Cross Section Pricing Intrinsic Value Options Option Price Stock Price

Cross Section Pricing Intrinsic Value Options Option Price Stock Price

Interest Rates Settlement Projects Computer software Options

Components of the Option Price 1 - Underlying stock price = Ps 2 - Striking or Exercise price = S 3 - Volatility of the stock returns (standard deviation of annual returns) = v 4 - Time to option expiration = t = days/ Time value of money (discount rate) = r 6 - PV of Dividends = D = (div) e -rt

Black-Scholes Option Pricing Model O C = P s [N(d 1 )] - S[N(d 2 )]e -rt

Black-Scholes Option Pricing Model O C = P s [N(d 1 )] - S[N(d 2 )]e -rt O C - Call Option Price P s - Stock Price N(d 1 ) - Cumulative normal density function of (d 1 ) S - Strike or Exercise price N(d 2 ) - Cumulative normal density function of (d 2 ) r - discount rate (90 day comm paper rate or risk free rate) t - time to maturity of option (days/365) v - volatility - annual standard deviation of returns

(d 1 )= ln + ( r + ) t PsSPsS v22v22 v t Cumulative Normal Density Function N(d 1 )=

(d 1 )= ln + ( r + ) t PsSPsS v22v22 v t Cumulative Normal Density Function (d 2 ) = d 1 -v t

Call Option Example What is the price of a call option given the following?. P = 36r = 10%v =.40 S = 40t = 90 days / 365

Call Option (d 1 ) = ln + ( r + ) t PsSPsS v22v22 v t (d 1 ) = N(d 1 ) = =.3794 Example What is the price of a call option given the following?. P = 36r = 10%v =.40 S = 40t = 90 days / 365

Call Option (d 2 ) = N(d 2 ) = =.3065 (d 2 ) = d 1 -v t Example What is the price of a call option given the following?. P = 36r = 10%v =.40 S = 40t = 90 days / 365

Call Option Example What is the price of a call option given the following?. P = 36r = 10%v =.40 S = 40t = 90 days / 365 O C = P s [N(d 1 )] - S[N(d 2 )]e -rt O C = 36[.3794] - 40[.3065]e - (.10)(.2466) O C = $ 1.70

Call Option Example What is the price of a call option given the following?. P = 36r = 10%v =.40 S = 40t = 90 / 365 days

Call Option Example (same option) What is the price of a call option given the following?. P = 41r = 10%v =.42 S = 40t = 30 days/ 365 (d 1 ) = ln + (.1 + ) 30/ /365 (d 1 ) =.3335N(d 1 ) =.6306

(d 2 ) =.2131 N(d 2 ) =.5844 (d 2 ) = d 1 -v t = (.0907) Call Option Example (same option) What is the price of a call option given the following?. P = 41r = 10%v =.42 S = 40t = 30 days/ 365

Call Option O C = P s [N(d 1 )] - S[N(d 2 )]e -rt O C = 41[.6306] - 40[.5844]e - (.10)(.0822) O C = $ 2.67 Example (same option) P = 41r = 10%v =.42 S = 40t = 30 days/ 365

Call Option Example (same option) P = 41r = 10%v =.42 S = 40t = 30 days/ 365

Call Option Example (same option) P = 41r = 10%v =.42 S = 40t = 30 days/ 365 Intrinsic Value = = 1 Time Premium = = 1.67 Profit to Date = =.94 Due to price shifting faster than decay in time premium

Call Option Example (same option) P = 41r = 10%v =.42 S = 40t = 30 days/ 365 Q: How do we lock in a profit? A: Sell the Call

Call Option Example (same option) P = 41r = 10%v =.42 S = 40t = 30 days/ 365

Call Option Example (same option) P = 41r = 10%v =.42 S = 40t = 30 days/ 365

Call Option Example (same option) P = 41r = 10%v =.42 S = 40t = 30 days/ 365

Put Option Black-Scholes O p = S[N(-d 2 )]e -rt - P s [N(-d 1 )] Put-Call Parity (general concept) Put Price = Oc + S - P - Carrying Cost + D Carrying cost = r x S x t Call + Se -rt = Put + P s Put = Call + Se -rt - P s

Put Option Example (same option) P = 41r = 10%v =.42 S = 40t = 30 days/ 365 Calculate the Value of The Put [N(-d 1 ) =.3694 [N(-d 2 )=.4156 Black-Scholes O p = S[N(-d 2 )]e -rt - P s [N(-d 1 )] O p = 40[.4156]e -.10(.0822) - 41[.3694] O p = 1.34

Example (same option) P = 41r = 10%v =.42 S = 40t = 30 days/ 365 Calculate the Value of The Put Put-Call Parity Put = Call + Se -rt - P s Put = e -.10(.0822) - 41 Put = = 1.34 Put Option

Put-Call Parity & American Puts P s - S < Call - Put < P s - Se -rt Call + S - P s > Put > Se -rt - P s + call Example - American Call > Put > e -.10(.0822) > Put > 1.34 With Dividends, simply add the PV of dividends

Volatility Calculate the Annualized variance of the daily relative price change Square root to arrive at standard deviation Standard deviation is the volatility

Implied Volatility O PriceVolumeImplied V Jan30C Jan35C Apr35C Apr40C Recalculate the volatility using volume & price deviation

Implied Volatility VolumeVolume Weights Jan30C5050/200 =.25 Jan35C9090/200 =.45 Apr35C5555/200 =.275 Apr40C55 / 200=

Implied Volatility Distance Factor (25% tolerance) Jan30C[(3/33)-.25] 2 /.25 2 =.41 Jan35C[(2/33)-.25] 2 /.25 2 =.57 Apr35C[(2/33)-.25] 2 /.25 2 =.57 Apr40C[(7/33)-.25] 2 /.25 2 =.02 Weight Adjusted Implied volatility = 298 =.41x.25x x.45x = x x

Expected Return Example P = 41 40C=2.67 Possible PriceProbProfitProbxProfit Expected Profit =

Expected Return Steps for Infinite Distribution of Outcomes 1 - convert annual V to time adjusted V V t = V (t.5 ) 2 - Prob(below a price q ) = N [ln(q/p) /V t ] 3 - Prob (above price q ) = 1 - Prob (below)

Expected Return Example V t =.42 (30/365).5 =.1204 Prob (<40) = N[ln(40/41) /.1204] = N[-.2051] =.4187 Prob (<42.67) = N[ln(42.67/41) /.1204] = N[.3316] =.6299 Example (same option) P = 41r = 10%v =.42 S = 40t = 30 days/ 365Call = 2.67

Expected Return Example (same option) P = 41r = 10%v =.42 S = 40t = 30 days/ 365Call = 2.67 $ % 58% 63%

Dividends Example Price = 36Ex-Div in 60 $0.72 t = 90/365r = 10% P D = e -.10(.1644) = Put-Call Parity Amer D+ C + S - P s > Put > Se -rt - P s + C + D Euro Put = Se -rt - P s + C + D + CC

Binomial Pricing Model

Binomial Pricing Outcome Trees Example - one month option Price = $20Possible outcomes = 22 or 18 21call = ?Monthly risk free rate = 1% Intrinsic 22 = 1 Intrinsic 18 = 0

T0T1ValueX Shares Pa=2222x -1 P=20 Pb=1818x 22x - 1 = 18x x=.25 at.25 shares A=B Binomial Pricing

at.25 shares A=B T1 Value = 22(.25) - 1 = 4.5 T0 Value = 20 (.25) - Call = 5 - Call (T0 Value) (1+ r) = 4.5 (5-call) (1.01) = 4.5 call =.5446 Binomial Pricing

Probability Up = p = (a - d)Prob Down = 1 - p (u - d) a = e r  t d =e -  [  t].5 u =e  [  t].5  t = time intervals as % of year Binomial Pricing

Example Price = 36  =.40 t = 90/365  t = 30/365 Strike = 40r = 10% a = u = d =.8917 Pu =.5075 Pd =.4925 Binomial Pricing

Binomial Pricing

50.78 = price Binomial Pricing

Binomial Pricing = price = intrinsic value

Binomial Pricing = price = intrinsic value

Binomial Pricing = price = intrinsic value

Binomial Pricing = price = intrinsic value

Project Select a Call option (w/ high vol & expires next month) Use spreadsheet to calc BS value for this Friday Calc volatility (include div if necessary) Calc Expected Return Probability Intervals Use spreadsheet to calc Binomial value. Use weekly intervals. Chart Black Scholes position Create a cross section price chart (showing time value decay) - Calculate option price at various stock prices for 0, 30, 60, 90 days. Include 1 page executive summary