Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 9. Binomial Trees : Hull, Ch. 12.

Transcription

1 Week 9 Binomial Trees : Hull, Ch

2 Binomial Trees Objective: To explain how the binomial model can be used to price options. 2

3 Binomial Trees 1. Introduction. 2. One Step Binomial Model. 3. Risk Neutral Valuation. 4. Two-Step Binomial Model. 5. American Options. 6. Delta. 7. Dividends and the Binomial Tree. 8. Determining u and d. 9. Exotic Options. 10. Futures vs Option Formulas. 3

4 1. Introduction 4

5 Introduction to the Binomial Model What do we know about Option Pricing? Lower/Upper bounds: No assumptions, Arbitrage opportunity, Not very precise e.g. 3 c 18) Put-Call parity: No assumptions, Arbitrage opportunity, Relative pricing formula, not like F 0 =S 0 e rt Here we propose an option pricing model to find the theoretical price or fair price for a given option. To get this stronger result, we need to impose some structure: Assumption on the dynamics of S. Organization: (1) Simple Example, (2) Generalization, (3) Applications 5

6 2. One Step Binomial Model 6

7 A Simple Binomial Model A stock price is currently $20. In three months it will be either $22 or $18. Stock price = $20 Stock Price = $22 Stock Price = $18 7

8 A Call Option A 3-month call option on the stock has a strike price of $21. Stock price = $20 Option Price =? Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 8

9 ) ( ) ( 0 0 d d u u kt T kt c p c p e c c E e c But k, pu and pd are unknown. k = expected return on a risky project. k = r + risk premium. Call Option Price Today 9 Jorge Cruz Lopez - Bus 316: Derivative Securities

10 Setting Up a Riskless Portfolio Consider the Portfolio: long D shares short 1 call option Portfolio is riskless when 22D 1 = 18D or D = D 1 18D Remember: A riskless portfolio is a portfolio that has a fixed (and known) payoff in the future. 10

11 Valuing the Portfolio The riskless portfolio is: long 0.25 shares short 1 call option Assume that the risk-free rate is 12%. The value of the portfolio in 3 months is: = 4.50 = The value of the portfolio today is: 4.5e = Notice that we can discount at the riskless rate because this is a riskless portfolio!. 11

12 Valuing the Option Therefore, the portfolio that is: long 0.25 shares short 1 option is worth today The value of the share position today is: D 20 = = So now we can imply the value of the option today. The value of the option c today is: V 0 = D S 0 c = c c = Pretty COOL, eh? 12

13 Generalization An option maturing in T years written on a stock that is currently worth S. where S ƒ ƒ is the current option price u is a constant > 1 ƒ u is the option price in the upper state d is a constant < 1 ƒ d is the option price in the lower state S u ƒ u S d ƒ d 13 Jorge Cruz Lopez - Bus 316: Derivative Securities

14 Generalization Consider the portfolio that is long D shares and short one option. The payoff at time T is: S u D ƒ u S d D ƒ d The portfolio is riskless when S u D ƒ u = S d D ƒ d or D ƒu S u fd S d 14

15 Generalization Value of the portfolio at time T (maturity) is: S u D ƒ u or S d D ƒ d From the riskless portfolio, the value of the portfolio today is: (S u D ƒ u )e rt From the initial position, another expression for the portfolio value today is: S D f Hence the option price today is: f = S D (S u D ƒ u )e rt 15

16 Generalization Substituting for D we obtain: ƒ = [ p ƒ u + (1 p )ƒ d ]e rt where p e u rt d d 16 Jorge Cruz Lopez - Bus 316: Derivative Securities

17 3. Risk Neutral Valuation 17

18 Risk-Neutral Valuation ƒ = [ p ƒ u + (1 p )ƒ d ]e -rt The variables p and (1 p ) can be interpreted as the risk-neutral probabilities of up and down movements. Therefore, the value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate. S ƒ S u ƒ u S d ƒ d 18 Jorge Cruz Lopez - Bus 316: Derivative Securities

19 Irrelevance of Stock s Expected Return IMPORTANT: Notice that the stock growth rate and the probabilities of the stock moving up or down are irrelevant. That is, the expected return on the stock is irrelevant. WHY? This is because we re valuing the option in relative to the current stock price. This price contains all relevant information about the future prospects of the stock. 19

20 Original Example Revisited Proof that Risk Neutral valuation gives the same result as the no arbitrage argument: S = 20 ƒ S u = 22 ƒ u = 1 S d = 18 ƒ d = 0 p e u rt d d e Jorge Cruz Lopez - Bus 316: Derivative Securities

21 Valuing the Option S ƒ S u = 22 ƒ u = 1 S d = 18 ƒ d = 0 The value of the option today is: e [ ] = No Arbitrage Approach and Risk-Neutral Approach give the same result. 21

22 Pricing a Put No-Arbitrage Approach. Risk-Neutral Approach. Example: p (K = 40, T = 3/12) S0 = 40 S0 d = 35 and S0 u = 45 r = 8% 22

23 Pricing a Put: No Arbitrage Approach The portfolio is riskless when D S ƒ u u f d S d f = S D (S u D ƒ u )e rt = 40(-0.5) [45(-0.5) - 0] e 0.08*0.25 =

24 Pricing a Put: Risk Neutral Approach S u = 45 ƒ u = 0 S = 40 ƒ S d = 35 ƒ d = 5 p rt e d u d e The value of the option today is: e [ ( )5] = Jorge Cruz Lopez - Bus 316: Derivative Securities

25 4. Two-Step Binomial Trees 25

26 A Two-Step Example Same as the previous call example where p = Let each time step be 3 months. The tree is recombining (u and d constant). 26

27 Reminder: K=21 So = 20 Valuing a Call Option: Step by Step A Value at node B = e ( ) = Value at node A = e ( ) = Instead, we can proceed directly B C E D F 27 Jorge Cruz Lopez - Bus 316: Derivative Securities

28 Valuing a Call Option: The Direct Way S f u S f u d S u 2 S f uu u d S f ud dt f d d 2 S f dd f u = e -rdt [pf uu + (1-p)f ud ] f d = e -rdt [pf ud + (1-p)f dd ] f = e -rdt [p f u + (1-p) f d ] f = e -rdt [p {e -rdt [pf uu + (1-p)f ud ]} + (1-p) {e -rdt [pf ud + (1-p)f dd ]}] f = e -r2dt [p 2 f uu + 2p(1-p)f ud + (1-p) 2 f dd ] Check: sum prob = 1 28

29 A Put Option Example K=52; T=2; r=5% A B C E D F Try it yourself! 29

30 5. American Options 30

31 American Options Recall: Any time that the payoff from early exercise exceeds the price of the option, it is optimal to exercise early. C < (S 0 K) EE P < (K - S 0 ) EE 31

32 When the Put Option is American K=52; T=2; r=5% European American A B C E D F A B C E D F At this point the payoff from early exercise is greater than the price of the option. Therefore, we have early exercise. 32

33 6. Delta 33

34 Delta Delta (D) is the ratio of the change in the price of a stock option to the change in the price of the underlying stock. In the binomial tree: D S ƒ u u f d S d The value of D varies from node to node. 34

35 7. Dividends and the Binomial Tree 35

36 Binomial Trees with Dividends With a percentage dividend (ds), the tree is still recombining. With a cash dividend ($D), the tree is not recombining anymore, so pricing becomes more complex. 36

37 8. Determining u and d 37

38 Determining u and d One way of matching the volatility is to set: u e s Dt d e s Dt 1 / u where s is the annual volatility and Dt is the length of the time step 38

39 9. Exotic Options 39

40 Applications: Exotic Options Pricing a Power Option European American Pricing a Chooser Option (Ch. 20) Pricing a Lookback Option (Ch. 20) 40

41 10. Futures vs Option Formulas 41

42 Difference Between Futures and Option Pricing Formulas? What should we do when on the Futures market we have: Fmarket S0 x exp(rt) What should we do when on the option market we have: cmarket cbinomial or pmarket pbinomial 42