1. Chapter 11 Binomial Trees

2. Outline A One-Step Binomial Model Risk-Neutral Valuation
Two-Step Binomial Model
American Options
Delta
Matching Volatility With u and d
Options On Other Assets

3. Binomial Trees
Binomial Tree representing different possible paths that might be followed by the stock price over the life of an option
In each time step, it has a certain probability of moving up by a certain percentage amount and a certain probability of moving down by a certain percentage amount

4. A one-step binomial model
A Simple Binomial Model
A 3-month call option on the stock has a strike price of 21.
A stock price is currently $20
In three months it will be either $22 or $18
Stock Price = $22
Option Price = $1
Stock price = $20
Stock Price = $18
Option Price = $0

5. Setting Up a Riskless Portfolio
Consider the Portfolio: long D shares short 1 call option
Portfolio is riskless when 22D – 1 = 18D => D = 0.25
A riskless portfolio is therefore=> Long : 025 shares
Short : 1 call option
22D – 1
18D

6. The riskless portfolio is: long 0.25 shares short 1 call option
Valuing the Portfolio
The riskless portfolio is:
long 0.25 shares
short 1 call option
The value of the portfolio in 3 months is
22 × 0.25 – 1 = 4.5 or 18 ×0.25=4.5
The value of the portfolio today is (if Rf=12%) e – 0.12×0.25 = 4.367

7. the portfolio today is Stock price today = $20
Valuing the Option
Stock price today = $20
Suppose the option price = f
the portfolio today is
0.25 × 20 – f = 5 – f
It follows that
5 – f =4.367 So
f= the current value
of option

8. Generalization
S0 = stock price u= percentage increase in the stock price d= percentage decrease in ƒ= option on stock price whose current price ƒu = payoff from the option(when price moves up) ƒd= payoff from the option(when price moves down) T= the duration of the option
S0u ƒu
S0d ƒd S0 ƒ

9. Generalization (continued)
Consider the portfolio that is long D shares and short 1 call option
The portfolio is riskless when S0uD – ƒu = S0dD – ƒd or
S0uD – ƒu
S0dD – ƒd

10. Generalization (continued)
Value of the portfolio at time T is (S0uD – ƒu)e–rT
The cost of setting up the portfolio is S0D – f
Hence S0D – ƒ = (S0uD – ƒu )e–rT
ƒ = S0D – (S0uD – ƒu )e–rT
Substituting for we obtain
ƒ = [ pƒu + (1 – p)ƒd ]e–rT ---(11.2) where
Delta值算出來後，我們就來看看投組在第T期的現值為(S0uD – ƒu)e–rT ，投組成本為S0D – f，
這兩式相等可得
ƒ = S0D – (S0uD – ƒu )e–rT，接下來再把

11. Generalization (continued)
Ex. (see Figure11.1)
u=1.1, d=0.9,r=0.12,T=0.25,fu=1, ƒd=0
ƒ = [ pƒu + (1 – p)ƒd ]e–rT
= [ × ×0 ]e–0.12×0.25
= 0.633
接下來有個例題大家可以看一下，我們可以先把P算出來，再把所有值代入一般式中，就可以求得答案。

12. The option pricing formula in equation(11
The option pricing formula in equation(11.2) does not involve the probabilities of stock price moving up or down.
The key reason is that we are not valuing the option in absolute terms. We are calculating its value in terms of the price of the underlying stock. The probabilities of future up or down movements are already incorporated into the stock price.
在11.2式的選擇權公式中並沒有牽涉到股價上漲或下跌的機率
主要原因是因為我們用的不是”獨立“的項目去評價選擇權，而是用”標的股票的價格”去計算選擇權價值，
未來股價上漲下跌的機率已經包含在”股價”裡面

13. Risk-Neutral Valuation
We assume p and 1-p as probabilities of up and down movements.
Expected option payoff = p × ƒu + (1 – p ) × ƒd
The expected stock price at time T is
E(ST) = pS0u + (1-p) S0d = pS0 (u-d) + S0d
substituting => E(ST)=S0erT
From this equation, we can see that the stock price grows on average
at the risk-free rate. Because setting the probability of the up
movement equal to p is therefore equivalent to assuming that the
return on the stock equals the risk-free rate.
雖然推導(11.2)式並不需對股價上漲或下跌的機率做任何假設，但一般會將(11.2)式中的變數 p 解釋為股價上漲的機率，1-p 為股價下跌的機率。
根據P及1-P我們可以算出選擇權的預期報酬
然後在T期的時候，股票預期報酬的推導如下
由上述式子我們可以看出股價平均而言是以無風險利率成長，因為設定股價上漲的機率為P就相當於是假設股票的報酬率為無風險利率

14. Risk-Neutral Valuation (continued)
In a risk-neutral world all individuals are indifferent to risk. In such a world , investors require no compensation for risk, and the expected return on all securities is the risk-free interest rate.
Risk-neutral valuation states that we can with complete impunity assume the world is risk neutral when pricing options.

15. Original Example Revisited
* European 3-month call option
*Rf=12%
Since p is the probability that gives a return on the stock equal to the risk-free rate. We can find it from
E(ST)=S0erT => 22p + 18(1 – p ) = 20e0.12 ×0.25 => p =
At the end of the three months, the call option has a
probability of being worth 1 and a
probability of being worth zero. So the expect value is
Expected option payoff = p × ƒu + (1 – p ) × ƒd
0.6523× ×0 =
In a risk-neutral world this should be discounted at the
risk-free rate. The value of the option today is
0.6523e–0.12×0.25= 0.633
S0u = 22
ƒu = 1
S0d = 18
ƒd = 0
S0=20 ƒ p
(1 – p )

16. Real world compare with Risk-Neutral world
It is not easy to know the correct discount rate to apply to the expected payoff in the real world.
Using risk-neutral valuation can solve this problem because we know that in a risk-neutral world the expected return on all assets is the risk-free rate.

17. Two-Step Binomial Model
Stock price=$20 , u=10% , d=10%
Each time step is 3 months
r=12%, K=21
(Figure Stock prices in a two-step tree) 20 22 18
24.2
19.8
16.2

18. Valuing a Call Option
(Figure 11.4) p=
Value at node B = e–0.12×0.25(0.6523× ×0) =
Value at node A = e–0.12×0.25(0.6523× ×0) =
*ƒ = [ pƒu + (1 – p)ƒd ]e–rT ---(11.2) where
24.2 D
3.2=max{ ,0} 22 B 20
1.2823
2.0257
19.8 A E
0=max{ ,0} 18 C
16.2 F
0=max{ ,0}

19. Generalization S0u2 ƒuu S0u ƒu S0ud S0 ƒud ƒ S0d ƒd S0d2 ƒdd
Figure Stock and option prices in general two-step tree
S0u2
ƒuu
S0u ƒu
S0ud
ƒud S0 ƒ
S0d ƒd
S0d2
ƒdd

20. Generalization (continued)
*The length of time step is Dt years
ƒ = e–r Dt[ pƒu + (1 – p)ƒd ] (1) (11.2)
(11.3)
ƒu = e–r Dt[ pƒuu + (1 – p)ƒud ] (2)
ƒd= e–r Dt[ pƒud + (1 – p)ƒdd ] (3)
ƒ = e–2rDt[ p2ƒuu +2p (1 – p)ƒud + (1 – p)2ƒdd ]

21. A Put Example (Figure 11.7)
K = 52, duration = 2yr, current price = $50 u=20%, d=20%, r = 5% 50
4.1923 60 40 72 48 32
1.4147
9.4636 A B C D E F
0=max{52-72,0}
4=max{52-48,0}
20=max{52-32,0}
ƒ = e–2rDt[ p2ƒuu +2p (1 – p)ƒud + (1 – p)2ƒdd ]
= e–2*0.05*1 [ ×0 + 2× (1 – ) ×4 +
(1 –0.6282)2× 20] =

22. American Options American options can be valued using a binomial tree
The procedure is to work back through the tree from the end to the beginning, testing at each node to see whether early exercise is optimal

23. American Options(Figure 11.8)
American Put option K = 52, duration = 2yr, current price = $50,u=20%, d=20%, r = 5% 50
5.0894 60 40 72 48 32
1.4147
12.0 A B C D E F
0=max{52-72,0}
max{1.4147,52-60}
4=max{52-48,0}
max{5.0894,52-50}
max{9.4636,52-40}
20=max{52-32,0}
Value at node B = e–-0.05×1(0.6282× ×4)=1.4147
Value at node C = e–-0.05×1(0.6282× ×20)=9.4636
Value at node A = e–-0.05×1(0.6282× ×12)=5.0894

24. Delta
Delta (D) is an important parameter in the pricing and hedging of option.
The delta (D) of stock option =

25. Delta (Figure 11.7) Delta At the end of the first time step is
At the end of the second time step is either
The two-step examples show that delta changes over time

26. Matching Volatility With u and d
In practice, when constructing a binomial tree to represent the movements in a stock price. We choose the parameters u and d to match the volatility of the stock price.
s = volatility
Dt = the length of the time step
This is the approach used by Cox, Ross, and Rubinstein

27. Options On Other Assets
Option on stocks paying a continuous dividend yield
Dividend yield at rate = q
Total return from dividends and capital gains in a risk-neutral world = r.
=> Capital gains return = r-q
The stock expected value after one time step of length Dt is S0e(r-q) Dt
pS0u+(1-p)S0d=S0e(r-q) Dt =>

28. Options On Other Assets

29. Options On Other Assets
Option on stock indices ( a= e(r-q) Dt )
European 6-month call option on an index level when index level is 810,K=800, rf=5%, σ=20%,q=2%
Node time:
Node time:
Node time:
810
53.39
895.19
100.66
732.92
5.06
989.34
189.34
810.00 10
663.17
0.00

30. Options On Other Assets
Option on currencies ( a= e(r-rf) Dt )
Three-step tree:American 3-month call.when the value of the currency is 0.61,K=0.6,rf=5%, σ=20%,foreign rf=7%
0.61
0.019
0.632
0.033
0.589
0.007
0.654
0.054
0.015
0.569
0.00
0.032
0.677
0.077
0.550

31. Options On Other Assets
Option on futures ( a= )
Three-step tree: American 9-month put. when the futures price is 31,K=30,rf=5%, σ=30% 31
2.84
36.02
0.93
26.68
4.54
41.85
1.76
22.97
7.03
3.32
48.62
19.77
10.23