Buy 1 Stock and sell 2 Calls… -$100+2C+$25/1.08=0 C= $ 38.43 (versus $ 26.85 at old volatility) It is logical that this Call is more expensive…

Change Strike X to $100 (was $ 125) Hedge ratio changes to 2/3 Buy 2 stocks sell 3 calls… -$200+3C+$100/1.08=0 C= $ 35.80 (versus $ 26.85 at X=$125) This Call should be more expensive… (it is not in the money at So=$100) You can further try out what will happen at Rf=10% or So=$95…

Black-Scholes Option Valuation Assuming that the risk free rate stays the same over the life of the option Assuming that the volatility of the underlying asset stays the same over the life of the option σ Assuming Option held to maturity…(European style option)

Without doing the math… Black-Scholes: value call= Current stock price*probability – present value of strike price*probability Note that if dividend=0 that: Co=So-Xe-rt*N(d2)=The adjusted intrinsic value= So-PV(X)

Class assignment: Black Scholes (10 min) Assume the BS option model: Call= Se-dt(N(d1))-Xe-rt(N(d2)) d1=(ln(S/X)+(r-d+σ2/2)t)/ (σ√t) d2=d1- σ√t If you use EXCEL for N(d1) and N(d2) use NORMSDIST function! stock price (S) $100 Strike price (X) $95 Rf ( r)=10% Dividend yield (d)=0 Time to expiration (t)= 1 quarter of a year Standard deviation =0.50 A)Calculate the theoretical value of a call option with strike price $95 maturity 0.25 year… B) if the volatility increases to 0.60 what happens to the value of the call? (calculate it)

answer A) Calculate: d1= ln(100/95)+(0.10-0+0.5^2/2)0.25/(0.5*(0.25^0.5))=0.43 Calculate d2= 0.43-0.5*(0.25^0.5)=0.18 From the normal distribution find: N(0.43)=0.6664 (interpolate) N(0.18)=0.5714 Co=$100*0.6664-$95*e -.10*0.25 *0.5714=$13.70 B) If the volatility is 0.6 then : D1= ln(100/95)+(0.10+0.36/2)0.25/(0.6*(0.25^0.5))=0.4043 D2= 0.4043-0.6(0.25^0.5)=0.1043 N(d1)=0.6570 N(d2)=0.5415 Co=$100*0.6570-$ 95*e -.10*0.25 *0.5415=$15.53 Higher volatility results in higher call premium!

In Excel…

Homework assignment: Black & Scholes Calculate the theoretical value of a call option for your company using BS Now compare the market value of that option How big is the difference? How can that difference be explained?

Implied Volatility… If we assume the market value is correct we set the BS calculation equal to the market price leaving open the volatility The volatility included in today’s market price for the option is the so called implied volatility Excel can help us to find the volatility (sigma)

Homework assignment: Implied Volatility Consider one option series of your company in which there is enough volume trading Use the BS model to calculate the implied volatility (leave sigma open and calculate back) Set the price of the option at the current market level

Implied Volatility Index - VIX Investor fear gauge…

The put-call parity… Relates prices of put and call options according to: P=C-So + PV(X) + PV(dividends) X= strike price of both call and put option PV(X)= present value of the claim to X dollars to be paid at expiration of the options Buy a call and write a put with same strike price…then set the Present Value of the pay off equal to C-P…

The put-call parity Assume: S= Selling Price P= Price of Put Option C= Price of Call Option X= strike price R= risk less rate T= Time then X*e^-rt= NPV of realizable risk less share price (P and C converge) S+P-C= X*e^-rt So P= C +(X*e^-rt - S) is the relationship between the price of the Put and the price of the Call

Class Assignment: Testing Put-Call Parity Consider the following data for a stock: Stock price: $110 Call price (t=0.5 X=$105): $14 Put price (t=0.5 X=$105) : $5 Risk free rate 5% (continuously compounded rate) 1) Are these prices for the options violating the parity rule? Calculate! 2) If violated how could you create an arbitrage opportunity out of this?

Answer: 1) Parity if: C-P=S-Xe-rT So $14-$5= $110-$105*e -0.5*5 So $9= $ 7.59….this is a violation of parity 2) Arbitrage: Buy the cheap position ($7.59) and sell the expensive position ($9) i.e. borrow the PV of the exercise price X, Buy the stock, sell call and buy put: Buy the cheap position: Borrow PV of X= Xe-rT= +$ 102.41 (cash in) Buy stock - $110 (cash out) Sell the expensive position: Sell Call: +$14 (cash in) Buy Put: -$5 (cash out) Total $1.41 If S<$105 the pay offs are S-$105-$ 0+($105-S)= $ 0 If S>$105 the pay offs are S-$105-(S-$105)-$0=$ 0

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