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Chapter 19a - A B C D E F G H I J Black-Scholes option...

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Black-Scholes option pricing formula S 50 Current stock price X 50 Exercise price r 10.00% Risk-free rate of interest T 1 Time to maturity of option (in years) Sigma 25% Stock volatility 0.5250 <-- (LN(S/X)+(r+0.5*sigma^2)*T)/(sigma*SQRT(T)) 0.2750 0.7002 0.6083 Call price 7.49 Put price 2.73 <-- call price - S + X*Exp(-r*T): by Put-Call parity Stock B-S Intrinsic Price Value Value 36 10.2246 14 38 8.7220 12 40 7.3615 10 42 6.1496 8 44 5.0870 6 46 4.1691 4 48 3.3872 2 50 2.7298 0 52 2.1835 0 54 1.7345 0 56 1.3692 0 58 1.0745 0 60 0.8389 0 62 0.6518 0 64 0.5043 0 66 0.3886 0 intrinsic value. If we could exercise early, the option would 68 0.2985 0 have increased value. Hence: Early exercise may be valuable for puts. Stock B-S Call Intrinsic Price Value Value 20 0.0010844 0 23 0.0083245 0 26 0.0398152 0 29 0.1354873 0 32 0.358385 0 35 0.7836194 0 38 1.4801745 0 41 2.4949258 0 44 3.8451327 0 47 5.5199716 0 50 7.4878954 0 53 9.7058247 3 56 12.127284 6 59 14.708159 9 62 17.409952 12 65 20.20104 15 68 23.056607 18 d 1 d 2 <-- d 1 - sigma*SQRT(T) N(d 1 ) <-- Uses formula NormSDist(d 1 ) N(d 2 ) <-- Uses formula NormSDist(d 2 ) <-- S*N(d 1 )-X*exp(-r*T)*N(d 2 ) When the stock price is low, the B-S value is less than the 35 40 45 50 55 60 65 70 0 2 4 6 8 10 12 14 16 Intrinsic Put Value versus Black-Scholes Value B-S Value Intrinsic Value St ock price S ($) Value ($) 10 20 30 40 50 60 70 80 0 5 10 15 20
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