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Financial Options & Option Valuation

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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  1. Financial Options & Option Valuation Session 4– Binomial Model & Black Scholes CORP FINC 5880 SUFE Spring 2014 Shanghai WITH ANSWERS ON CLASS ASSIGNMENTS

  2. What determines option value? • Stock Price (S) • Exercise Price (Strike Price) (X) • Volatility (σ) • Time to expiration (T) • Interest rates (Rf) • Dividend Payouts (D)

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

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

  5. Your answer should be:

  6. 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:

  7. 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

  8. 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!

  9. 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!

  10. 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)

  11. 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….

  12. 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?

  13. Answer…

  14. 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)

  15. 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)

  16. 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)

  17. 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!

  18. In Excel…

  19. 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

  20. Answer… Without dividend With dividend Conclude: real close to market price and dividend has small impact

  21. Or let’s find volatility of facebook stock

  22. 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…

  23. So how about Twitter?

  24. The May X=$45 Call…P= $4.40

  25. 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?

  26. 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)

  27. 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

  28. Implied Volatility Index - VIX Investor fear gauge…

  29. 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!

  30. 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

  31. 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…

  32. 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

  33. 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?

  34. 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

  35. 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

  36. If you want to know more about the MATH behind the BS model • http://www.youtube.com/watch?v=mqRjn3-kPvA

  37. 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)

  38. Example… • Results Calc type Value • 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

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