Pricing of American options University of Oulu - Department of Finance Spring 2018
Volatility-based binomial price process uuuus 0 = 26.51 uuus 0 = 24.71 uus 0 = us 0 = S 0 = ds 0 = dds 0 = ddds 0 = 16.19 dddds 0 = 15.09
Binomial price process with risk-neutral probabilities = er t d = 0.525 = 1 = 0.475 24.71 16.19 26.51 15.09 uuuu 1 p 4 u p0 d = 0.0760 uuud, uudu uduu, duuu 4 p 3 u p1 d = 0.2749 uudd, udud, uddu duud, dudu, dduu 6 p 2 u p2 d = 0.3731 uddd, dudd ddud, dddu 4 p 1 u p3 d = 0.2251 dddd 1 p 0 u p4 d = 0.0509
Expected price and variance in a binomial tree Upward movement multiplicator : u = e σ t Downward movement multiplicator: d = e σ t Price after an upward movement: us Price after a downward movement: ds Probability of an upward movement: = er t d Probability of a downward movement: = 1 Geometric Brownian motion: S = rs t + σsɛ t E( S) = rs t Std( S) = σs t Var( S) = σ 2 S 2 t Taylor series approximations: (1) e r t = 1 + r t + (r t)2 2! + (r t)3 3! + (r t)4 4! +... 1 + r t (2) u = e σ t (3) d = e σ t = 1 + σ t + (σ t) 2 2! + (σ t) 3 3! = 1 σ t + (σ t) 2 (σ t) 3 2! 3! + (σ t) 4 4! + (σ t) 4 4! +... 1 + σ t + σ2 t... 1 σ t + σ2 t 2 2
Expected price and variance in a binomial tree Ê( S) = Ê(S) S = puus + ds S = [ u + (1 )d 1 ] S = [ () + d 1 ] S [ e r t d ] = () + d 1 S = ( ) e r t 1 S = (1 + r t 1)S = rs t (1) Var( S) = E( S 2 ) E( S) 2 = (us) 2 + (ds) 2 [ us + ds ] 2 = (us) 2 + (1 )(ds) 2 [ us + (1 )ds ] 2 = [ () 2 p 2 u ()2] S 2 = [ e r t d ()2 (e ] r t d) 2 () 2 () 2 S 2 [ = e r t (u + d) 1 e 2r t ] S 2 (1) (2) (3) (1) [ ( = (1 + r t) 1 + σ t + σ2 t + 1 σ ) ] t + σ2 t 1 (1 + r t) 2 S 2 2 2 = = [ ( (1 + r t) 2 + σ 2 ) t 1 (1 + r t) 2] S 2 [ 2 + σ 2 t + 2r t + rσ 2 ( t) 2 1 1 2r t r 2 ( t) 2] S 2 = σ 2 S 2 t
Pricing of a European put option n P S T p T P p T 1 0.0760 26.51 0.00 0.00 4 0.2749 0.00 0.00 6 0.3731 1.00 0.37 4 0.2251 3.63 0.82 1 0.0509 15.09 5.91 0.30 S 0 = σ = 0.20 T = 0.5 X = 21 1.0000 1.49 p 0 = Xe rt N( d 2 ) S 0 N( d 1 ) = 1.41 Ê[max(0, X S T )] = 1.49 p 0 = e 0.048 0.5 1.49 = 1.45 24.71 16.19 26.51 15.09 uuuu 1 p 4 u p0 d = 0.0760 uuud, uudu uduu, duuu 4 p 3 u p1 d = 0.2749 uudd, udud, uddu duud, dudu, dduu 6 p 2 u p2 d = 0.3731 uddd, dudd ddud, dddu 4 p 1 u p3 d = 0.2251 dddd 1 p 0 u p4 d = 0.0509
Binomial price with twenty-five time-steps σ = 0.20 T = 0.5 n = 25 t = 0.02 u = e σ t = 1.029 d = 1 u = e σ t = 0.972 = er t d = 0.510 = 1 = 0.490
Pricing of the option with twenty-five time-steps n P S T p T P p T 1 0.0000 40.87 0.00 0.00 25 0.0000 38.61 0.00 0.00 300 0.0000 36.47 0.00 0.00 2300 0.0001 34.45 0.00 0.00 12650 0.0005 32.54 0.00 0.00 53130 0.0021 30.74 0.00 0.00 177100 0.0068 29.03 0.00 0.00 480700 0.0178 27.43 0.00 0.00 1081575 0.0384 25.91 0.00 0.00 2042975 0.0697 24.47 0.00 0.00 3268760 0.1071 23.12 0.00 0.00 4457400 0.1404 21.84 0.00 0.00 5200300 0.1573 20.63 0.37 0.06 5200300 0.1512 19.48 1.52 0.23 4457400 0.1245 18.40 2.60 0.32 3268760 0.0877 17.39 3.61 0.32 2042975 0.0527 16.42 4.58 0.24 1081575 0.0268 15.51 5.49 0.15 480700 0.0114 14.65 6.35 0.07 177100 0.0040 13.84 7.16 0.03 53130 0.0012 13.07 7.93 0.01 12650 0.0003 12.35 8.65 0.00 2300 0.0000 11.67 9.33 0.00 300 0.0000 11.02 9.98 0.00 25 0.0000 10.41 10.59 0.00 1 0.0000 9.83 11.17 0.00 Binomial tree with 4 time steps: p 0 = e 0.048 0.5 1.49 = 1.45 Binomial tree with 25 time steps: p 0 = e 0.048 0.5 1.43 = 1.40 Binomial tree with 125 time steps: p 0 = 1.41 Black-Scholes: p 0 = Xe rt N( d 2 ) S 0 N( d 1 ) = 1.41 1.0000 1.43
50-step binomial tree
100-step binomial tree
European put option Step 1 24.71 26.51 X = 21 0.00 = max(0, 21 26.51) 0.00 = max(0, 21 ) 1.00 = max(0, 21 ) = er t d = 0.525 = 1 = 0.475 16.19 3.63 = max(0, 21 ) 15.09 5.91 = max(0, 21 15.09)
European put option Step 2 = er t d = 0.525 = 1 = 0.475 24.71 0.00 0.47 2.24 16.19 4.68 0.00 e 0.048 0.125 ( 0.00 + 0.00) = 0.00 0.00 e 0.048 0.125 ( 0.00 + 1.00) = 0.47 1.00 e 0.048 0.125 ( 1.00 + 3.63) = 2.24 3.63 e 0.048 0.125 ( 3.63 + 5.91) = 4.68 5.91 X = 21
European put option Step 3 X = 21 = er t d = 0.525 = 1 = 0.475 0.22 1.30 3.38 0.00 e 0.048 0.125 ( 0.00 + 0.47) = 0.22 0.47 e 0.048 0.125 ( 0.47 + 2.24) = 1.30 2.24 e 0.048 0.125 ( 2.24 + 4.68) = 3.38 4.68
European put option Step 4 X = 21 = er t d = 0.525 = 1 = 0.475 0.73 2.27 0.22 e 0.048 0.125 ( 0.22 + 1.30) = 0.73 1.30 e 0.048 0.125 ( 1.30 + 3.38) = 2.27 3.38
European put option Step 5 X = 21 1.45 0.73 e 0.048 0.125 ( 0.73 + 2.27) = 1.45 2.27 = er t d = 0.525 = 1 = 0.475
American put option Step 1 24.71 26.51 X = 21 0.00 = max(0, 21 26.51) 0.00 = max(0, 21 ) 1.00 = max(0, 21 ) = er t d = 0.525 = 1 = 0.475 16.19 3.63 = max(0, 21 ) 15.09 5.91 = max(0, 21 15.09)
American put option Step 2 X = 21 = er t d = 0.525 = 1 = 0.475 24.71 0.00 0.47 2.36 16.19 4.81 0.00 e 0.048 0.125 ( 0.00 + 0.00) = 0.00 max(0, 21 24.71) = 0.00 0.00 e 0.048 0.125 ( 0.00 + 1.00) = 0.47 max(0, 21 21.64) = 0.00 1.00 e 0.048 0.125 ( 1.00 + 3.63) = 2.24 max(0, 21 ) = 2.36 3.63 e 0.048 0.125 ( 3.63 + 5.91) = 4.68 max(0, 21 16.19) = 4.81 5.91
American put option Step 3 X = 21 = er t d = 0.525 = 1 = 0.475 0.22 1.36 3.63 0.00 e 0.048 0.125 ( 0.00 + 0.47) = 0.22 max(0, 21 ) = 0.00 0.47 e 0.048 0.125 ( 0.47 + 2.36) = 1.36 max(0, 21 ) = 1.00 2.36 e 0.048 0.125 ( 2.36 + 4.81) = 3.50 max(0, 21 ) = 3.63 4.81
American put option Step 4 X = 21 = er t d = 0.525 = 1 = 0.475 0.76 2.42 0.22 e 0.048 0.125 ( 0.22 + 1.36) = 0.76 max(0, 21 ) = 0.00 1.36 e 0.048 0.125 ( 1.36 + 3.63) = 2.42 max(0, 21 ) = 2.36 3.63
American put option Step 5 X = 21 1.54 0.76 e 0.048 0.125 ( 0.76 + 2.42) = 1.54 2.42 max(0, 21 ) = 1.00 Binomial tree with 4 time steps: p 0 = 1.54 Binomial tree with 25 time steps: p 0 = 1.50 Binomial tree with 125 time steps: p 0 = 1.50 = er t d = 0.525 = 1 = 0.475
SAS-code Pricing of American options data binomial; N = 5; S0 = ; SIGMA = 0.20; T = 0.50; X = 21.00; R = 0.048; DT = T/(N-1); U = exp(+sigma*sqrt(dt)); D = exp(-sigma*sqrt(dt)); PU = (exp(r*dt)-d)/(u-d); PD = 1 - PU; Initialization of the dataset. Number of horizontal nodes. Initial stock price. Volatility of the stock price. Time-to-maturity. Strike price. Risk-free rate. Time-interval. Upward price movement coefficient. Downward price movement coefficient. Probability of an upward price movement. Probability of a downward price movement.
SAS-code Pricing of American options array S[251,251]; array P[251,251]; S[1,1] = S0; do I = 2 to N; do J = 1 to I; if J = 1 then S[I] = U*S[I-1,J]; if J > 1 then S[I] = D*S[I-1,J-1]; end; end; do J = 1 to N; P[N,J] = max(0,x-s[n,j]); end; do K = 1 to N-1; I = N-K; do J = 1 to I; P[I,J] = max(x-s[i,j], exp(-r*dt)*(pu*p[i+1,j]+pd*p[i+1,j+1])); end; end; PRICE = P[1,1]; Initialization of the stock price matrix. Initialization of the option price matrix. Initial price. Evaluation of horizontal levels. Evaluation of vertical levels. Stock price at upper nodes. Stock price at other nodes. End of vertical loop. End of horizontal loop. Evaluation of vertical levels (at the maturity). Put option payoff at the maturity End of vertical loop. Backward evaluation of horizontal levels. Calculation of the horizontal index value. Evaluation of vertical levels. Value of early exercise. Expected value of a future exercise. End of vertical loop. End of horizontal loop. Current price of the put option.