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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

- Stock Price (S)
- Exercise Price (Strike Price) (X)
- Volatility (σ)
- Time to expiration (T)
- Interest rates (Rf)
- Dividend Payouts (D)

- 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:

- 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

- Then you can make arbitrage profits:
- Risk free $6.80…no matter what happens to share price!

- 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)

- 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….

- 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?

- 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)

- 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)

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!

- 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

- 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…

- 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?

- 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)

- 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

Investor fear gauge…

- 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))

- 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…

- 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

- 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 ($9) i.e. borrow the PV of the exercise price X, Buy the stock, sell call and buy put:
- Buy the cheap position:
- Borrow PV of X= Xe-rT= +$ 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 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

- http://www.youtube.com/watch?v=mqRjn3-kPvA

- 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)

- 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