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Introduction to Binomial Trees Chapter 12 1
Introduction to Binomial Trees
Chapter 12
1
A Simple Binomial Model
l A stock price is currently $20.l In three months it will be either $22 or $18.
Stock Price = $22
Stock Price = $18
Stock price = $20
2
Stock Price = $22Option Price = $1
Stock Price = $18Option Price = $0
Stock price = $20Option Price=?
A Call Option (Figure 12.1, page 274)
A 3-month call option on the stock has a strike price of 21.
3
Up Move
Down Move
Setting Up a Riskless Portfolio
l For a portfolio that is long Δ shares and short 1 call option, the terminal values are:
l Portfolio is riskless when 22Δ – 1 = 18Δ or when Δ = 0.25
4
22Δ – 1
18Δ
Up Move
Down Move
ΔS0-C
Valuing the Portfolio(Risk-Free Rate is 12%)
l The riskless portfolio is: l long 0.25 sharesl short 1 call option
l The value of the portfolio in 3 months is: 22 ×0.25 – 1 = 4.50or 18×0.25 = 4.50
l The value of the portfolio today is: 4.5e – 0.12×0.25 = 4.3670
5
Valuing the Option
l The portfolio that is: l long 0.25 sharesl short 1 call option
is worth today 4.367 = ΔS0 - C
l The value of the Δ shares today is: ΔS0 = 0.25 ×20 = 5.000
l The value C of the call option today is therefore: C =ΔS0 – 4.367 = 5.000 – 4.367 = 0.633
6
Stock Price = $22Option Price = $0
Stock Price = $18Option Price = $3
Stock price = $20Option Price=?
A Put Option
A 3-month put option on the stock has a strike price of 21.
7
Up Move
Down Move
Setting Up a Riskless Portfolio:
l For a portfolio that is long Δ shares and short 1 put option, the terminal values are:
l Portfolio is riskless when 22Δ = 18Δ −3 or when Δ = -0.75
8
22Δ
18Δ−3
Up Move
Down Move
ΔS0-P
Valuing the Portfolio (Rf = 12%)
l The riskless portfolio is: l long –0.75 shares (i.e. short 0.75 shares)l short 1 put option
l The value of the portfolio in 3 months is:l 22 x (–0.75) = –16.50l Or 18 x (–0.75) – 3 = –16.50
l Since riskless, the value of the portfolio today is: –16.50 e – 0.12×0.25 = –16.012
9
Valuing the Option
l The portfolio that is:l long -0.75 shares (i.e. short 0.75 shares)l short 1 put option
is worth today –16.012 = ΔS0 - P
l The value of the Δ shares today is: ΔS0 = -0.75 ×20 = -15.000
l The value P of the put option today is therefore:P =ΔS0 + 16.012 = -15.000 + 16.012 P = 1.012
10
Remarks #1
l Note that for the call option, the riskless portfolio is constructed by buying Δ shares and selling/writing the call option.
l Since Δ is always positive for a call option, buying Δshares is the same as buying |Δ| shares.
l For the put option, the riskless portfolio is constructed by buying Δ shares and selling/writing the put option.
l Since Δ is always negative for a put option, buying Δshares is the same as shorting |Δ| shares.
11
Remarks #2
l Note that Δ does not have to be solved for each time as the “plug-in” figure that makes the portfolio riskless.
l It will always be the ratio of the difference in the option prices to the difference in the stock prices in the next period (with a small adjustment if dividends occur).
l In other words:l Δ = (Cu-Cd)/(Su-Sd) for the calll Δ = (Pu-Pd)/(Su-Sd) for the put
l Δ reflects the slope of the option pricing function.
12
Remarks #3
l Δ represents by how much the option changes in value given a change in the price of the underlying asset.
l If the stock price changes by one dollar, the option price approximately changes by Δ dollars.
l Example: Δ=0.30 for a call option and Δ=-0.25 for a put. l If the stock goes up by $1, the call appreciates by $0.30 (30 cents)l If the stock goes up by $1, the put depreciates by $0.25 (25 cents)
l Therefore, if instead of owning one share of the stock, you only own a fraction of a share, Δ share, the dollar movement of that Δ share matches that of the option.
l Being long one and short the other creates a riskless asset.13
Remarks #4
l Also note that in order to create the riskless portfolio, all you need is to be long on one side and short on the other.
l Therefore, for the call option, a riskless portfolio could also have been created by buying the call and shorting Δ shares, i.e. by composing the C-ΔS portfolio.
l For the put option, a riskless portfolio could also have been created by buying the put and shorting Δ shares, i.e. by composing the P-ΔS portfolio.
l But recall that for the put, Δ is negative. The portfolio can thus also be expressed as P+|Δ|S.
l In plain English, you “buy the put and short a negative number of shares”, i.e. you “buy the put and buy a positive number of shares”.
l One can also check graphically that the positions being added (C, P, ΔS, -ΔS, or |Δ|S) have opposite slopes: thus they offset each other.
14
Remarks #5
l Recall that in the case of the call option, we managed to replicate a risk-free payment, as if a risk-free investment B were returned with interest to you at expiration, hence giving you a payoff of BerT.
l Therefore we have: ΔST – CT = BerT and so ΔS0 – C = B.l Thus we have today: C = ΔS0 – B (and CT = ΔST – BerT )l This means that we can replicate the Call option future payoffs by
combining an investment of Δ shares in the stock today and shorting a risk-free investment (at time 0, the amount B shorted/borrowed is the present value of the risk-free payment made at expiration).
l The call option price today is thus the cost of this strategy.l In our previous example, Δ = 0.25, S0 = 20, and B = 4.5e-0.12x0.25
l Consequently the call option price is C = (0.25)(20) - 4.5e-0.12x0.25
l And so the call option price is C = 0.633 (same value as before)
15
Remarks #6
l Recall that in the case of the put option, we also managed to replicate a risk-free payment, as if a risk-free loan B were to be paid back by you at expiration with interest, hence costing you a payment of BerT.
l Therefore we have: ΔST – PT = –BerT and so ΔS0 – P = –B.l Thus we have today: P = ΔS0 + B (and PT = ΔST + BerT )l This means that we can replicate the Put option future payoffs by
combining a shorting of |Δ| shares in the stock today and investing in a risk-free bond (at time 0, the amount B bought/invested is the present value of the risk-free payment received at expiration).
l The put option price today is thus the cost of this strategy.l In our previous example, Δ = -0.75, S0 = 20, and B = 16.5e-0.12x0.25
l Consequently the put option price is P = (-0.75)(20) + 16.5e-0.12x0.25
l And so the put option price is P = 1.012 (same value as before)
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Additional Example
l The current stock price is $40, the riskless rate is 8%, and in one year the stock will either be worth $50 or $30.
l There is both a call option and a put option on that stock, with a strike price K of $40.
l What is the price of the call? l What is the price of the put?
l In each case, how much would you need to borrow or invest in order to replicate the call/put option payoffs? (while having an investment of delta shares of the stock)
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Additional Example: answers
l Call option price: C = $6.153
l Amount borrowed to replicate the Call: $13.847
l Put option price: P = $3.078
l Amount invested to replicate the Put: $23.078
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Graphical Example
19
Generalization (Figure 12.2, page 275)
A derivative lasts for a time T and its value is dependent on the price of a stock.
Suƒu
Sdƒd
Sƒ
20
Up Move
Down Move
Generalization (continued)
l Value of a portfolio that is long Δ shares and short 1 derivative:
l The portfolio is riskless when S0uΔ– ƒu = S0dΔ– ƒd
or, by rearranging, when:
21
dSuSfdu
00 −−=Δ ƒ
S0uΔ – ƒu
S0dΔ – ƒd
Up Move
Down Move
Generalization (continued)
l The value of the portfolio at time T is: Su Δ – ƒu
l The value of the portfolio today is: (Su Δ – ƒu )e–rT
l Another expression for the portfolio value today is: S Δ – f
l Hence we have: ƒ = S Δ – (Su Δ – ƒu )e–rT
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Generalization (continued)
l But Δ can be expressed as:
Therefore, substituting Δ , we obtain:ƒ = [ pƒu + (1 – p)ƒd ]e–rT
where
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p e du d
rT
= −−
dSuSfdu
00 −−=Δ ƒ
p as a Probabilityl It is natural to interpret p and 1−p as the probabilities
of up and down movements.
l The value of a derivative is then its expected payoff discounted at the risk-free rate.
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S0uƒu
S0dƒd
S0ƒ
Risk-Neutral Valuationl When the probability of up and down movements are p
and 1-p , the expected stock price at time T is S0erT
l This is because the expected stock price at time T is:
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0 0
0 0
0
0
( ) (1 )
( ) 1
( )( )
( )
T
rT rT
T
rT rT
T
rTT
E S pS u p S d
e d e dE S S u S du d u d
S ue ud ud deE Su d
E S S e
= + −
⎛ ⎞− −= + −⎜ ⎟− −⎝ ⎠− + −=
−=
Risk-Neutral Valuation
l This shows that the stock price earns the risk-free rate.
l Binomial trees illustrate the general result that to value a derivative, we can assume that the expected return on the underlying asset is the risk-free rate, and that the discount rate is also the risk-free rate.
l This is known as using “risk-neutral valuation”.
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Irrelevance of the Stock’s Expected Return
l When we are valuing an option in terms of the underlying stock, the expected return on the stock is irrelevant.
l The key reason is that the probabilities of future up or down movements are already incorporated into the price of the stock.
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Original Example Revisited
l Since p is a risk-neutral probability: 20e0.12 ×0.25 = 22p + 18(1 – p ); so p = 0.6523
l Alternatively, we can use the formula:
6523.09.01.19.00.250.12
=−−=
−−=
×edudep
rT
Su = 22
Sd = 18
Sƒ
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Valuing the Call Option Using Risk-Neutral Valuation
The value of the Call option is: C=e–0.12×0.25 [0.6523×1 + 0.3477×0] C = 0.633
Su = 22ƒu = 1
Sd = 18ƒd = 0
Sƒ
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Valuing the Put Option Using Risk-Neutral Valuation
The value of the Put option is: P=e–0.12×0.25 [0.6523×0 + 0.3477×3] P = 1.012
Su = 22ƒu = 0
Sd = 18ƒd = 3
Sƒ
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Real World vs. Risk-Neutral World
l The probability of an up movement in a risk-neutral world is not the same as the probability of an up movement in the real world.
l Suppose that in the real world, the expected return on the stock is 16%, and q is the probability of an up movement in the real world.
l We have: 22q + 18(1-q) = 20e0.16x3/12 so that q = 0.7041l The expected payoff from the option in the real world is
then: qx1 + (1-q)x0 = 0.7041l Unfortunately it is difficult to know the correct discount
rate to apply to the expected payoff (It must be the option expected discount rate, higher than the stock’s)
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A Two-Step ExampleFigure 12.3, page 280
l Each time step is 3 monthsl K=21, r =12% (like before)
20
22
18
24.2
19.8
16.2
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Valuing a Call OptionFigure 12.4, page 280
l Value at node B = e–0.12×0.25(0.6523×3.2 + 0.3477×0) = 2.0257
l Value at node A = e–0.12×0.25(0.6523×2.0257 + 0.3477×0)= 1.2823
201.2823
22
18
24.23.2
19.80.0
16.20.0
2.0257
0.0
A
B
C
D
E
F
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A Put Option ExampleFigure 12.7, page 283
K = 52, time step =1yr
r = 5%, u =1.2, d = 0.8, p = 0.6282
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504.1923
60
40
720
484
3220
1.4147
9.4636
What Happens When the Put Option is American (Figure 12.8, page 284)
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505.0894
60
40
720
484
3220
1.4147
12.0CThe American feature
increases the value at node C from 9.4636 to 12.0000.
This increases the value of the option from 4.1923 to 5.0894.
Delta
l Delta (Δ) is the ratio of the change in the price of a stock option to the change in the price of the underlying stock.
l The value of Δ varies from node to node.
l It is the first of the “Greek Letters”, or measures of sensitivity of option prices with respect to the various underlying variables (stock price, interest rate, volatility, etc…).
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Choosing u and d
l One way of matching the volatility is to set:
where σ is the volatility, andΔt is the length of the time step.
l This is the approach used by Cox, Ross, and Rubinstein (1979)
t
t
eud
euΔσ−
Δσ
==
=
1
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Assets Other than Non-Dividend Paying Stocksl For options on stock indices, currencies
and futures, the basic procedure for constructing the tree is the same except that the calculation of p differs slightly.
l A continuous dividend yield q, a foreign risk-free rate rf, or the domestic risk-free rate must be subtracted from the domestic risk-free rate in the formula for p.
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Generalizing the Probability p of an Up Move
( )
( )
for a nondividend-paying stock
for a stock index where is the dividend yield on the index
for a currency where is the foreign
risk-free rate
1 for a futures
f
r t
r q t
r r tf
a dpu d
a e
a e q
a e r
a
Δ
− Δ
− Δ
−=−
=
=
=
= contract
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Increasing the Time Steps
l In practice, at least 30 time steps are necessary to give good option values.
l But that minimum number of steps also depends on the option’s time to expiration.
l The software DerivaGem allows up to 500 time steps to be used.
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The Black-Scholes-Merton Model
l The Black-Scholes-Merton model can be derived by looking at what happens to the price of a European call option as the time step tends to zero.
l For European options, the Black-Scholes price is the “binomial tree price” when the tree is taken to the limit with infinitely small steps.
l The outline of the proof can be found in the Appendix to Chapter 12.
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