Financial Options & Option Valuation Financial Options & Option Valuation

Session 4– Binomial Model & Black Scholes Session 4– Binomial Model & Black Scholes

CORP FINC 5880 SUFE Spring 2014 ShanghaiCORP FINC 5880 SUFE Spring 2014 Shanghai

WITH ANSWERS ON CLASS ASSIGNMENTSWITH ANSWERS ON CLASS ASSIGNMENTS

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 ↑ Then…..?

Time to expiration↑

Then…..?

Interest rate ↑ Then…..?

Dividend payout ↑ Then…..?

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↑

Then…..? Go up.

Interest rate ↑ Then…..? Go up.

Dividend payout ↑ Then…..? Go down.

Your answer should be:

Call Putt

So up up down

X up down up

Rf up up down

D up down up

Time up up up

STDEV up up up

Binomial Option Pricing• Assume a stock price can only take two possible values at expiration• Up (u=2) or down (d=0.5)• Suppose the stock now sells at $100 so at expiration u=$200 d=$50• If we buy a call with strike $125 on this stock this call option also has

only two possible results• up=$75 or down=$ 0• Replication means:• Compare this to buying 1 share and borrow $46.30 at Rf=8%• The pay off of this are:

Strategy Today CF Future CF if St>X (200)

Future CF if ST<X(50)

Buy Stock -$100 +$200 +$50

Write 2 Calls +2C - $150 $ 0

Borrow PV(50) +$50/1.08 - $50 - $50

TOTAL +2C-$53.70(=$0) $0 (fair game) $0 (fair game)

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 -$43.20

Pay 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 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 -$50 -$200

Lending

$50 at 8%

-$50 +$54 +$54

Pay off $ 0 $4 $ 4

Breaking Up in smaller periods

• Lets say a stock can go up/down every half year ;if up +10% if down -5%

• If you invest $100 today• After half year it is u1=$110 or d1=$95• After the next half year we can now have:• U1u2=$121 u1d2=$104.50 d1u2= $104.50 or

d1d2=$90.25…• We 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) outcomes…what kind of output would you get?

• What kind of statistical distribution evolves?

Answer…

Probability Calculation Result

3 up moves 1/8=12.5% $100*(1.05)^3 $ 115.76

2 up and 1 down moves

3/8=37.5% $100*1.05^2*0.97 $ 106.94

1 up and 2 down moves

3/8=37.5% $100*1.05*0.97^2 $98.79

3 down moves 1/8=12.5% $100*0.97^3 $91.27

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 Assume the BS option model: Co= So e-dt(N(d1)) - X e-rt(N(d2))d1=(ln(S/X)+(r-d+σ2/2)t)/ (σ√t)d2=d1- σ√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; σ=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 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!

Let’s try a real option;• Apple Inc. yesterday closed at just below $525 at $524.94

• The call with strike $520 expiring 25 April (Friday) was priced $14.10

• Note that this option is almost $5 in the money

• The market values the time value of less than one week at $14.10 - $5= $9.10

• Rf= 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

Answer…Option valuationBS Model

INPUTS

Stock Price $ 524.94Exercise Price $ 520Interest Rate decimal 0.0272Dividend Yield decimal 0.00

Time to Expiration decimal 0.019178082Standard Deviation decimal 0.40

PROCESS

d1 0.207803248

d2 0.152409266

NORM d1 0.582308702

NORM d2 0.560567926

CALL $ 14.33PUT $ 9.12

Option valuationBS Model

INPUTS

Stock Price $ 524.94Exercise Price $ 520Interest Rate decimal 0.0272Dividend Yield decimal 0.02

Time to Expiration decimal 0.019178082Standard Deviation decimal 0.40

PROCESS

d1 0.199840363

d2 0.144446382

NORM d1 0.579197283

NORM d2 0.557426003

CALL $ 14.20PUT $ 9.22

Without dividend With dividend

Conclude: real close to market price and dividend has small impact

Or let’s find volatility of facebook stock

Facebook…So= $58.94 (yesterday)

• The X=$58.50 call (19) May 2014

• Is 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 Rate decimal 0.0273Dividend Yield decimal 0.00

Time to Expiration decimal0.08493150

7Standard Deviation 0.750

PROCESS

d10.15417688

5

d2

-0.06439569

5

NORM d10.56126486

6

NORM d20.47432757

9

CALL $ 5.40PUT $ 4.82

So how about Twitter?

The May X=$45 Call…P= $4.40Option valuationBS Model

INPUTS

Stock Price $ 45.01Exercise Price $ 45Interest Rate decimal 0.0273Dividend Yield decimal 0.00

Time to Expiration decimal 0.084931507Standard Deviation 0.833

PROCESS

d1 0.131847002

d2 -0.110914276

NORM d1 0.552447346

NORM d2 0.455842162

CALL $ 4.40PUT $ 4.29

Homework assignment 9: 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)

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…

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= $100• Rf=10%• T= one quarter• Dividend yield=0• A) 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!

Answer:

• BS European option:• P= Xe-rt(1-N(d2))-Se-dt(1-N(d1))

• A) So: $95*e-.10*0.25*(1-0.5714) - $100(1-.6664)= $ 6.35

• B) Using call put parity:• P=C+PV(X)-S= $13.70+$95e -.10*.25 -$100= $ 6.35

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

Black Scholes

• The Black-Scholes model is used to calculate a theoretical call price (ignoring dividends paid during the life of the option) using the five key determinants of an option's price: stock price, strike price, volatility, time to expiration, and short-term (risk free) interest rate.

Myron Scholes and Fischer Black

If you want to know more about the MATH behind the BS model

• http://www.youtube.com/watch?v=mqRjn3-kPvA

Some spreadsheets will show you the option Greeks;

• Delta (Delta (δδ)):: Measures how much the Measures how much the premium changes if the underlying share premium changes if the underlying share price rises with $ 1.- (positive for Call price rises with $ 1.- (positive for Call options and negative for Put options)options and negative for Put options)

• Gamma (Gamma (γγ):): Measures how sensitive Measures how sensitive delta is for changes in the underlying delta is for changes in the underlying asset price (important for risk managers)asset price (important for risk managers)

• Vega (Vega (νν):): Measures how much the Measures how much the premium changes if the volatility rises premium changes if the volatility rises with 1%; higher volatility usually means with 1%; higher volatility usually means higher option premiahigher option premia

• Theta (Theta (θθ):): Measrures how much the Measrures how much the premium falls when the option draws one premium falls when the option draws one day closer to expiryday closer to expiry

• Rho (Rho (ρρ):): Measrures how much the Measrures how much the premium changes if the riskless rate premium changes if the riskless rate rises with 1% (positive for call options rises with 1% (positive for call options and negative for put options) and negative for put options)

Black Scholes Option Pricing Model

INPUTS Symbols

Stock Price Now S $102.50Standard Deviation Annual σ 86.07%Riskfree rate Annual r 5.47%Exercise Price E $100.00Time to Maturity in Years T 0.3556

OUTPUTS

d1 0.342635d2 -0.17062N(d1) 0.634064N(d2) 0.432262

Cal Price C $22.60

-d1 -0.34264 -d2 0.170619 N(-d1) 0.365936 N(-d2) 0.567738

Put Price P $18.17

Example…• Results Calc typeValue• Price P 0.25517 Price of the call option

• Delta D 0.28144 Premium changes with $ 0.28144 if share price is up $1

• Gamma G 0.21606 Sensitivity of delta for changes in price of share

• Vega V 0.01757 Premium will go up with $ 0.01757 if volatility is up 1%

• Theta T -0.00419 1 day closer to expiry the premium will fall $ 0.00419

• Rho R 0.00597 If the risk less rate is up 1% the premium will increase $ 0.00597