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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

WITH 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)


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


Your answer should be:


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!


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)


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…


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)


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…


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…

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…


So how about Twitter?


The May X=$45 Call…P= $4.40


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 (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


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