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Financial Options & Option Valuation. Session 4– Binomial Model & Black Scholes CORP FINC 5880 SUFE Spring 2014 Shanghai WITH ANSWERS ON CLASS ASSIGNMENTS. What determines option value?. Stock Price (S) Exercise Price (Strike Price) (X) Volatility ( σ ) Time to expiration (T)

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Financial Options & Option Valuation

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## Financial Options & Option Valuation

Session 4– Binomial Model & Black Scholes

CORP FINC 5880 SUFE Spring 2014 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)

### 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:

### 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!

### 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!

• 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)

### 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?

### 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 stock

N(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) \$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)

• 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

Without dividend

With dividend

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

### Or let’s find volatility of facebook stock

• 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…

### 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!

• 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?

• 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:

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

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

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

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

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

### Example…

• ResultsCalc typeValue

• Price P0.25517 Price of the call option

• Delta D0.28144 Premium changes with \$ 0.28144 if share price is up \$1

• Gamma G0.21606 Sensitivity of delta for changes in price of share

• Vega V0.01757 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 R0.00597 If the risk less rate is up 1% the premium will increase \$ 0.00597