Chapter 13 Binomial Trees. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull

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1 Chapter 13 Biomial Trees 1

2 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

3 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

4 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

5 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 ! The value of the portfolio today is 4.5e

6 Valuig the Optio! The portfolio that is log 0.25 shares short 1 optio is worth 4.367! The value of the shares is ( )! The value of the optio is therefore ( ) 6

7 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

8 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

9 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

10 Geeralizatio (cotiued) Substitutig for Δ we obtai ƒ [ pƒ u + (1 p)ƒ d ]e rt where p rt e d u d 10

11 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

12 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

13 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 p + 18(1 p ) so that p Alteratively: p rt e d u d e

14 Valuig the Optio Usig Risk-Neutral Valuatio S 0 20 ƒ S 0 u 22 ƒ u 1 S 0 d 18 ƒ d 0 The value of the optio is e ( )

15 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

16 A Two-Step Example Figure 13.3, page ! K21, r 12%! Each time step is 3 moths 16

17 Valuig a Call Optio Figure 13.4, page A Value at ode B e ( ) Value at ode A e ( ) B 17

18 A Put Optio Example Figure 13.7, page K 52, time step 1yr r 5%, u 1.32, d 0.8, p

19 What Happes Whe the Put Optio is America (Figure 13.8, page 285) The America feature icreases the value at ode C from to This icreases the value of the optio from to C

20 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

21 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

22 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

23 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

24 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

25 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! )! (! ) ( where 25

26 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

Binomial Model. Stock Price Dynamics. The Key Idea Riskless Hedge

Binomial Model. Stock Price Dynamics. The Key Idea Riskless Hedge Biomial Model Stock Price Dyamics The value of a optio at maturity depeds o the price of the uderlyig stock at maturity. The value of the optio today depeds o the expected value of the optio at maturity