Financial Options & Option Valuation Financial Options & Option Valuation

Session 4 – Binomial Model & Black Scholes Session 4 – Binomial Model & Black Scholes CORP FINC 5880 Shanghai 2015 -MOOCCORP FINC 5880 Shanghai 2015 -MOOC

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

Session 4: Assignment Valuation Options

• Take an option traded of your company’s stock on the option market

• Value the option with both the Binomial Model and the BS model

• Now compare the market price (quotation) of the option

• Compare your results and explain the differences…