1. 1 Lec 13A: Black and Scholes OPM (Stock pays no dividends) (Hull,Ch 13) From Binomial model to Black-Scholes Suppose S 0 = 40, and r = 24%/yr (c.c.). Question: What is the price of a C E (K=$40, T= 1-yr) ? One simple solution: Use the Binomial OPM (with R-N probabilities). Case I. Break 1 year period into 1 period Let u = 1.35, d = 0.74, then R-N prob = 0.87, T = 1 Stock Price TreeR-N Prob Call Values (K=$40) 53.99 ➀ p = 0.87 13.99 40C 0 = ? 29.63 ➁ 1 - p = 0.13 0 Then, C 0 = e -0.24 [ 13.99(0.87) + 0(0.13) ] = $9.59 Lec 13A Black & Scholes OPM

2. 2 Case II. Break 1 year into 2 sub-periods. Set u = 1.236, d = 0.8089; then p = 0.7454 Stock Price Tree R-N Prob Values at Expiration 61.138 ➀ p 2 = 0.556 ➀ 21.14 ➀ 49.45 40 40 ➁ 2p(1-p) = 0.380 ➁ C 0 = ? 0 ➁ 32.35 26.17 ➂ (1-p) 2 = 0.065 ➂ 0 ➂ Call Value C 0 = e -0.24 [ 21.14(0.556) + 0(0.380) + 0(0.065) ] = $9.24

3. Lec 13A Black & Scholes OPM3 Case III. Break 1 year into 3 sub-periods. u =e 0.30√(1/3) =e σ√T/n = 1.189, d= 1/u i.e. d=e -0.30√(1/3) =e -σ√T/n = 0.841, and p ={e 0.24(1/3) -d}/{u-d} = {1.0833 -0.841} /{1.189-0.841}= 0.696 Call Value C 0 = e -0.24 [ 27.26(0.3372) + 7.56(0.4418) + 0 ] = $9.86 Stock Price TreeR-N ProbCall Values 67.26 p 3 = 0.337227.26 56.56 47.56 3p 2 (1-p) = 0.44187.56 40 C 0 = ? 33.64 3p(1-p) 2 = 0.19290 28.29 23.79 (1-p) 3 = 0.02810

4. Lec 13A Black & Scholes OPM4 Limit Case: Black-Scholes European Call Option formula (p. 4) as n → ∞, the binomial model becomes C 0 = S 0 N(d 1 ) – K e -rT N(d 2 ) d 1 = [ ln(S 0 /K) +(r+σ 2 /2) T ]/(σ √T), and d 2 = [ ln(S 0 /K) +(r-σ 2 /2) T ]/(σ √T) T = time to expiration in years (or fraction of a year) σ 2 = annualized variance of stock returns N(d 1 )= area under graph of standard normal cumulative prob ( -∞ to d 1 ) N(d 2 )= area under graph of standard normal cumul. probability (-∞ to d 2 ) in Excel =NORMSDIST(d 1 ) r= c.c. risk-free rate

5. Lec 13A Black & Scholes OPM5 Limit Case: Black-Scholes European Call Option formula (p. 4) C 0 = S 0 N(d 1 ) – K e -rT N(d 2 ) Example: Use same data as in cases I, II etc. to price a European Call S 0 = $40, K = $40, T = 1 (1 year), σ 2 = 0.09/year ⇒ σ = 0.30/yr d 1 = [ ln(40/40) +(0.24+0.09/2) 1 ]/(0.30 √1) = 0.950 d 2 = [ ln(40/40) +(0.24-0.09/2) 1 ]/(0.30 √1) = 0.65 N(d 1 ) = Area up to 0.95 = 0.8289, N(d 2 ) = Area up to 0.65 = 0.74215 ∴ C 0 = 40 (0.8289) - 40(e -0.24 (1) ) (0.74215) = $9.81 Compare to Binomial prices: n = 1, Binomial price = $9.59 n = 2, Binomial price = $9.24 n = 3, Binomial price = $9.86n = 4, Binomial price = $9.51 NOT BAD!!

6. Lec 13A Black & Scholes OPM6 Black-Scholes model for European Puts (p. 4) Recall the Put-Call relationship: +S +P = +C +B. Solve for the price of the put: P 0 = S 0 [1-N(d 1 ) ] – K e -rT [1-N(d 2 ) ] Alternatively, use the fact that N(K) + N(-K) = 1, then P 0 = -S 0 N(-d 1 ) + K e -rT N(-d 2 ) Example: Use same data to price a European Put S 0 = $40, K = $40, T = 1 (1 year), σ 2 = 0.09/year ⇒ σ = 0.30/yr P 0 = -40(1-0.8289) + 40(e -0.24 (1) ) (1-0.74215) = $1.27

7. Lec 13A Black & Scholes OPM7