Financial Risk Management

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ÍΕρρίκος Κακριδής
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1 Pricing of American options University of Oulu - Department of Finance Spring 2017

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 = 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 = 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 ( ) =

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

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

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

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

20 American put option Step 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 Option value as a function of the number of steps

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Financial Risk Management 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