Financial Options & Financial Options & Option Valuation Option Valuation

FINC 5880 week 3 Binomial Model & Black ScholesFINC 5880 week 3 Binomial Model & Black Scholes

FALL 2013 ShanghaiFALL 2013 Shanghai

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.

Try to guestimate…for a put option price… (5 min)

Stock Price ↑ Then put premium will?

Exercise Price ↑

Then…..?

Volatility ↑ Then…..?

Time to expiration↑

Then…..?

Interest rate ↑ Then…..?

Dividend payout ↑ Then…..?

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 (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…

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

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

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

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

Playing around with Black Scholes

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

http://www.spreadsheetmodeling.com/Black%20Scholeshttp://www.spreadsheetmodeling.com/Black%20Scholes%20Option%20Pricing%20-%20Basics.htm%20Option%20Pricing%20-%20Basics.htm

Testing EbaY premiumsINPUTS     Symbols  

 EbaY $ 35 Call October 2005      

Stock Price Now S $35.37

Standard Deviation Annual σ 22.18%

Riskfree rate Annual r 4.17%

Exercise Price   E $35.00

Time to Maturity in Years T 0.254795

       

OUTPUTS      

     

d1       0.244807

d2     0.132849

N(d1)     0.596697

N(d2)     0.552843

Cal Price   C $1.96

       

-d1     -0.24481

-d2     -0.13285

N(-d1)     0.403303

N(-d2)     0.447157

Put Price     P $1.22

Quote: $ 2.75Quote: $ 2.75

Quote: $ 2.00Quote: $ 2.00

At STDEV At STDEV 34% the 34% the premium premium gets close gets close to the to the quotequote

Adjusting the STDEV…INPUTS     Symbols  

       

Stock Price Now S $35.37

Standard Deviation Annual σ 33.00%

Riskfree rate Annual r 4.17%

Exercise Price   E $35.00

Time to Maturity in Years T 0.254795

       

OUTPUTS      

     

d1       0.210203

d2     0.043628

N(d1)     0.583245

N(d2)     0.517399

Cal Price   C $2.71

       

-d1     -0.2102

-d2     -0.04363

N(-d1)     0.416755

N(-d2)     0.482601

Put Price     P $1.97

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    

Inputs    

Calculation_type   P

Pricing_model (European or American) 1

Option_type   C

Strike   13.00

http://www.hoadley.net/options/options.htmhttp://www.hoadley.net/options/options.htm   12.50

Deal_date   14-Jan-00

Expiry_date   25-Mar-00

Volatility   30%

  5.75%

     

Optional arguments:    

Dividend_type   D

     

Dividend_details: Amt Ex-div

Dividend 1 0.20 01-Feb-00

Dividend 2 0.50 01-Mar-00

     

Binomial_steps   100

     

Results Calc type Value

Price P 0.25517

Delta D 0.28144

Gamma G 0.21606

Vega V 0.01757

Theta T -0.00419

Rho R 0.00597

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

Assignment Options FINC 5880• Download the Black Scholes Download the Black Scholes

model and calculate the model and calculate the theoretical Call and Put prices theoretical Call and Put prices of an option series of your of an option series of your company; now compare the company; now compare the calculated premium with the calculated premium with the real premium. Why are they real premium. Why are they different? different?

• Download the add in for Excel Download the add in for Excel fromfrom http://www.hoadley.net/options/options.htm

• This download will help you to This download will help you to calculate the Greeks ; Delta, calculate the Greeks ; Delta, Gamma, Theta, Vega, Rho for your Gamma, Theta, Vega, Rho for your company’s optionscompany’s options

• What are their values and what do What are their values and what do they mean? they mean?

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