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### Transcript of Session 4– Binomial Model & Black Scholes CORP FINC 5880 SUFE Spring 2014 Shanghai WITH...

• Financial Options & Option Valuation

Session 4 Binomial Model & Black Scholes CORP FINC 5880 SUFE Spring 2014 ShanghaiWITH ANSWERS ON CLASS ASSIGNMENTS

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

• Try to guestimatefor a call option price (5 min)

Stock Price Then call premium will?Exercise Price Then..?Volatility Then..?Time to expiration Then..?

Interest rate Then..?

Dividend payout Then..?

• Answer Try to guestimatefor a call option price (5 min)

Stock Price Then call premium will? Go upExercise Price Then..? Go down.Volatility Then..? Go up.Time to expiration Then..? Go up.

Interest rate Then..? Go up.

Dividend payout Then..? Go down.

CallPuttSo upupdownX updownupRf upupdownD updownupTime upupupSTDEV upupup

• Binomial Option PricingAssume a stock price can only take two possible values at expirationUp (u=2) or down (d=0.5)Suppose the stock now sells at \$100 so at expiration u=\$200 d=\$50If we buy a call with strike \$125 on this stock this call option also has only two possible resultsup=\$75 or down=\$ 0Replication means:Compare this to buying 1 share and borrow \$46.30 at Rf=8%The pay off of this are:

StrategyToday CFFuture CF if St>X (200)Future CF if ST

• Binomial model Key to this analysis is the creation of a perfect hedgeThe hedge ratio for a two state option like this is:

H= (Cu-Cd)/(Su-Sd)=(\$75-\$0)/(\$200-\$50)=0.5Portfolio 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 off0.5shares-1call (written)=\$ 23.15With 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 \$ 26.85Then you can make arbitrage profits:Risk free \$6.80no matter what happens to share price!

Cash flowAt S=\$50At S=\$200Write 2 options\$60\$ 0-\$150Buy 1 share-\$100\$50\$200Borrow\$40 at 8%\$40-\$43.20-\$43.20Pay off\$ 0\$ 6.80\$ 6.80

• 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 flowAt S=\$50At S=\$200.2 options???.. 1 share???Borrow/Lend\$ ? at 8%???Pay off???

• AnswerThen you can make arbitrage profits:Risk free \$4 no matter what happens to share price!The PV of \$4=\$3.70Or \$ 1.85 per option (exactly the amount by which the option was under priced!: \$26.85-\$25=\$1.85)

Cash flowAt \$50At \$200Buy 2 options-\$50\$ 0+\$150sell 1 share\$100-\$50-\$200Lending\$50 at 8%-\$50+\$54+\$54Pay off\$ 0\$4\$ 4

• Breaking Up in smaller periodsLets say a stock can go up/down every half year ;if up +10% if down -5%If you invest \$100 todayAfter half year it is u1=\$110 or d1=\$95After the next half year we can now have:U1u2=\$121 u1d2=\$104.50 d1u2= \$104.50 or d1d2=\$90.25We are creating a distribution of possible outcomes with \$104.50 more probable than \$121 or \$90.25.

• Class assignment: Binomial model(5 min)If up=+5% and down=-3% calculate how many outcomes there can be if we invest 3 periods (two outcomes only per period) starting with \$100.Give the probability for each outcome

Imagine we would do this for 365 (daily) outcomeswhat kind of output would you get?What kind of statistical distribution evolves?

ProbabilityCalculationResult3 up moves1/8=12.5%\$100*(1.05)^3\$ 115.762 up and 1 down moves 3/8=37.5%\$100*1.05^2*0.97\$ 106.941 up and 2 down moves3/8=37.5%\$100*1.05*0.97^2\$98.793 down moves1/8=12.5%\$100*0.97^3\$91.27

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

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

• Class assignment: Black Scholes Assume the BS option model: Co= So e-dt(N(d1)) - X e-rt(N(d2))d1=(ln(S/X)+(r-d+2/2)t)/ (t)d2=d1- t

In which: Co= Current Call Option Value; So= Current Stock Price; d= dividend yield; N(d)= the probability that a random draw from a standard Normal distribution will be less than d; X=Exercise Price of the option; e=the basis of natural log function; r=the risk free interest rate (opportunity cost); t=time to expirations of the option IN YEARS; ln=natural log function LN(x) in excel; =b Standard deviation of the annualized continuously compounded rate of return of the underlying stockN(d1)= a conditional probability of how far in the money the call option will be at expiration if and only if St>X; N(d2)= the probability that St will be at or above X

If you use EXCEL for N(d1) and N(d2) use NORMSDIST function!TRY THIS:stock price (S) \$100Strike price (X) \$95Rf ( r)=10% Dividend yield (d)=0Time to expiration (t)= 1 quarter of a yearStandard deviation =0.50A)Calculate the theoretical value of a call option with strike price \$95 maturity 0.25 yearB) if the volatility increases to 0.60 what happens to the value of the call? (calculate it)

• answerA) Calculate: d1= ln(100/95)+(0.10-0+0.5^2/2)0.25/(0.5*(0.25^0.5))=0.43Calculate d2= 0.43-0.5*(0.25^0.5)=0.18From 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))=0.4043D2= 0.4043-0.6(0.25^0.5)=0.1043N(d1)=0.6570N(d2)=0.5415Co=\$100*0.6570-\$ 95*e -.10*0.25 *0.5415=\$15.53

Higher volatility results in higher call premium!

• Sheet1

Option valuation

BS Model

ThenThen

INPUTSCP

IF

Stock Price\$100UpUPDOWN

Exercise Price\$95UpDOWNUP

Interest Ratedecimal0.10UpUPDOWN

Dividend Yielddecimal0.00UpDOWNUP

Time to Expirationdecimal0.25UpUPUP

Standard Deviationdecimal0.50UpUPUP

PROCESS

d10.4301731776

d20.1801731776

NORM d10.6664651641

NORM d20.5714916925

CALL\$13.70

PUT\$6.35

• Lets try a real option;Apple Inc. yesterday closed at just below \$525 at \$524.94The call with strike \$520 expiring 25 April (Friday) was priced \$14.10Note that this option is almost \$5 in the moneyThe market values the time value of less than one week at \$14.10 - \$5= \$9.10Rf= 2.72% STDEV=almost 40% t=7/365 days

1) Assume first that Apple does not pay a dividend how does the BS model price this option?2) Now assume the dividend yield for Apple Inc. at 2.3% recalculate the option value with BS

• AnswerWithout dividendWith dividendConclude: real close to market price and dividend has small impact

Option valuationBS Model

INPUTS

Stock Price\$524.94Exercise Price\$520Interest Ratedecimal0.0272Dividend Yielddecimal0.00Time to Expirationdecimal0.019178082Standard Deviationdecimal0.40

PROCESS

d10.207803248d20.152409266NORM d10.582308702NORM d20.560567926

CALL\$14.33PUT\$9.12

Option valuationBS Model

INPUTS

Stock Price\$524.94Exercise Price\$520Interest Ratedecimal0.0272Dividend Yielddecimal0.02Time to Expirationdecimal0.019178082Standard Deviationdecimal0.40

PROCESS

d10.199840363d20.144446382NORM d10.579197283NORM d20.557426003

CALL\$14.20PUT\$9.22

• Or lets find volatility of facebook stock

• FacebookSo= \$58.94 (yesterday)The X=\$58.50 call (19) May 2014Is priced \$ 5.40

With BS we can estimate the implicit volatility

Note that this is significantly higher than Apple

Option valuationBS Model

INPUTS

Stock Price\$58.94Exercise Price\$58.5Interest Ratedecimal0.0273Dividend Yielddecimal0.00Time to Expirationdecimal0.084931507Standard Deviation0.750

PROCESS

d10.154176885d2-0.064395695NORM d10.561264866NORM d20.474327579

CALL\$5.40PUT\$4.82

• The May X=\$45 CallP= \$4.40

Option valuationBS Model

INPUTS

Stock Price\$45.01Exercise Price\$45Interest Ratedecimal0.0273Dividend Yielddecimal0.00Time to Expirationdecimal0.084931507Standard Deviation0.833 PROCESS

d10.131847002d2-0.110914276NORM d10.552447346NORM d20.455842162

CALL\$4.40PUT\$4.29

• Homework assignment 9: Black & ScholesCalculate the theoretical value of a call option for your company using BSNow compare the market value of that optionHow big is the difference?How can that difference be explained?

• Implied VolatilityIf we assume the market value is correct we set the BS calculation equal to the market price leaving open the volatilityThe volatility included in todays market price for the option is the so called implied volatilityExcel can help us to find the volatility (sigma)

• Implied VolatilityConsider one option series of your company in which there is enough volume tradingUse 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 - VIXInvestor fear gauge

• Class assignment:Black Scholes put option valuation (10 min)P= Xe-rt(1-N(d2))-Se-dt(1-N(d1))

Say strike price=\$95 Stock price= \$100Rf=10%T= one quarterDividend yield=0A) Calculate the put value with BS? (use the normal distribution in your book pp 516-517)B) Show that if you use the call-put parity:P=C+PV(X)-S where PV(X)= Xe-rt and C= \$ 13.70 and that the value of the put is the same!

• A