Chapter 13 Biomial Trees 1
A Simple Biomial Model! A stock price is curretly $20! I 3 moths it will be either $22 or $18 Stock price $20 Stock Price $22 Stock Price $18 2
A Call Optio (Figure 13.1, page 275) A 3-moth call optio o the stock has a strike price of 21. Stock Price $22 Optio Price $1 Stock price $20 Optio Price? Stock Price $18 Optio Price $0 3
Settig Up a Riskless Portfolio! For a portfolio that is log Δ shares ad a short 1 call optio values are 22Δ 1 18Δ! Portfolio is riskless whe 22Δ 1 18Δ or Δ 0.25 4
Valuig the Portfolio (Risk-Free Rate is 12%)! The riskless portfolio is: log 0.25 shares short 1 call optio! The value of the portfolio i 3 moths is 22 0.25 1 4.50! The value of the portfolio today is 4.5e 0.12 0.25 4.3670 5
Valuig the Optio! The portfolio that is log 0.25 shares short 1 optio is worth 4.367! The value of the shares is 5.000 ( 0.25 20 )! The value of the optio is therefore 0.633 ( 5.000 0.633 4.367 ) 6
Geeralizatio (Figure 13.2, page 276) A derivative lasts for time T ad is depedet o a stock S 0 ƒ S 0 u ƒ u S 0 d ƒ d 7
Geeralizatio (cotiued)! Value of a portfolio that is log Δ shares ad short 1 derivative: S 0 uδ ƒ u S 0 dδ ƒ d! The portfolio is riskless whe S 0 uδ ƒ u S 0 dδ ƒ d or Δ ƒ u S0u S f d 0 d 8
Geeralizatio (cotiued)! Value of the portfolio at time T is S 0 uδ ƒ u! Value of the portfolio today is (S 0 uδ ƒ u )e rt! Aother expressio for the portfolio value today is S 0 Δ f! Hece ƒ S 0 Δ (S 0 uδ ƒ u )e rt 9
Geeralizatio (cotiued) Substitutig for Δ we obtai ƒ [ pƒ u + (1 p)ƒ d ]e rt where p rt e d u d 10
p as a Probability! It is atural to iterpret p ad 1-p as probabilities of up ad dow movemets! The value of a derivative is the its expected payoff i a risk-eutral world discouted at the risk-free rate S 0 u ƒ u S 0 ƒ (1 p ) S 0 d ƒ d 11
Risk-Neutral Valuatio! Whe the probability of a up ad dow movemets are p ad 1-p the expected stock price at time T is S 0 e rt! This shows that the stock price ears the risk-free rate! Biomial trees illustrate the geeral result that to value a derivative we ca assume that the expected retur o the uderlyig asset is the risk-free rate ad discout at the risk-free rate! This is kow as usig risk-eutral valuatio 12
Origial Example Revisited S 0 20 ƒ (1 p ) S 0 u 22 ƒ u 1 S 0 d 18 ƒ d 0 p is the probability that gives a retur o the stock equal to the risk-free rate: 20e 0.12 0.25 22p + 18(1 p ) so that p 0.6523 Alteratively: p rt e d u d e 0.25 0.9 1.1 0.9 0.12 0.6523 13
Valuig the Optio Usig Risk-Neutral Valuatio S 0 20 ƒ 0.6523 0.3477 S 0 u 22 ƒ u 1 S 0 d 18 ƒ d 0 The value of the optio is e 0.12 0.25 (0.6523 1 + 0.3477 0) 0.633 14
Irrelevace of Stock s Expected Retur! Whe we are valuig a optio i terms of the price of the uderlyig asset, the probability of up ad dow movemets i the real world are irrelevat! This is a example of a more geeral result statig that the expected retur o the uderlyig asset i the real world is irrelevat 15
A Two-Step Example Figure 13.3, page 281 22 24.2 20 19.8 18 16.2! K21, r 12%! Each time step is 3 moths 16
Valuig a Call Optio Figure 13.4, page 281 20 1.2823 A Value at ode B 22 2.0257 18 0.0 24.2 3.2 19.8 0.0 16.2 0.0 e 0.12 0.25 (0.6523 3.2 + 0.3477 0) 2.0257 Value at ode A e 0.12 0.25 (0.6523 2.0257 + 0.3477 0) 1.2823 B 17
A Put Optio Example Figure 13.7, page 284 50 4.1923 60 1.4147 40 9.4636 72 0 48 4 32 20 K 52, time step 1yr r 5%, u 1.32, d 0.8, p 0.6282 18
What Happes Whe the Put Optio is America (Figure 13.8, page 285) 50 5.0894 The America feature icreases the value at ode C from 9.4636 to 12.0000. This icreases the value of the optio from 4.1923 to 5.0894. 60 1.4147 40 12.0 C 72 0 48 4 32 20 19
Delta! Delta (Δ) is the ratio of the chage i the price of a stock optio to the chage i the price of the uderlyig stock! The value of Δ varies from ode to ode 20
Choosig u ad d Oe way of matchig the volatility is to set u e σ Δt d 1 u e σ Δt where σ is the volatility ad Δt is the legth of the time step. This is the approach used by Cox, Ross, ad Rubistei 21
Girsaov s Theorem! Volatility is the same i the real world ad the risk-eutral world! We ca therefore measure volatility i the real world ad use it to build a tree for the a asset i the risk-eutral world 22
Assets Other tha No-Divided Payig Stocks! For optios o stock idices, currecies ad futures the basic procedure for costructig the tree is the same except for the calculatio of p 23
The Probability of a Up Move p a u d d a e rδt for a odivided payigstock ( rq) Δt a e for a stock idex where yield o the idex q is the divided ( rr Δt a e for a currecy where risk - free rate f ) r f is the foreig a 1 for a futures cotract 24
Provig Black-Scholes-Merto from Biomial Trees (Appedix to Chapter 13) Optio is i the moey whe > α where so that rt K d u S p p e c 0 0 ) 0, max( ) (1! )! (! T K S σ α 2 ) l( 2 0 >α >α rt p p U d u p p U KU U S e c ) (1! )! (! ) (1! )! (! ) ( 2 1 2 1 0 where 25
Provig Black-Scholes-Merto from Biomial Trees cotiued! The expressio for U 1 ca be writte U where! * * rt! * * ( p ) ( 1 p ) e ( p ) ( p ) 1 [ pu + (1 p) d] 1 > α ( )!! > α ( )!! p * pu pu + (1 p) d! Both U 1 ad U 2 ca ow be evaluated i terms of the cumulative biomial distributio! We ow let the umber of time steps ted to ifiity ad use the result that a biomial distributio teds to a ormal distributio 26