Financial Risk Management

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "Financial Risk Management"

Ῥαφαὴλ Ἀμήνὄφις Ανδρέου
5 χρόνια πριν
Προβολές:

1 Pricing of American options University of Oulu - Department of Finance Spring 2018

2 Volatility-based binomial price process uuuus 0 = uuus 0 = uus 0 = us 0 = S 0 = ds 0 = dds 0 = ddds 0 = dddds 0 = 15.09

3 Binomial price process with risk-neutral probabilities = er t d = = 1 = uuuu 1 p 4 u p0 d = uuud, uudu uduu, duuu 4 p 3 u p1 d = uudd, udud, uddu duud, dudu, dduu 6 p 2 u p2 d = uddd, dudd ddud, dddu 4 p 1 u p3 d = dddd 1 p 0 u p4 d =

4 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! 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! σ t + σ2 t... 1 σ t + σ2 t 2 2

5 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 = = [ ( (1 + r t) 2 + σ 2 ) t 1 (1 + r t) 2] S 2 [ 2 + σ 2 t + 2r t + rσ 2 ( t) r t r 2 ( t) 2] S 2 = σ 2 S 2 t

6 Pricing of a European put option n P S T p T P p T S 0 = σ = 0.20 T = 0.5 X = 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 = uuuu 1 p 4 u p0 d = uuud, uudu uduu, duuu 4 p 3 u p1 d = uudd, udud, uddu duud, dudu, dduu 6 p 2 u p2 d = uddd, dudd ddud, dddu 4 p 1 u p3 d = dddd 1 p 0 u p4 d =

7 Binomial price with twenty-five time-steps σ = 0.20 T = 0.5 n = 25 t = 0.02 u = e σ t = d = 1 u = e σ t = = er t d = = 1 = 0.490

8 Pricing of the option with twenty-five time-steps n P S T p T P p T Binomial tree with 4 time steps: p 0 = e = 1.45 Binomial tree with 25 time steps: p 0 = e = 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 ) =

9 50-step binomial tree

10 100-step binomial tree

11 European put option Step X = = max(0, ) 0.00 = max(0, 21 ) 1.00 = max(0, 21 ) = er t d = = 1 = = max(0, 21 ) = max(0, )

12 European put option Step 2 = er t d = = 1 = e ( ) = e ( ) = e ( ) = e ( ) = X = 21

13 European put option Step 3 X = 21 = er t d = = 1 = e ( ) = e ( ) = e ( ) =

14 European put option Step 4 X = 21 = er t d = = 1 = e ( ) = e ( ) =

15 European put option Step 5 X = e ( ) = = er t d = = 1 = 0.475

16 American put option Step X = = max(0, ) 0.00 = max(0, 21 ) 1.00 = max(0, 21 ) = er t d = = 1 = = max(0, 21 ) = max(0, )

17 American put option Step 2 X = 21 = er t d = = 1 = e ( ) = 0.00 max(0, ) = e ( ) = 0.47 max(0, ) = e ( ) = 2.24 max(0, 21 ) = e ( ) = 4.68 max(0, ) =

18 American put option Step 3 X = 21 = er t d = = 1 = e ( ) = 0.22 max(0, 21 ) = e ( ) = 1.36 max(0, 21 ) = e ( ) = 3.50 max(0, 21 ) =

19 American put option Step 4 X = 21 = er t d = = 1 = e ( ) = 0.76 max(0, 21 ) = e ( ) = 2.42 max(0, 21 ) =

20 American put option Step 5 X = e ( ) = 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 = = 1 = 0.475

21 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.

22 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.

Σχετικά έγγραφα

Financial Risk Management

Financial Risk Management Pricing of American options University of Oulu - Department of Finance Spring 2017 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