Session 6 – Binomial Model & Black Scholes CORP FINC Shanghai ANS

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 premiumPut premium So upupdown X updownup Rf upupdown D updownup Time upup STDEV upup

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: StrategyToday CFFuture 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 TOTAL+2C-$53.70(=$0) $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 $25 with certainty…. So now solve: Hedged portfolio value=present value certain pay off 0.5shares-1call (written)=$ 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 $ 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-$43.20 Pay off$ 0$ 6.80

Class assignment: What if the option is under-priced? Say $25 instead of $ (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!: $ $25=$1.85) Cash flow At $50At $200 Buy 2 options -$50$ 0+$150 sell 1 share $100-$50-$200 Lending $50 at 8% -$50+$54 Pay off$ 0$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=$121 u1d2=$ d1u2= $ or d1d2=$90.25… We are creating a distribution of possible outcomes with $ 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… ProbabilityCalculationResult 3 up moves1/8=12.5%$100*(1.05)^3$ up and 1 down moves 3/8=37.5%$100*1.05^2*0.97$ up and 2 down moves 3/8=37.5%$100*1.05*0.97^2$ down moves1/8=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+σ 2 /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)+( ^2/2)0.25/(0.5*(0.25^0.5))=0.43 Calculate d2= *(0.25^0.5)=0.18 From the normal distribution find: N(0.43)= (interpolate) N(0.18)= Co=$100* $95*e -.10*0.25 *0.5714=$13.70 B) If the volatility is 0.6 then : D1= ln(100/95)+( /2)0.25/(0.6*(0.25^0.5))= D2= (0.25^0.5)= N(d1)= N(d2)= Co=$100* $ 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 ) B) Show that if you use the call-put parity: P=C+PV(X)-S where PV(X)= Xe -rt and C= $ 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*( ) - $100( )= $ 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 ($9) i.e. borrow the PV of the exercise price X, Buy the stock, sell call and buy put: Buy the cheap position: Borrow PV of X= Xe -rT = +$ (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)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)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 premiaVega (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 expiryTheta (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)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 P Price of the call option Delta D Premium changes with $ if share price is up $1 Gamma G Sensitivity of delta for changes in price of share Vega V Premium will go up with $ if volatility is up 1% Theta T day closer to expiry the premium will fall $ Rho R If the risk less rate is up 1% the premium will increase $