Financial Options & Option Valuation Financial Options & Option Valuation

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

CORP FINC Shanghai 2015 - ANSCORP FINC Shanghai 2015 - ANS

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 premium Put premium

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: • 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!

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

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

Class Assignment: Valuation Options

1) You are determining the Value of a Call Option with the Binomial model for Wal Mart with X=$75

You know: So=$80 St(u)=$90 St(d)=$70

Maturity 1 year and Rf=2% per year

What is the theoretical value of this Call Option?

(Show your calculation!)

2) Somebody is telling you that you made a mistake and that St(u)=$95 and St(d)=$65 i.e. the STDEV(return) is higher than in 1;

Recalculate the theoretical value of the Call Option with X=$75 under these new assumptions with the Binomial model.

3) Assume that your answer under 2 is “correct”; now use the Black Scholes model ; if the value of the Call is what you found in 2) what is the implied volatility of Wal Mart stock according to BS? Copy the values of d1 and d2 as well as N(d1) and N(d2) to your answer sheet! You can assume a dividend yield of 2.4%...

4) If somebody told you that the market value of this option is $10 how can you execute ARBITRAGE and gain a RISK FREE profit? How much profit can you realize? Show your detailed actions and COMPUTATIONS.

Solutions…

CF Today St=$90 St=$70

Buy 3 Stocks -$240 +$270 +$210

Write 4 Calls +4C -$60 $0

Borrow PV($210) +$210/1.02=$205.88 -$210 -$210

TOTAL $0 $0 $0

The hedge ratio: (Cu-Cd)/(Su-Sd)=$15/$20=3/4 (buy 3 stocks write 4 calls)-$240+4C+$205.88 =$0 so C=$8.53

And if Su=$95 and Sd=$65

CF Today St=$95 St=$65

Buy 2 Stocks -$160 +$190 +$130

Write 3 Calls +3C -$60 $0

Borrow PV($130) +$130/1.02=$127.45 -$130 -$130

TOTAL $0 $0 $0

The hedge ratio: (Cu-Cd)/(Su-Sd)=$20/$30=2/3 (buy 2 stocks write 3 calls)-$260+3C+$127.45 =$0 so C=$10.85

The volatility of the Wal Mart stock is higher so this should indeed result in a higher value for the same option…

Black Scholes inputs..Option valuationBS Model

INPUTS

Stock Price $ 80Exercise Price $ 75Interest Rate decimal 0.02Dividend Yield decimal 0.02Time to Expiration decimal 1

Standard Deviation decimal 0.27730

PROCESS

d1 0.356964176

d2 0.079664176

NORM d1 0.639440688

NORM d2 0.531747824

CALL $ 10.85PUT $ 6.26

Arbitrage

Action NOW St=$95 St=$65

Buy 3 Calls X=$75 (market price $10)

-$30 +$60 $0

Sell 2 stocks WMT +$160 -$190 -$130

Lend/INVEST $130 - $130 +$130(1.02) +$130(1.02)

TOTAL $0 $ 2.60 +$2.60

Note that if the option is cheaper in the market than what it should be according to the valuation you will buy it! You can realize a risk free profit of $2.60 for 3 calls or $2.60/3 for every call the PV of this is exactly the price difference of $0.85. ($10.85-$10)

Q.E.D.